Integrand size = 24, antiderivative size = 24 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\frac {75 c^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{448 a^2}+\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{672 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}}{56 a^2}-\frac {25 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{224 a}-\frac {25 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{336 a}-\frac {5 x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{84 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^{5/2}}{7 a^2 c}-\frac {75 c^3 \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )}{896 a}-\frac {25 c^2 \text {Int}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}},x\right )}{1344 a}-\frac {c \text {Int}\left (\frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}},x\right )}{112 a}-\frac {25 c^3 \text {Int}\left (\frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}},x\right )}{224 a} \]
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Not integrable
Time = 0.25 (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 = {} \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^{5/2}}{7 a^2 c}-\frac {5 \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2} \, dx}{14 a} \\ & = \frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}}{56 a^2}-\frac {5 x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{84 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^{5/2}}{7 a^2 c}-\frac {c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx}{112 a}-\frac {(25 c) \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2} \, dx}{84 a} \\ & = \frac {25 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{672 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}}{56 a^2}-\frac {25 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{336 a}-\frac {5 x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{84 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^{5/2}}{7 a^2 c}-\frac {c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx}{112 a}-\frac {\left (25 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}} \, dx}{1344 a}-\frac {\left (25 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx}{112 a} \\ & = \frac {75 c^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{448 a^2}+\frac {25 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{672 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}}{56 a^2}-\frac {25 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{224 a}-\frac {25 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}{336 a}-\frac {5 x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}}{84 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^{5/2}}{7 a^2 c}-\frac {c \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\arctan (a x)}} \, dx}{112 a}-\frac {\left (25 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}} \, dx}{1344 a}-\frac {\left (75 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx}{896 a}-\frac {\left (25 c^3\right ) \int \frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx}{224 a} \\ \end{align*}
Not integrable
Time = 5.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx \]
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Not integrable
Time = 3.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
\[\int x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2} \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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