Integrand size = 22, antiderivative size = 22 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2},x\right ) \]
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Not integrable
Time = 0.02 (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 = {} \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx=\int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx \\ \end{align*}
Not integrable
Time = 0.87 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx=\int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx \]
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Not integrable
Time = 5.70 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[\int \frac {\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {3}{2}}}{x^{2}}d x\]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 4.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx=c \left (\int a^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{x^{2}}\, dx\right ) \]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 192.31 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\left (c\,a^2\,x^2+c\right )}{x^2} \,d x \]
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