Integrand size = 23, antiderivative size = 23 \[ \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx=\text {Int}\left (\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)},x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 23, 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 = {} \[ \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx=\int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx \\ \end{align*}
Not integrable
Time = 0.37 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx=\int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx \]
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Not integrable
Time = 13.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
\[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\arctan \left (a x \right )}d x\]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)} \, dx=\int \sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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