Integrand size = 24, antiderivative size = 24 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^{3/2}}{7 a^2 c}-\frac {3 \text {Int}\left (\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)},x\right )}{14 a} \]
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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 = {} \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^{3/2}}{7 a^2 c}-\frac {3 \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx}{14 a} \\ \end{align*}
Not integrable
Time = 6.94 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx \]
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Not integrable
Time = 13.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
\[\int x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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