Integrand size = 20, antiderivative size = 20 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx=\int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 0.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx=\int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx \]
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Not integrable
Time = 1.42 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {x \left (a^{2} c \,x^{2}+c \right )}{\arctan \left (a x \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 1.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx=c \left (\int \frac {x}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx + \int \frac {a^{2} x^{3}}{\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx\right ) \]
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Exception generated. \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )}{\arctan (a x)^{3/2}} \, dx=\int \frac {x\,\left (c\,a^2\,x^2+c\right )}{{\mathrm {atan}\left (a\,x\right )}^{3/2}} \,d x \]
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