Optimal. Leaf size=80 \[ b \text{Unintegrable}\left (x^{-2 p-6} \tan ^{-1}(c x) \left (d+e x^2\right )^p,x\right )-\frac{a x^{-2 p-5} \left (d+e x^2\right )^{p+1} \text{Hypergeometric2F1}\left (-\frac{3}{2},1,\frac{1}{2} (-2 p-3),-\frac{e x^2}{d}\right )}{d (2 p+5)} \]
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Rubi [A] time = 0.143778, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^{-6-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^{-6-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx &=a \int x^{-6-2 p} \left (d+e x^2\right )^p \, dx+b \int x^{-6-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ &=b \int x^{-6-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx+\left (a \left (d+e x^2\right )^p \left (1+\frac{e x^2}{d}\right )^{-p}\right ) \int x^{-6-2 p} \left (1+\frac{e x^2}{d}\right )^p \, dx\\ &=-\frac{a x^{-5-2 p} \left (d+e x^2\right )^p \left (1+\frac{e x^2}{d}\right )^{-p} \, _2F_1\left (\frac{1}{2} (-5-2 p),-p;\frac{1}{2} (-3-2 p);-\frac{e x^2}{d}\right )}{5+2 p}+b \int x^{-6-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ \end{align*}
Mathematica [A] time = 3.42695, size = 0, normalized size = 0. \[ \int x^{-6-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.881, size = 0, normalized size = 0. \begin{align*} \int{x}^{-6-2\,p} \left ( e{x}^{2}+d \right ) ^{p} \left ( a+b\arctan \left ( cx \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (c x\right ) + a\right )}{\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \arctan \left (c x\right ) + a\right )}{\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (c x\right ) + a\right )}{\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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