Optimal. Leaf size=50 \[ -\frac{x f^a g^c \left (-x^n (b \log (f)+d \log (g))\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-x^n (b \log (f)+d \log (g))\right )}{n} \]
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Rubi [A] time = 0.0444385, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2287, 2208} \[ -\frac{x f^a g^c \left (-x^n (b \log (f)+d \log (g))\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-x^n (b \log (f)+d \log (g))\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2287
Rule 2208
Rubi steps
\begin{align*} \int f^{a+b x^n} g^{c+d x^n} \, dx &=\int \exp \left (a \log (f)+c \log (g)+x^n (b \log (f)+d \log (g))\right ) \, dx\\ &=-\frac{f^a g^c x \Gamma \left (\frac{1}{n},-x^n (b \log (f)+d \log (g))\right ) \left (-x^n (b \log (f)+d \log (g))\right )^{-1/n}}{n}\\ \end{align*}
Mathematica [A] time = 0.0216559, size = 50, normalized size = 1. \[ -\frac{x f^a g^c \left (-x^n (b \log (f)+d \log (g))\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-x^n (b \log (f)+d \log (g))\right )}{n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.352, size = 0, normalized size = 0. \begin{align*} \int{f}^{a+b{x}^{n}}{g}^{c+d{x}^{n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24461, size = 68, normalized size = 1.36 \begin{align*} -\frac{f^{a} g^{c} x \Gamma \left (\frac{1}{n}, -{\left (b \log \left (f\right ) + d \log \left (g\right )\right )} x^{n}\right )}{\left (-{\left (b \log \left (f\right ) + d \log \left (g\right )\right )} x^{n}\right )^{\left (\frac{1}{n}\right )} n} \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 (f^{b x^{n} + a} g^{d x^{n} + c}, 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 f^{b x^{n} + a} g^{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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