3.363 \(\int F^{a+b (c+d x)^n} \, dx\)

Optimal. Leaf size=50 \[ -\frac{F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

[Out]

-((F^a*(c + d*x)*Gamma[n^(-1), -(b*(c + d*x)^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^n^(-1)))

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Rubi [A]  time = 0.0065579, antiderivative size = 50, 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 = {2208} \[ -\frac{F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n),x]

[Out]

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

Mathematica [A]  time = 0.0090424, size = 50, normalized size = 1. \[ -\frac{F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n),x]

[Out]

-((F^a*(c + d*x)*Gamma[n^(-1), -(b*(c + d*x)^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^n^(-1)))

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b \left ( dx+c \right ) ^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^n),x)

[Out]

int(F^(a+b*(d*x+c)^n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n),x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^n*b + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n),x, algorithm="fricas")

[Out]

integral(F^((d*x + c)^n*b + a), 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*(d*x+c)**n),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^n),x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^n*b + a), x)