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

Optimal. Leaf size=54 \[ -\frac{F^a (c+d x)^2 \left (-b \log (F) (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

[Out]

-((F^a*(c + d*x)^2*Gamma[2/n, -(b*(c + d*x)^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(2/n)))

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Rubi [A]  time = 0.0207883, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2218} \[ -\frac{F^a (c+d x)^2 \left (-b \log (F) (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0116291, size = 54, normalized size = 1. \[ -\frac{F^a (c+d x)^2 \left (-b \log (F) (c+d x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b \log (F) (c+d x)^n\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((F^a*(c + d*x)^2*Gamma[2/n, -(b*(c + d*x)^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(2/n)))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b \left ( dx+c \right ) ^{n}} \left ( dx+c \right ) \, 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),x)

[Out]

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

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

[Out]

integrate((d*x + c)*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 ({\left (d x + c\right )} 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),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

[Out]

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