Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {4}{a^2 c x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {8}{c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {16 \text {Int}\left (\frac {1}{x^5 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right )}{a^2}+44 \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right )+40 a^2 \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right ) \]
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Not integrable
Time = 0.54 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}-\frac {2 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx}{a}-(4 a) \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {4}{a^2 c x^4 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {8}{c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+16 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+28 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx+\frac {16 \int \frac {1}{x^5 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a^2}+\left (40 a^2\right ) \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx \\ \end{align*}
Not integrable
Time = 16.58 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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