Integrand size = 23, antiderivative size = 23 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=-\frac {3 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{4 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}+\frac {3}{8} c \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )+\frac {1}{2} c \text {Int}\left (\frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]
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Not integrable
Time = 0.08 (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 = {} \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{4 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}+\frac {1}{8} (3 c) \int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx+\frac {1}{2} c \int \frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}
Not integrable
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx \]
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Not integrable
Time = 13.71 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
\[\int \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {a^{2} c \,x^{2}+c}d x\]
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Exception generated. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 26.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
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Exception generated. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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