Integrand size = 20, antiderivative size = 20 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\frac {5 c \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{32 a^2}-\frac {5 c x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{24 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}}{4 a^2}-\frac {5 c \text {Int}\left (\frac {1}{\sqrt {\arctan (a x)}},x\right )}{64 a}-\frac {5 c \text {Int}\left (\arctan (a x)^{3/2},x\right )}{12 a} \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 20, 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 = {} \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}}{4 a^2}-\frac {5 \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx}{8 a} \\ & = \frac {5 c \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{32 a^2}-\frac {5 c x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{24 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^{5/2}}{4 a^2}-\frac {(5 c) \int \frac {1}{\sqrt {\arctan (a x)}} \, dx}{64 a}-\frac {(5 c) \int \arctan (a x)^{3/2} \, dx}{12 a} \\ \end{align*}
Not integrable
Time = 1.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx \]
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Not integrable
Time = 1.93 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int x \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 26.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=c \left (\int x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 111.29 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]
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Not integrable
Time = 0.52 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^{5/2} \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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