Optimal. Leaf size=200 \[ -\frac {2^{1+p+q} \sqrt {c} \left (-\frac {\sqrt {c} \left (e-\frac {\sqrt {c e^2-4 a f^2}}{\sqrt {c}}+2 f x\right )}{\sqrt {c e^2-4 a f^2}}\right )^{-1-p-q} \left (a+\frac {c e x}{f}+c x^2\right )^p \left (\frac {a f}{c}+e x+f x^2\right )^{1+q} \, _2F_1\left (-p-q,1+p+q;2+p+q;\frac {\sqrt {c} \left (e+\frac {\sqrt {c e^2-4 a f^2}}{\sqrt {c}}+2 f x\right )}{2 \sqrt {c e^2-4 a f^2}}\right )}{\sqrt {c e^2-4 a f^2} (1+p+q)} \]
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Rubi [A]
time = 0.09, antiderivative size = 200, normalized size of antiderivative = 1.00, 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 = {982, 638}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 638
Rule 982
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*}
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Mathematica [A]
time = 0.28, size = 172, normalized size = 0.86 \begin {gather*} \frac {2^{-1+p+q} \left (\frac {a f}{c}+x (e+f x)\right )^q \left (a+\frac {c x (e+f x)}{f}\right )^p \left (-\sqrt {c e^2-4 a f^2}+\sqrt {c} (e+2 f x)\right ) \left (1+\frac {\sqrt {c} (e+2 f x)}{\sqrt {c e^2-4 a f^2}}\right )^{-p-q} \, _2F_1\left (-p-q,1+p+q;2+p+q;\frac {1}{2}-\frac {\sqrt {c} (e+2 f x)}{2 \sqrt {c e^2-4 a f^2}}\right )}{\sqrt {c} f (1+p+q)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (a +\frac {c e x}{f}+c \,x^{2}\right )^{p} \left (\frac {a f}{c}+e x +f \,x^{2}\right )^{q}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (e\,x+f\,x^2+\frac {a\,f}{c}\right )}^q\,{\left (a+c\,x^2+\frac {c\,e\,x}{f}\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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