3.751 \(\int f^{a+b x^n} g^{c+d x^n} \, dx\)

Optimal. Leaf size=50 \[ -\frac{x f^a g^c \left (-x^n (b \log (f)+d \log (g))\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-x^n (b \log (f)+d \log (g))\right )}{n} \]

[Out]

-((f^a*g^c*x*Gamma[n^(-1), -(x^n*(b*Log[f] + d*Log[g]))])/(n*(-(x^n*(b*Log[f] +
d*Log[g])))^n^(-1)))

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Rubi [A]  time = 0.076897, 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 \[ -\frac{x f^a g^c \left (-x^n (b \log (f)+d \log (g))\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-x^n (b \log (f)+d \log (g))\right )}{n} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^n)*g^(c + d*x^n),x]

[Out]

-((f^a*g^c*x*Gamma[n^(-1), -(x^n*(b*Log[f] + d*Log[g]))])/(n*(-(x^n*(b*Log[f] +
d*Log[g])))^n^(-1)))

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Rubi in Sympy [A]  time = 4.37221, size = 53, normalized size = 1.06 \[ - \frac{x \left (x^{n} \left (- b \log{\left (f \right )} - d \log{\left (g \right )}\right )\right )^{- \frac{1}{n}} \Gamma{\left (\frac{1}{n},x^{n} \left (- b \log{\left (f \right )} - d \log{\left (g \right )}\right ) \right )} e^{a \log{\left (f \right )} + c \log{\left (g \right )}}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b*x**n)*g**(c+d*x**n),x)

[Out]

-x*(x**n*(-b*log(f) - d*log(g)))**(-1/n)*Gamma(1/n, x**n*(-b*log(f) - d*log(g)))
*exp(a*log(f) + c*log(g))/n

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Mathematica [A]  time = 0.054879, size = 0, normalized size = 0. \[ \int f^{a+b x^n} g^{c+d x^n} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[f^(a + b*x^n)*g^(c + d*x^n),x]

[Out]

Integrate[f^(a + b*x^n)*g^(c + d*x^n), x]

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \[ \int{f}^{a+b{x}^{n}}{g}^{c+d{x}^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b*x^n)*g^(c+d*x^n),x)

[Out]

int(f^(a+b*x^n)*g^(c+d*x^n),x)

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Maxima [A]  time = 1.01942, size = 68, normalized size = 1.36 \[ -\frac{f^{a} g^{c} x \Gamma \left (\frac{1}{n}, -{\left (b \log \left (f\right ) + d \log \left (g\right )\right )} x^{n}\right )}{\left (-{\left (b \log \left (f\right ) + d \log \left (g\right )\right )} x^{n}\right )^{\left (\frac{1}{n}\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*g^(d*x^n + c),x, algorithm="maxima")

[Out]

-f^a*g^c*x*gamma(1/n, -(b*log(f) + d*log(g))*x^n)/((-(b*log(f) + d*log(g))*x^n)^
(1/n)*n)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (f^{b x^{n} + a} g^{d x^{n} + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*g^(d*x^n + c),x, algorithm="fricas")

[Out]

integral(f^(b*x^n + a)*g^(d*x^n + c), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b*x**n)*g**(c+d*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{b x^{n} + a} g^{d x^{n} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*g^(d*x^n + c),x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)*g^(d*x^n + c), x)