Integrand size = 22, antiderivative size = 22 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx=\text {Int}\left (\frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}},x\right ) \]
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Not integrable
Time = 0.02 (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 = {} \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx=\int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx \\ \end{align*}
Not integrable
Time = 3.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx=\int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx \]
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Not integrable
Time = 1.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[\int \frac {a^{2} c \,x^{2}+c}{x \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 7.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx=c \left (\int \frac {1}{x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {a^{2} x}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx\right ) \]
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Exception generated. \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 294.37 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx=\int { \frac {a^{2} c x^{2} + c}{x \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 0.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {c+a^2 c x^2}{x \arctan (a x)^{5/2}} \, dx=\int \frac {c\,a^2\,x^2+c}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}} \,d x \]
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