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

Optimal. Leaf size=20 \[ -\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0438938, antiderivative size = 20, 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} \[ -\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x))/(c + d*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0048202, size = 20, normalized size = 1. \[ -\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.079, size = 22, normalized size = 1.1 \begin{align*}{\frac{{F}^{a}}{d}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.60219, size = 42, normalized size = 2.1 \begin{align*} -\frac{F^{a}{\rm Ei}\left (\frac{b \log \left (F\right )}{d x + c}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-F^a*Ei(b*log(F)/(d*x + c))/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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