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

Optimal. Leaf size=39 \[ -\frac{x^2 f^a \left (-b \log (f) x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b \log (f) x^n\right )}{n} \]

[Out]

-((f^a*x^2*Gamma[2/n, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(2/n)))

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Rubi [A]  time = 0.0145723, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2218} \[ -\frac{x^2 f^a \left (-b \log (f) x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x,x]

[Out]

-((f^a*x^2*Gamma[2/n, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(2/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 \, dx &=-\frac{f^a x^2 \Gamma \left (\frac{2}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-2/n}}{n}\\ \end{align*}

Mathematica [A]  time = 0.0057416, size = 39, normalized size = 1. \[ -\frac{x^2 f^a \left (-b \log (f) x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n)*x,x]

[Out]

-((f^a*x^2*Gamma[2/n, -(b*x^n*Log[f])])/(n*(-(b*x^n*Log[f]))^(2/n)))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{f}^{a+b{x}^{n}}x\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.26472, size = 55, normalized size = 1.41 \begin{align*} -\frac{f^{a} x^{2} \Gamma \left (\frac{2}{n}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac{2}{n}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-f^a*x^2*gamma(2/n, -b*x^n*log(f))/((-b*x^n*log(f))^(2/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, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^(b*x^n + a)*x, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b*x**n)*x,x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^n + a)*x, x)