Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)},x\right ) \]
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Not integrable
Time = 0.08 (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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx=\int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx \\ \end{align*}
Not integrable
Time = 1.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx=\int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx \]
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Not integrable
Time = 25.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{x \arctan \left (a x \right )}d x\]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{x \arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 6.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{x \operatorname {atan}{\left (a x \right )}}\, dx \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{x \arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 46.64 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{x \arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{x \arctan (a x)} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x\,\mathrm {atan}\left (a\,x\right )} \,d x \]
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