Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx=\text {Int}\left (\frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3},x\right ) \]
[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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx \\ \end{align*}
Not integrable
Time = 2.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx \]
[In]
[Out]
Not integrable
Time = 11.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {x^{m} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\arctan \left (a x \right )^{3}}d x\]
[In]
[Out]
Not integrable
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
[In]
[Out]
Not integrable
Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Not integrable
Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^{5/2}}{\arctan (a x)^3} \, dx=\int \frac {x^m\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]
[In]
[Out]