Integrand size = 21, antiderivative size = 21 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx \\ \end{align*}
Not integrable
Time = 0.85 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx \]
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Not integrable
Time = 1.50 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
\[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{3}}{\arctan \left (a x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 7.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.24 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx=c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx\right ) \]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 285.31 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Not integrable
Time = 0.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\arctan (a x)^{5/2}} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {atan}\left (a\,x\right )}^{5/2}} \,d x \]
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