Integrand size = 23, antiderivative size = 23 \[ \int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\frac {a d^2 x \sqrt {d+e x^2}}{16 e}+\frac {1}{8} a d x^3 \sqrt {d+e x^2}+\frac {1}{6} a x^3 \left (d+e x^2\right )^{3/2}-\frac {a d^3 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{16 e^{3/2}}+b \text {Int}\left (x^2 \left (d+e x^2\right )^{3/2} \arctan (c x),x\right ) \]
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Not integrable
Time = 0.13 (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 = {} \[ \int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = a \int x^2 \left (d+e x^2\right )^{3/2} \, dx+b \int x^2 \left (d+e x^2\right )^{3/2} \arctan (c x) \, dx \\ & = \frac {1}{6} a x^3 \left (d+e x^2\right )^{3/2}+b \int x^2 \left (d+e x^2\right )^{3/2} \arctan (c x) \, dx+\frac {1}{2} (a d) \int x^2 \sqrt {d+e x^2} \, dx \\ & = \frac {1}{8} a d x^3 \sqrt {d+e x^2}+\frac {1}{6} a x^3 \left (d+e x^2\right )^{3/2}+b \int x^2 \left (d+e x^2\right )^{3/2} \arctan (c x) \, dx+\frac {1}{8} \left (a d^2\right ) \int \frac {x^2}{\sqrt {d+e x^2}} \, dx \\ & = \frac {a d^2 x \sqrt {d+e x^2}}{16 e}+\frac {1}{8} a d x^3 \sqrt {d+e x^2}+\frac {1}{6} a x^3 \left (d+e x^2\right )^{3/2}+b \int x^2 \left (d+e x^2\right )^{3/2} \arctan (c x) \, dx-\frac {\left (a d^3\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{16 e} \\ & = \frac {a d^2 x \sqrt {d+e x^2}}{16 e}+\frac {1}{8} a d x^3 \sqrt {d+e x^2}+\frac {1}{6} a x^3 \left (d+e x^2\right )^{3/2}+b \int x^2 \left (d+e x^2\right )^{3/2} \arctan (c x) \, dx-\frac {\left (a d^3\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{16 e} \\ & = \frac {a d^2 x \sqrt {d+e x^2}}{16 e}+\frac {1}{8} a d x^3 \sqrt {d+e x^2}+\frac {1}{6} a x^3 \left (d+e x^2\right )^{3/2}-\frac {a d^3 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{16 e^{3/2}}+b \int x^2 \left (d+e x^2\right )^{3/2} \arctan (c x) \, dx \\ \end{align*}
Not integrable
Time = 13.88 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{2} \,d x } \]
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Not integrable
Time = 63.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 95.98 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{2} \,d x } \]
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Not integrable
Time = 0.81 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^2 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2} \,d x \]
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