Integrand size = 24, antiderivative size = 24 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{a^2 c}-\frac {3 \text {Int}\left (\frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}},x\right )}{2 a} \]
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Not integrable
Time = 0.08 (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 = {} \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{a^2 c}-\frac {3 \int \frac {\sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{2 a} \\ \end{align*}
Not integrable
Time = 1.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]
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Not integrable
Time = 7.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
\[\int \frac {x \arctan \left (a x \right )^{\frac {3}{2}}}{\sqrt {a^{2} c \,x^{2}+c}}d x\]
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Exception generated. \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 36.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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Exception generated. \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 151.32 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{\frac {3}{2}}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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