Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x},x\right ) \]
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Not integrable
Time = 0.03 (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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx \\ \end{align*}
Not integrable
Time = 1.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx \]
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Not integrable
Time = 5.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {3}{2}}}{x}d x\]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 17.76 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx=c^{3} \left (\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx\right ) \]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3}{x} \,d x \]
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