Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}},x\right ) \]
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Not integrable
Time = 0.03 (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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx \\ \end{align*}
Not integrable
Time = 2.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx \]
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Not integrable
Time = 1.48 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{3}}{x \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 8.49 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.83 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx=c^{3} \left (\int \frac {1}{x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{3}}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{5}}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx\right ) \]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.63 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{x \arctan (a x)^{5/2}} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}} \,d x \]
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