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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.204414, 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 \[ f^{\frac{c}{a}} \text{ExpIntegralEi}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )-\text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right ) \]

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)))]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.5461, size = 34, normalized size = 0.83 \[ f^{\frac{c}{a}} \operatorname{Ei}{\left (- \frac{b c x \log{\left (f \right )}}{a \left (a + b x\right )} \right )} - \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{a + b x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(c/(b*x+a))/x,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0188304, size = 41, normalized size = 1. \[ f^{\frac{c}{a}} \text{ExpIntegralEi}\left (-\frac{b c x \log (f)}{a^2+a b x}\right )-\text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right ) \]

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 = 0.027, size = 47, normalized size = 1.2 \[ -{f}^{{\frac{c}{a}}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}}+{\frac{c\ln \left ( f \right ) }{a}} \right ) +{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c/(b*x+a))/x,x)

[Out]

-f^(c/a)*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. \[ \int \frac{f^{\frac{c}{b x + a}}}{x}\,{d x} \]

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 = 0.261513, size = 55, normalized size = 1.34 \[ f^{\frac{c}{a}}{\rm Ei}\left (-\frac{b c x \log \left (f\right )}{a b x + a^{2}}\right ) -{\rm Ei}\left (\frac{c \log \left (f\right )}{b x + a}\right ) \]

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. \[ \int \frac{f^{\frac{c}{a + b x}}}{x}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{\frac{c}{b x + a}}}{x}\,{d x} \]

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)