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

Optimal. Leaf size=71 \[ \frac{F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{f}-\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{f} \]

[Out]

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

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Rubi [A]  time = 0.405028, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2222, 2210, 2228, 2178} \[ \frac{F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{f}-\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x))/(e + f*x),x]

[Out]

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

Rule 2222

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

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]

Rule 2228

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/(((e_.) + (f_.)*(x_))*((g_.) + (h_.)*(x_))), x_Symbol] :> -Dist[
d/(f*(d*g - c*h)), Subst[Int[F^(a - (b*h)/(d*g - c*h) + (d*b*x)/(d*g - c*h))/x, x], x, (g + h*x)/(c + d*x)], x
] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.123814, size = 66, normalized size = 0.93 \[ \frac{F^a \left (F^{\frac{b f}{c f-d e}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )-\text{Ei}\left (\frac{b \log (F)}{c+d x}\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.162, size = 106, normalized size = 1.5 \begin{align*}{\frac{{F}^{a}}{f}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}} \right ) }-{\frac{1}{f}{F}^{{\frac{acf-ade+bf}{cf-de}}}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}}-\ln \left ( F \right ) a-{\frac{-\ln \left ( F \right ) acf+\ln \left ( F \right ) ade-\ln \left ( F \right ) bf}{cf-de}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57847, size = 185, normalized size = 2.61 \begin{align*} \frac{F^{\frac{a d e -{\left (a c + b\right )} f}{d e - c f}}{\rm Ei}\left (\frac{{\left (b d f x + b d e\right )} \log \left (F\right )}{c d e - c^{2} f +{\left (d^{2} e - c d f\right )} x}\right ) - F^{a}{\rm Ei}\left (\frac{b \log \left (F\right )}{d x + c}\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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