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

Optimal. Leaf size=22 \[ -\frac{F^a \text{Ei}\left (\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0447805, 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} \[ -\frac{F^a \text{Ei}\left (\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0058154, size = 22, normalized size = 1. \[ -\frac{F^a \text{Ei}\left (\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.026, size = 23, normalized size = 1.1 \begin{align*}{\frac{{F}^{a}}{2\,d}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{ \left ( dx+c \right ) ^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*F^a*Ei(1,-b*ln(F)/(d*x+c)^2)

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

[Out]

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

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Fricas [A]  time = 1.61045, size = 69, normalized size = 3.14 \begin{align*} -\frac{F^{a}{\rm Ei}\left (\frac{b \log \left (F\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c),x, algorithm="fricas")

[Out]

-1/2*F^a*Ei(b*log(F)/(d^2*x^2 + 2*c*d*x + c^2))/d

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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)**2)/(d*x+c),x)

[Out]

Timed out

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

[Out]

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