3.365 \(\int \frac{F^{a+b (c+d x)^n}}{(c+d x)^2} \, dx\)

Optimal. Leaf size=52 \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.0345223, antiderivative size = 52, 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} \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \frac{F^{a+b (c+d x)^n}}{(c+d x)^2} \, dx &=-\frac{F^a \Gamma \left (-\frac{1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{\frac{1}{n}}}{d n (c+d x)}\\ \end{align*}

Mathematica [A]  time = 0.0120298, size = 52, normalized size = 1. \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{n}}}{ \left ( dx+c \right ) ^{2}}}\, 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)^2,x)

[Out]

int(F^(a+b*(d*x+c)^n)/(d*x+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^n*b + a)/(d*x + c)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{F^{{\left (d x + c\right )}^{n} b + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral(F^((d*x + c)^n*b + a)/(d^2*x^2 + 2*c*d*x + c^2), 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)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^n*b + a)/(d*x + c)^2, x)