Solve ∫ (from 1 to r) of 3ax wrt x/b^2+c^2x^2dx

$$\int _ { 1 } ^ { r } \frac { 3 a x d x } { b ^ { 2 } + c ^ { 2 } x ^ { 2 } } d x$$

Evaluate

$$\left\{\begin{matrix}\frac{3ad\left(-|c|b^{2}\arctan(\frac{r|c|}{|b|})+r|b|c^{2}-|b|c^{2}+\arctan(\frac{|c|}{|b|})|c|b^{2}\right)}{|b|c^{4}},&c\neq 0\text{ and }b\neq 0\\\frac{ad\left(r^{3}-1\right)}{b^{2}},&c=0\\\frac{ad\left(c^{2}r^{3}-3b^{2}\right)}{\left(bc\right)^{2}},&b=0\text{ and }c=0\\\frac{ad\left(3rb^{2}-c^{2}\right)}{\left(bc\right)^{2}},&c=0\text{ and }b=0\\\frac{3ad\left(r-1\right)}{c^{2}},&b=0\end{matrix}\right.$$

Differentiate w.r.t. a

$$\left\{\begin{matrix}\frac{3d\left(-|c|b^{2}\arctan(\frac{r|c|}{|b|})+r|b|c^{2}-|b|c^{2}+\arctan(\frac{|c|}{|b|})|c|b^{2}\right)}{|b|c^{4}},&c\neq 0\text{ and }b\neq 0\\\frac{d\left(r^{3}-1\right)}{b^{2}},&c=0\\\frac{d\left(c^{2}r^{3}-3b^{2}\right)}{\left(bc\right)^{2}},&b=0\text{ and }c=0\\\frac{d\left(3rb^{2}-c^{2}\right)}{\left(bc\right)^{2}},&c=0\text{ and }b=0\\\frac{3d\left(r-1\right)}{c^{2}},&b=0\end{matrix}\right.$$