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

Optimal. Leaf size=61 \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.034471, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^n)*(c + d*x)^m,x]

[Out]

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

Mathematica [A]  time = 0.0209261, size = 61, normalized size = 1. \[ -\frac{F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^n)*(c + d*x)^m,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b \left ( dx+c \right ) ^{n}} \left ( dx+c \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} 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)*(d*x+c)^m,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x + c\right )}^{m} 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)*(d*x+c)^m,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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*(d*x+c)**n)*(d*x+c)**m,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} 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)*(d*x+c)^m,x, algorithm="giac")

[Out]

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