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

Optimal. Leaf size=22 \[ \frac{F^a \text{Ei}\left (b (c+d x)^n \log (F)\right )}{d n} \]

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]])/(d*n)

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Rubi [A]  time = 0.0358306, antiderivative size = 22, 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 = {2210} \[ \frac{F^a \text{Ei}\left (b (c+d x)^n \log (F)\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]])/(d*n)

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{F^{a+b (c+d x)^n}}{c+d x} \, dx &=\frac{F^a \text{Ei}\left (b (c+d x)^n \log (F)\right )}{d n}\\ \end{align*}

Mathematica [A]  time = 0.0018404, size = 22, normalized size = 1. \[ \frac{F^a \text{Ei}\left (b (c+d x)^n \log (F)\right )}{d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]])/(d*n)

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Maple [A]  time = 0., size = 26, normalized size = 1.2 \begin{align*} -{\frac{{F}^{a}{\it Ei} \left ( 1,-b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) \right ) }{dn}} \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]

-1/d/n*F^a*Ei(1,-b*(d*x+c)^n*ln(F))

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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}}{d x + c}\,{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(F^((d*x + c)^n*b + a)/(d*x + c), x)

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Fricas [A]  time = 1.57954, size = 49, normalized size = 2.23 \begin{align*} \frac{F^{a}{\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right )}{d n} \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]

F^a*Ei((d*x + c)^n*b*log(F))/(d*n)

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

Integral(F**(a + b*(c + d*x)**n)/(c + d*x), x)

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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}}{d x + c}\,{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(F^((d*x + c)^n*b + a)/(d*x + c), x)