Integrand size = 22, antiderivative size = 22 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\text {Int}\left (x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 22, 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 = {} \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx \\ \end{align*}
Not integrable
Time = 3.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx \]
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Not integrable
Time = 3.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[\int x^{2} \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 41.87 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=c \left (\int x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{2} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]
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Exception generated. \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 108.42 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{2} \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]
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Not integrable
Time = 0.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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