3.770 \(\int f^{x^n} x^m \, dx\)

Optimal. Leaf size=41 \[ -\frac{x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-x^n\right )\right )}{n} \]

[Out]

-((x^(1 + m)*Gamma[(1 + m)/n, -(x^n*Log[f])])/(n*(-(x^n*Log[f]))^((1 + m)/n)))

________________________________________________________________________________________

Rubi [A]  time = 0.015585, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2218} \[ -\frac{x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-x^n\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^x^n*x^m,x]

[Out]

-((x^(1 + m)*Gamma[(1 + m)/n, -(x^n*Log[f])])/(n*(-(x^n*Log[f]))^((1 + m)/n)))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int f^{x^n} x^m \, dx &=-\frac{x^{1+m} \Gamma \left (\frac{1+m}{n},-x^n \log (f)\right ) \left (-x^n \log (f)\right )^{-\frac{1+m}{n}}}{n}\\ \end{align*}

Mathematica [A]  time = 0.0084263, size = 41, normalized size = 1. \[ -\frac{x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-x^n\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^x^n*x^m,x]

[Out]

-((x^(1 + m)*Gamma[(1 + m)/n, -(x^n*Log[f])])/(n*(-(x^n*Log[f]))^((1 + m)/n)))

________________________________________________________________________________________

Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{f}^{{x}^{n}}{x}^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(x^n)*x^m,x)

[Out]

int(f^(x^n)*x^m,x)

________________________________________________________________________________________

Maxima [A]  time = 1.11877, size = 57, normalized size = 1.39 \begin{align*} -\frac{x^{m + 1} \Gamma \left (\frac{m + 1}{n}, -x^{n} \log \left (f\right )\right )}{\left (-x^{n} \log \left (f\right )\right )^{\frac{m + 1}{n}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(x^n)*x^m,x, algorithm="maxima")

[Out]

-x^(m + 1)*gamma((m + 1)/n, -x^n*log(f))/((-x^n*log(f))^((m + 1)/n)*n)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (f^{\left (x^{n}\right )} x^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(x^n)*x^m,x, algorithm="fricas")

[Out]

integral(f^(x^n)*x^m, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{x^{n}} x^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(x**n)*x**m,x)

[Out]

Integral(f**(x**n)*x**m, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{\left (x^{n}\right )} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(x^n)*x^m,x, algorithm="giac")

[Out]

integrate(f^(x^n)*x^m, x)