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

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)))

________________________________________________________________________________________

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.

[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)))

Rule 2287

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

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.

[In]

Integrate[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)))

________________________________________________________________________________________

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.

[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)

________________________________________________________________________________________

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.

[In]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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.

[In]

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

[Out]

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