Integrand size = 22, antiderivative size = 22 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=-\frac {2 x}{3 a c \arctan (a x)^{3/2}}+\frac {2 \text {Int}\left (\frac {1}{\arctan (a x)^{3/2}},x\right )}{3 a c} \]
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Not integrable
Time = 0.03 (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 = {} \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{3 a c \arctan (a x)^{3/2}}+\frac {2 \int \frac {1}{\arctan (a x)^{3/2}} \, dx}{3 a c} \\ \end{align*}
Not integrable
Time = 0.91 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=\int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx \]
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Not integrable
Time = 1.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[\int \frac {x}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 3.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {x}{a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c} \]
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Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 184.65 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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