Solve ( 2 x ^ 2 + 5 x ^ 2 + 1 x + 3 ) / x + 2 wrt x = ∫ ( 2 x ^ 2 + x -6+15/x+2)dx

$$\frac { ( 2 x ^ { 2 } + 5 x ^ { 2 } + 1 x + 3 ) } { x + 2 } d x = \int ( 2 x ^ { 2 } + x - 6 + \frac { 15 } { x + 2 } ) d x$$

Solve for d

$$\left\{\begin{matrix}d=\frac{90x\ln(x+2)+180\ln(x+2)+4x^{4}+11x^{3}-30x^{2}-72x}{6x\left(7x^{2}+x+3\right)}+\frac{С\left(x+2\right)}{7x^{3}+x^{2}+3x}\text{, }&x\neq \frac{-1+\sqrt{83}i}{14}\text{ and }x\neq \frac{-\sqrt{83}i-1}{14}\text{ and }x\neq 0\text{ and }x\neq -2\\d\in \mathrm{C}\text{, }&\left(x=0\text{ and }С=15\ln(2)\right)\text{ or }\left(x=\frac{-\sqrt{83}i-1}{14}\text{ and }С_{1}=-15\ln(\frac{-\sqrt{83}i+27}{14})-\frac{1865i\sqrt{83}}{4116}-\frac{1151}{4116}\right)\text{ or }\left(x=\frac{-1+\sqrt{83}i}{14}\text{ and }С_{2}=-15\ln(\frac{27+\sqrt{83}i}{14})+\frac{1865i\sqrt{83}}{4116}-\frac{1151}{4116}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+2.

$$\left(2x^{2}+5x^{2}+1x+3\right)dx=\left(x+2\right)\int 2x^{2}+x-6+\frac{15}{x+2}\mathrm{d}x$$

Combine 2x^{2} and 5x^{2} to get 7x^{2}.

$$\left(7x^{2}+1x+3\right)dx=\left(x+2\right)\int 2x^{2}+x-6+\frac{15}{x+2}\mathrm{d}x$$

Use the distributive property to multiply 7x^{2}+1x+3 by d.

$$\left(7x^{2}d+1xd+3d\right)x=\left(x+2\right)\int 2x^{2}+x-6+\frac{15}{x+2}\mathrm{d}x$$

Use the distributive property to multiply 7x^{2}d+1xd+3d by x.

$$7dx^{3}+dx^{2}+3dx=\left(x+2\right)\int 2x^{2}+x-6+\frac{15}{x+2}\mathrm{d}x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x^{2}+x-6 times \frac{x+2}{x+2}.

$$7dx^{3}+dx^{2}+3dx=\left(x+2\right)\int \frac{\left(2x^{2}+x-6\right)\left(x+2\right)}{x+2}+\frac{15}{x+2}\mathrm{d}x$$

Since \frac{\left(2x^{2}+x-6\right)\left(x+2\right)}{x+2} and \frac{15}{x+2} have the same denominator, add them by adding their numerators.

$$7dx^{3}+dx^{2}+3dx=\left(x+2\right)\int \frac{\left(2x^{2}+x-6\right)\left(x+2\right)+15}{x+2}\mathrm{d}x$$

Do the multiplications in \left(2x^{2}+x-6\right)\left(x+2\right)+15.

$$7dx^{3}+dx^{2}+3dx=\left(x+2\right)\int \frac{2x^{3}+4x^{2}+x^{2}+2x-6x-12+15}{x+2}\mathrm{d}x$$

Combine like terms in 2x^{3}+4x^{2}+x^{2}+2x-6x-12+15.

$$7dx^{3}+dx^{2}+3dx=\left(x+2\right)\int \frac{2x^{3}+5x^{2}-4x+3}{x+2}\mathrm{d}x$$

Use the distributive property to multiply x+2 by \int \frac{2x^{3}+5x^{2}-4x+3}{x+2}\mathrm{d}x.

$$7dx^{3}+dx^{2}+3dx=x\int \frac{2x^{3}+5x^{2}-4x+3}{x+2}\mathrm{d}x+2\int \frac{2x^{3}+5x^{2}-4x+3}{x+2}\mathrm{d}x$$

Combine all terms containing d.

$$\left(7x^{3}+x^{2}+3x\right)d=x\int \frac{2x^{3}+5x^{2}-4x+3}{x+2}\mathrm{d}x+2\int \frac{2x^{3}+5x^{2}-4x+3}{x+2}\mathrm{d}x$$

The equation is in standard form.

$$\left(7x^{3}+x^{2}+3x\right)d=-15x\ln(\frac{1}{|x+2|})-30\ln(\frac{1}{|x+2|})+\frac{2x^{4}}{3}+Сx+\frac{11x^{3}}{6}-5x^{2}+2С_{1}-12x$$

Divide both sides by 7x^{3}+x^{2}+3x.

$$\frac{\left(7x^{3}+x^{2}+3x\right)d}{7x^{3}+x^{2}+3x}=\frac{-15x\ln(\frac{1}{|x+2|})-30\ln(\frac{1}{|x+2|})+\frac{2x^{4}}{3}+Сx+\frac{11x^{3}}{6}-5x^{2}+2С_{1}-12x}{7x^{3}+x^{2}+3x}$$

Dividing by 7x^{3}+x^{2}+3x undoes the multiplication by 7x^{3}+x^{2}+3x.

$$d=\frac{-15x\ln(\frac{1}{|x+2|})-30\ln(\frac{1}{|x+2|})+\frac{2x^{4}}{3}+Сx+\frac{11x^{3}}{6}-5x^{2}+2С_{1}-12x}{7x^{3}+x^{2}+3x}$$

Divide -30\ln(\frac{1}{|x+2|})-5x^{2}+\frac{11x^{3}}{6}+2С-12x-15x\ln(\frac{1}{|x+2|})+\frac{2x^{4}}{3}+xС by 7x^{3}+x^{2}+3x.

$$d=\frac{-90x\ln(\frac{1}{|x+2|})-180\ln(\frac{1}{|x+2|})+4x^{4}+11x^{3}+6Сx-30x^{2}+12С_{1}-72x}{6x\left(7x^{2}+x+3\right)}$$