Integrand size = 24, antiderivative size = 24 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\text {Int}\left (\frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3},x\right ) \]
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Not integrable
Time = 0.04 (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 = {} \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx \\ \end{align*}
Not integrable
Time = 1.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx \]
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Not integrable
Time = 4.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{x \left (a^{2} c \,x^{2}+c \right )^{3}}d x\]
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Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 6.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \]
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Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 87.54 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3} x} \,d x } \]
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Not integrable
Time = 0.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{3/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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