3.174 \(\int f^{a+b x^n} x^m \, dx\)

Optimal. Leaf size=46 \[ -\frac{f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (f) x^n\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.0264467, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ -\frac{f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^m,x]

[Out]

-((f^a*x^(1 + m)*Gamma[(1 + m)/n, -(b*x^n*Log[f])])/(n*(-(b*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^{a+b x^n} x^m \, dx &=-\frac{f^a x^{1+m} \Gamma \left (\frac{1+m}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-\frac{1+m}{n}}}{n}\\ \end{align*}

Mathematica [A]  time = 0.0123183, size = 46, normalized size = 1. \[ -\frac{f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x^m,x]

[Out]

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

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{f}^{a+b{x}^{n}}{x}^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b*x^n)*x^m,x)

[Out]

int(f^(a+b*x^n)*x^m,x)

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Maxima [A]  time = 1.22754, size = 63, normalized size = 1.37 \begin{align*} -\frac{f^{a} x^{m + 1} \Gamma \left (\frac{m + 1}{n}, -b x^{n} \log \left (f\right )\right )}{\left (-b 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^(a+b*x^n)*x^m,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^(b*x^n + a)*x^m, x)

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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)*x**m,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{b x^{n} + a} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^n + a)*x^m, x)