3.221 \(\int \frac{f^{\frac{c}{a+b x}}}{x} \, dx\)

Optimal. Leaf size=41 \[ f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )-\text{Ei}\left (\frac{c \log (f)}{a+b x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.131148, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2222, 2210, 2228, 2178} \[ f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )-\text{Ei}\left (\frac{c \log (f)}{a+b x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x))/x,x]

[Out]

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

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

Mathematica [A]  time = 0.0309776, size = 41, normalized size = 1. \[ f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a^2+b x a}\right )-\text{Ei}\left (\frac{c \log (f)}{a+b x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x))/x,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.112, size = 47, normalized size = 1.2 \begin{align*} -{f}^{{\frac{c}{a}}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}}+{\frac{c\ln \left ( f \right ) }{a}} \right ) +{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c/(b*x + a))/x, x)

________________________________________________________________________________________

Fricas [A]  time = 1.56656, size = 89, normalized size = 2.17 \begin{align*} f^{\frac{c}{a}}{\rm Ei}\left (-\frac{b c x \log \left (f\right )}{a b x + a^{2}}\right ) -{\rm Ei}\left (\frac{c \log \left (f\right )}{b x + a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c/(a + b*x))/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c/(b*x + a))/x, x)