Integrand size = 25, antiderivative size = 25 \[ \int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=-\frac {a x^{-1-2 p} \left (d+e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1,\frac {1}{2} (1-2 p),-\frac {e x^2}{d}\right )}{d (1+2 p)}+b \text {Int}\left (x^{-2-2 p} \left (d+e x^2\right )^p \arctan (c x),x\right ) \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 25, 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 = {} \[ \int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = a \int x^{-2-2 p} \left (d+e x^2\right )^p \, dx+b \int x^{-2-2 p} \left (d+e x^2\right )^p \arctan (c x) \, dx \\ & = b \int x^{-2-2 p} \left (d+e x^2\right )^p \arctan (c x) \, dx+\left (a \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p}\right ) \int x^{-2-2 p} \left (1+\frac {e x^2}{d}\right )^p \, dx \\ & = -\frac {a x^{-1-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),-\frac {e x^2}{d}\right )}{1+2 p}+b \int x^{-2-2 p} \left (d+e x^2\right )^p \arctan (c x) \, dx \\ \end{align*}
Not integrable
Time = 2.52 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx \]
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Not integrable
Time = 0.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
\[\int x^{-2-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2} \,d x } \]
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Timed out. \[ \int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\text {Timed out} \]
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Not integrable
Time = 0.59 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2} \,d x } \]
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Not integrable
Time = 3.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 2} \,d x } \]
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Not integrable
Time = 0.89 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int x^{-2-2 p} \left (d+e x^2\right )^p (a+b \arctan (c x)) \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+2}} \,d x \]
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