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 = 0.13, 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 = {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 624
Rule 968
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 = 0.24, 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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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {c f x^{2} + c e x + a f}{c}\right )^{q} \left (\frac {c f x^{2} + c e x + a f}{f}\right )^{p}, x\right ) \]
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 \[ \int {\left (c x^{2} + \frac {c e x}{f} + a\right )}^{p} {\left (f x^{2} + e x + \frac {a f}{c}\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+\frac {c e x}{f}+a \right )^{p} \left (f \,x^{2}+e x +\frac {a f}{c}\right )^{q}\, dx \]
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 \[ \int {\left (c x^{2} + \frac {c e x}{f} + a\right )}^{p} {\left (f x^{2} + e x + \frac {a f}{c}\right )}^{q}\,{d x} \]
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 \[ \int {\left (e\,x+f\,x^2+\frac {a\,f}{c}\right )}^q\,{\left (a+c\,x^2+\frac {c\,e\,x}{f}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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