Integrand size = 24, antiderivative size = 24 \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{3 a^2 c}-\frac {\text {Int}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}},x\right )}{6 a} \]
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Not integrable
Time = 0.07 (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 = {} \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{3 a^2 c}-\frac {\int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}} \, dx}{6 a} \\ \end{align*}
Not integrable
Time = 3.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx \]
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Not integrable
Time = 9.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
\[\int x \sqrt {a^{2} c \,x^{2}+c}\, \sqrt {\arctan \left (a x \right )}d x\]
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Exception generated. \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.87 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\int x \sqrt {c \left (a^{2} x^{2} + 1\right )} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\int x\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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