Solve ∫ a^x+x^2/a^xda

$$\int a ^ { x } + \frac { x ^ { 2 } } { a ^ { x } } d a$$

Evaluate

$$\left\{\begin{matrix}\frac{x^{2}a^{1-x}-x\ln(a)+\ln(a)}{1-x}+С,&x=-1\\\frac{x^{3}\ln(a)+x^{2}\ln(a)+a^{x+1}}{x+1}+С,&x=1\\\frac{ax^{3}+ax^{2}-xa^{2x+1}+a^{2x+1}}{\left(1-x^{2}\right)a^{x}}+С,&|x|\neq 1\end{matrix}\right.$$

Differentiate w.r.t. x

$$\left\{\begin{matrix}\frac{xa^{1-x}\left(x^{2}\ln(a)-x\ln(a)-x+2\right)}{\left(x-1\right)^{2}},&x=-1\\\frac{x\ln(a)a^{x+1}+\ln(a)a^{x+1}+2x^{3}\ln(a)-a^{x+1}+4x^{2}\ln(a)+2x\ln(a)}{\left(x+1\right)^{2}},&x=1\\\frac{a^{1-x}\left(x^{3}\ln(a)a^{2x}+x^{5}\ln(a)-x^{2}\ln(a)a^{2x}+x^{4}\ln(a)-x^{2}a^{2x}-x\ln(a)a^{2x}-x^{3}\ln(a)+\ln(a)a^{2x}+2xa^{2x}-x^{2}\ln(a)-a^{2x}-x^{4}+3x^{2}+2x\right)}{\left(x^{2}-1\right)^{2}},&|x|\neq 1\end{matrix}\right.$$