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 = 0.18, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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 2204
Rule 2234
Rule 2287
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.43, size = 135, normalized size = 1.42 \[ -\frac {\sqrt {\pi } \sqrt {-c \log \relax (f) - f \log \relax (g)} \operatorname {erf}\left (\frac {{\left ({\left (2 \, c x + b\right )} \log \relax (f) + {\left (2 \, f x + e\right )} \log \relax (g)\right )} \sqrt {-c \log \relax (f) - f \log \relax (g)}}{2 \, {\left (c \log \relax (f) + f \log \relax (g)\right )}}\right ) e^{\left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \relax (f)^{2} - 2 \, {\left (2 \, c d - b e + 2 \, a f\right )} \log \relax (f) \log \relax (g) + {\left (e^{2} - 4 \, d f\right )} \log \relax (g)^{2}}{4 \, {\left (c \log \relax (f) + f \log \relax (g)\right )}}\right )}}{2 \, {\left (c \log \relax (f) + f \log \relax (g)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 131, normalized size = 1.38 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f) - f \log \relax (g)} {\left (2 \, x + \frac {b \log \relax (f) + e \log \relax (g)}{c \log \relax (f) + f \log \relax (g)}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f)^{2} - 4 \, a c \log \relax (f)^{2} - 4 \, c d \log \relax (f) \log \relax (g) - 4 \, a f \log \relax (f) \log \relax (g) + 2 \, b e \log \relax (f) \log \relax (g) - 4 \, d f \log \relax (g)^{2} + e^{2} \log \relax (g)^{2}}{4 \, {\left (c \log \relax (f) + f \log \relax (g)\right )}}\right )}}{2 \, \sqrt {-c \log \relax (f) - f \log \relax (g)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int f^{c \,x^{2}+b x +a} g^{f \,x^{2}+e x +d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 90, normalized size = 0.95 \[ \frac {\sqrt {\pi } f^{a} g^{d} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - f \log \relax (g)} x - \frac {b \log \relax (f) + e \log \relax (g)}{2 \, \sqrt {-c \log \relax (f) - f \log \relax (g)}}\right ) e^{\left (-\frac {{\left (b \log \relax (f) + e \log \relax (g)\right )}^{2}}{4 \, {\left (c \log \relax (f) + f \log \relax (g)\right )}}\right )}}{2 \, \sqrt {-c \log \relax (f) - f \log \relax (g)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 130, normalized size = 1.37 \[ -\frac {f^a\,g^d\,\sqrt {\pi }\,{\mathrm {e}}^{-\frac {b^2\,{\ln \relax (f)}^2}{4\,\left (c\,\ln \relax (f)+f\,\ln \relax (g)\right )}-\frac {e^2\,{\ln \relax (g)}^2}{4\,\left (c\,\ln \relax (f)+f\,\ln \relax (g)\right )}-\frac {b\,e\,\ln \relax (f)\,\ln \relax (g)}{2\,\left (c\,\ln \relax (f)+f\,\ln \relax (g)\right )}}\,\mathrm {erf}\left (\frac {x\,\left (c\,\ln \relax (f)+f\,\ln \relax (g)\right )\,2{}\mathrm {i}+b\,\ln \relax (f)\,1{}\mathrm {i}+e\,\ln \relax (g)\,1{}\mathrm {i}}{2\,\sqrt {c\,\ln \relax (f)+f\,\ln \relax (g)}}\right )\,1{}\mathrm {i}}{2\,\sqrt {c\,\ln \relax (f)+f\,\ln \relax (g)}} \]
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 \[ \int f^{a + b x + c x^{2}} g^{d + e x + f x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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