Integrand size = 22, antiderivative size = 22 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)},x\right ) \]
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Not integrable
Time = 0.06 (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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx \\ \end{align*}
Not integrable
Time = 2.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx \]
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Not integrable
Time = 19.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[\int \frac {x \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\arctan \left (a x \right )}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{\arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 4.74 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx=\int \frac {x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\operatorname {atan}{\left (a x \right )}}\, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{\arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 47.17 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{\arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^{3/2}}{\arctan (a x)} \, dx=\int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{\mathrm {atan}\left (a\,x\right )} \,d x \]
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