Optimal. Leaf size=50 \[ -\frac {x f^a g^c \left (-x^n (b \log (f)+d \log (g))\right )^{-1/n} \Gamma \left (\frac {1}{n},-x^n (b \log (f)+d \log (g))\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 2208
Rule 2287
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 = 0.03, size = 50, normalized size = 1.00 \[ -\frac {x f^a g^c \left (-x^n (b \log (f)+d \log (g))\right )^{-1/n} \Gamma \left (\frac {1}{n},-x^n (b \log (f)+d \log (g))\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{b x^{n} + a} g^{d x^{n} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{b x^{n} + a} g^{d x^{n} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int f^{b \,x^{n}+a} g^{d \,x^{n}+c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.25, size = 50, normalized size = 1.00 \[ -\frac {f^{a} g^{c} x \Gamma \left (\frac {1}{n}, -{\left (b \log \relax (f) + d \log \relax (g)\right )} x^{n}\right )}{\left (-{\left (b \log \relax (f) + d \log \relax (g)\right )} x^{n}\right )^{\left (\frac {1}{n}\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int f^{a+b\,x^n}\,g^{c+d\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________