Integrand size = 24, antiderivative size = 24 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\text {Int}\left (x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2},x\right ) \]
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Not integrable
Time = 0.04 (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 = {} \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \\ \end{align*}
Not integrable
Time = 0.77 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \]
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Not integrable
Time = 8.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int x^{m} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{m} \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 53.60 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{m} \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]
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Not integrable
Time = 0.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x^m\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]
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