Integrand size = 22, antiderivative size = 22 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\frac {3 c^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{56 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{448 a^2}+\frac {5 c^3 \left (1+a^2 x^2\right )^3 \sqrt {\arctan (a x)}}{448 a^2}-\frac {c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{14 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{56 a}-\frac {5 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}}{112 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^{5/2}}{8 a^2}-\frac {3 c^3 \text {Int}\left (\frac {1}{\sqrt {\arctan (a x)}},x\right )}{112 a}-\frac {9 c^2 \text {Int}\left (\frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}},x\right )}{896 a}-\frac {5 c \text {Int}\left (\frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\arctan (a x)}},x\right )}{896 a}-\frac {c^3 \text {Int}\left (\arctan (a x)^{3/2},x\right )}{7 a} \]
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Not integrable
Time = 0.14 (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 = {} \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^{5/2}}{8 a^2}-\frac {5 \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2} \, dx}{16 a} \\ & = \frac {5 c^3 \left (1+a^2 x^2\right )^3 \sqrt {\arctan (a x)}}{448 a^2}-\frac {5 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}}{112 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^{5/2}}{8 a^2}-\frac {(5 c) \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\arctan (a x)}} \, dx}{896 a}-\frac {(15 c) \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2} \, dx}{56 a} \\ & = \frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{448 a^2}+\frac {5 c^3 \left (1+a^2 x^2\right )^3 \sqrt {\arctan (a x)}}{448 a^2}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{56 a}-\frac {5 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}}{112 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^{5/2}}{8 a^2}-\frac {(5 c) \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\arctan (a x)}} \, dx}{896 a}-\frac {\left (9 c^2\right ) \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx}{896 a}-\frac {\left (3 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x)^{3/2} \, dx}{14 a} \\ & = \frac {3 c^3 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}{56 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}{448 a^2}+\frac {5 c^3 \left (1+a^2 x^2\right )^3 \sqrt {\arctan (a x)}}{448 a^2}-\frac {c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}{14 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}{56 a}-\frac {5 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^{3/2}}{112 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^{5/2}}{8 a^2}-\frac {(5 c) \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\arctan (a x)}} \, dx}{896 a}-\frac {\left (9 c^2\right ) \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx}{896 a}-\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {\arctan (a x)}} \, dx}{112 a}-\frac {c^3 \int \arctan (a x)^{3/2} \, dx}{7 a} \\ \end{align*}
Not integrable
Time = 0.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx \]
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Not integrable
Time = 2.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[\int x \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {5}{2}}d x\]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 96.84 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.09 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=c^{3} \left (\int x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int 3 a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int 3 a^{4} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{6} x^{7} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 104.84 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]
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Not integrable
Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2} \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]
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