Optimal. Leaf size=95 \[ \frac{\sqrt{\pi } f^a g^d \exp \left (-\frac{(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+f \log (g))+e \log (g)}{2 \sqrt{c \log (f)+f \log (g)}}\right )}{2 \sqrt{c \log (f)+f \log (g)}} \]
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Rubi [A] time = 0.180561, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2287, 2234, 2204} \[ \frac{\sqrt{\pi } f^a g^d \exp \left (-\frac{(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+f \log (g))+e \log (g)}{2 \sqrt{c \log (f)+f \log (g)}}\right )}{2 \sqrt{c \log (f)+f \log (g)}} \]
Antiderivative was successfully verified.
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Rule 2287
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int f^{a+b x+c x^2} g^{d+e x+f x^2} \, dx &=\int \exp \left (a \log (f)+d \log (g)+x (b \log (f)+e \log (g))+x^2 (c \log (f)+f \log (g))\right ) \, dx\\ &=\left (\exp \left (-\frac{(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}\right ) f^a g^d\right ) \int \exp \left (\frac{(b \log (f)+e \log (g)+2 x (c \log (f)+f \log (g)))^2}{4 (c \log (f)+f \log (g))}\right ) \, dx\\ &=\frac{\exp \left (-\frac{(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}\right ) f^a g^d \sqrt{\pi } \text{erfi}\left (\frac{b \log (f)+e \log (g)+2 x (c \log (f)+f \log (g))}{2 \sqrt{c \log (f)+f \log (g)}}\right )}{2 \sqrt{c \log (f)+f \log (g)}}\\ \end{align*}
Mathematica [A] time = 0.0646792, size = 93, normalized size = 0.98 \[ \frac{\sqrt{\pi } f^a g^d \exp \left (-\frac{(b \log (f)+e \log (g))^2}{4 (c \log (f)+f \log (g))}\right ) \text{Erfi}\left (\frac{\log (f) (b+2 c x)+\log (g) (e+2 f x)}{2 \sqrt{c \log (f)+f \log (g)}}\right )}{2 \sqrt{c \log (f)+f \log (g)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{f}^{c{x}^{2}+bx+a}{g}^{f{x}^{2}+ex+d}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07988, size = 122, normalized size = 1.28 \begin{align*} \frac{\sqrt{\pi } f^{a} g^{d} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - f \log \left (g\right )} x - \frac{b \log \left (f\right ) + e \log \left (g\right )}{2 \, \sqrt{-c \log \left (f\right ) - f \log \left (g\right )}}\right ) e^{\left (-\frac{{\left (b \log \left (f\right ) + e \log \left (g\right )\right )}^{2}}{4 \,{\left (c \log \left (f\right ) + f \log \left (g\right )\right )}}\right )}}{2 \, \sqrt{-c \log \left (f\right ) - f \log \left (g\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55353, size = 385, normalized size = 4.05 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-c \log \left (f\right ) - f \log \left (g\right )} \operatorname{erf}\left (\frac{{\left ({\left (2 \, c x + b\right )} \log \left (f\right ) +{\left (2 \, f x + e\right )} \log \left (g\right )\right )} \sqrt{-c \log \left (f\right ) - f \log \left (g\right )}}{2 \,{\left (c \log \left (f\right ) + f \log \left (g\right )\right )}}\right ) e^{\left (-\frac{{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 2 \,{\left (2 \, c d - b e + 2 \, a f\right )} \log \left (f\right ) \log \left (g\right ) +{\left (e^{2} - 4 \, d f\right )} \log \left (g\right )^{2}}{4 \,{\left (c \log \left (f\right ) + f \log \left (g\right )\right )}}\right )}}{2 \,{\left (c \log \left (f\right ) + f \log \left (g\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x + c x^{2}} g^{d + e x + f x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19354, size = 177, normalized size = 1.86 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) - f \log \left (g\right )}{\left (2 \, x + \frac{b \log \left (f\right ) + e \log \left (g\right )}{c \log \left (f\right ) + f \log \left (g\right )}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) \log \left (g\right ) - 4 \, a f \log \left (f\right ) \log \left (g\right ) + 2 \, b e \log \left (f\right ) \log \left (g\right ) - 4 \, d f \log \left (g\right )^{2} + e^{2} \log \left (g\right )^{2}}{4 \,{\left (c \log \left (f\right ) + f \log \left (g\right )\right )}}\right )}}{2 \, \sqrt{-c \log \left (f\right ) - f \log \left (g\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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