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

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

[Out]

-x^(1+m)*GAMMA((1+m)/n,-x^n*ln(f))/n/((-x^n*ln(f))^((1+m)/n))

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 1.00 \[ -\frac {x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-x^n \log (f)\right )}{n} \]

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)))

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fricas [F]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{\left (x^{n}\right )} x^{m}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\left (x^{n}\right )} x^{m}\,{d x} \]

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)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int f^{x^{n}} x^{m}\, dx \]

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)

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maxima [A]  time = 1.01, size = 42, normalized size = 1.02 \[ -\frac {x^{m + 1} \Gamma \left (\frac {m + 1}{n}, -x^{n} \log \relax (f)\right )}{\left (-x^{n} \log \relax (f)\right )^{\frac {m + 1}{n}} n} \]

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)

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mupad [B]  time = 3.78, size = 71, normalized size = 1.73 \[ \frac {f^{x^n}\,x^{m+1}\,{\mathrm {e}}^{-\frac {x^n\,\ln \relax (f)}{2}}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left (x^n\,\ln \relax (f)\right )}{{\left (x^n\,\ln \relax (f)\right )}^{\frac {m+n+1}{2\,n}}\,\left (m+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(f^(x^n)*x^(m + 1)*exp(-(x^n*log(f))/2)*whittakerM(1 - (m + n + 1)/(2*n), (m + n + 1)/(2*n) - 1/2, x^n*log(f))
)/((x^n*log(f))^((m + n + 1)/(2*n))*(m + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{x^{n}} x^{m}\, dx \]

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)

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