Integrand size = 26, antiderivative size = 26 \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\text {Int}\left (x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2},x\right ) \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 26, 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 = {} \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx \\ \end{align*}
Not integrable
Time = 1.70 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx \]
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Not integrable
Time = 12.81 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
\[\int x^{m} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{m} \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int x^m \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int x^m\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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