Integrand size = 26, antiderivative size = 26 \[ \int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=-\frac {a \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{4 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{3 c x}+\frac {1}{8} a^2 \text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )-\frac {5}{4} a^3 \text {Int}\left (\frac {\sqrt {\arctan (a x)}}{x \sqrt {c+a^2 c x^2}},x\right ) \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{3 c x^3}+\frac {1}{2} a \int \frac {\sqrt {\arctan (a x)}}{x^3 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {\arctan (a x)^{3/2}}{x^2 \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {a \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{4 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{3 c x}+\frac {1}{8} a^2 \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx-\frac {1}{4} a^3 \int \frac {\sqrt {\arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx-a^3 \int \frac {\sqrt {\arctan (a x)}}{x \sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}
Not integrable
Time = 18.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx \]
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Not integrable
Time = 15.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
\[\int \frac {\arctan \left (a x \right )^{\frac {3}{2}}}{x^{4} \sqrt {a^{2} c \,x^{2}+c}}d x\]
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Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 58.85 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{x^{4} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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Exception generated. \[ \int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 147.84 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {3}{2}}}{\sqrt {a^{2} c x^{2} + c} x^{4}} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\arctan (a x)^{3/2}}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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