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

Optimal. Leaf size=35 \[ -\frac{x f^a \left (-b \log (f) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (f) x^n\right )}{n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0043448, antiderivative size = 35, 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 = {2208} \[ -\frac{x f^a \left (-b \log (f) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n),x]

[Out]

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

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

Mathematica [A]  time = 0.0045952, size = 35, normalized size = 1. \[ -\frac{x f^a \left (-b \log (f) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (f) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^n),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{f}^{a+b{x}^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.24017, size = 47, normalized size = 1.34 \begin{align*} -\frac{f^{a} x \Gamma \left (\frac{1}{n}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\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),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^(b*x^n + a), x)

________________________________________________________________________________________

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)

[Out]

Exception raised: TypeError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^n + a), x)