Integrand size = 26, antiderivative size = 26 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=-\frac {2 x^2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+2 a \text {Int}\left (\frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}},x\right ) \]
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Not integrable
Time = 0.28 (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 = {} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {4 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{a}+(2 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2 x^2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+(2 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{a c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+(2 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+(2 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x^2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+(2 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx \\ \end{align*}
Not integrable
Time = 3.66 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx \]
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Not integrable
Time = 3.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
\[\int \frac {x^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 32.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 209.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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