Integrand size = 24, antiderivative size = 24 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx=\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{5 a^2 c}-\frac {3 \text {Int}\left (\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)},x\right )}{10 a} \]
[Out]
Not integrable
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx=\int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{5 a^2 c}-\frac {3 \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx}{10 a} \\ \end{align*}
Not integrable
Time = 3.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx=\int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx \]
[In]
[Out]
Not integrable
Time = 15.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
\[\int x \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {3}{2}}d x\]
[In]
[Out]
Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Not integrable
Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
[In]
[Out]