Optimal. Leaf size=200 \[ -\frac{\sqrt{c} 2^{p+q+1} \left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^{q+1} \left (-\frac{\sqrt{c} \left (-\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}+e+2 f x\right )}{\sqrt{c e^2-4 a f^2}}\right )^{-p-q-1} \, _2F_1\left (-p-q,p+q+1;p+q+2;\frac{\sqrt{c} \left (e+2 f x+\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}\right )}{2 \sqrt{c e^2-4 a f^2}}\right )}{(p+q+1) \sqrt{c e^2-4 a f^2}} \]
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Rubi [A] time = 0.134029, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {968, 624} \[ -\frac{\sqrt{c} 2^{p+q+1} \left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^{q+1} \left (-\frac{\sqrt{c} \left (-\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}+e+2 f x\right )}{\sqrt{c e^2-4 a f^2}}\right )^{-p-q-1} \, _2F_1\left (-p-q,p+q+1;p+q+2;\frac{\sqrt{c} \left (e+2 f x+\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}\right )}{2 \sqrt{c e^2-4 a f^2}}\right )}{(p+q+1) \sqrt{c e^2-4 a f^2}} \]
Antiderivative was successfully verified.
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Rule 968
Rule 624
Rubi steps
\begin{align*} \int \left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^q \, dx &=\left (\left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^{-p}\right ) \int \left (\frac{a f}{c}+e x+f x^2\right )^{p+q} \, dx\\ &=-\frac{2^{1+p+q} \sqrt{c} \left (-\frac{\sqrt{c} \left (e-\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}+2 f x\right )}{\sqrt{c e^2-4 a f^2}}\right )^{-1-p-q} \left (a+\frac{c e x}{f}+c x^2\right )^p \left (\frac{a f}{c}+e x+f x^2\right )^{1+q} \, _2F_1\left (-p-q,1+p+q;2+p+q;\frac{\sqrt{c} \left (e+\frac{\sqrt{c e^2-4 a f^2}}{\sqrt{c}}+2 f x\right )}{2 \sqrt{c e^2-4 a f^2}}\right )}{\sqrt{c e^2-4 a f^2} (1+p+q)}\\ \end{align*}
Mathematica [A] time = 0.238339, size = 172, normalized size = 0.86 \[ \frac{2^{p+q-1} \left (\sqrt{c} (e+2 f x)-\sqrt{c e^2-4 a f^2}\right ) \left (a+\frac{c x (e+f x)}{f}\right )^p \left (\frac{a f}{c}+x (e+f x)\right )^q \left (\frac{\sqrt{c} (e+2 f x)}{\sqrt{c e^2-4 a f^2}}+1\right )^{-p-q} \, _2F_1\left (-p-q,p+q+1;p+q+2;\frac{1}{2}-\frac{\sqrt{c} (e+2 f x)}{2 \sqrt{c e^2-4 a f^2}}\right )}{\sqrt{c} f (p+q+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 3.482, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{\frac{cex}{f}}+c{x}^{2} \right ) ^{p} \left ({\frac{af}{c}}+ex+f{x}^{2} \right ) ^{q}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + \frac{c e x}{f} + a\right )}^{p}{\left (f x^{2} + e x + \frac{a f}{c}\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\frac{c f x^{2} + c e x + a f}{c}\right )^{q} \left (\frac{c f x^{2} + c e x + a f}{f}\right )^{p}, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + \frac{c e x}{f} + a\right )}^{p}{\left (f x^{2} + e x + \frac{a f}{c}\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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