Integrand size = 24, antiderivative size = 24 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {2 a^3 \arctan (a x)^{7/2}}{7 c}+\frac {\text {Int}\left (\frac {\arctan (a x)^{5/2}}{x^4},x\right )}{c}-\frac {a^2 \text {Int}\left (\frac {\arctan (a x)^{5/2}}{x^2},x\right )}{c} \]
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Not integrable
Time = 0.13 (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 = {} \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^{5/2}}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\arctan (a x)^{5/2}}{x^4} \, dx}{c} \\ & = a^4 \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx+\frac {\int \frac {\arctan (a x)^{5/2}}{x^4} \, dx}{c}-\frac {a^2 \int \frac {\arctan (a x)^{5/2}}{x^2} \, dx}{c} \\ & = \frac {2 a^3 \arctan (a x)^{7/2}}{7 c}+\frac {\int \frac {\arctan (a x)^{5/2}}{x^4} \, dx}{c}-\frac {a^2 \int \frac {\arctan (a x)^{5/2}}{x^2} \, dx}{c} \\ \end{align*}
Not integrable
Time = 3.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx \]
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Not integrable
Time = 4.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {\arctan \left (a x \right )^{\frac {5}{2}}}{x^{4} \left (a^{2} c \,x^{2}+c \right )}d x\]
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Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 7.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \]
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Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 109.26 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]
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Not integrable
Time = 0.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{5/2}}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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