3.222 \(\int \frac{f^{\frac{c}{a+b x}}}{x^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{b c \log (f) f^{\frac{c}{a}} \text{ExpIntegralEi}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )}{a^2}-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x} \]

[Out]

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

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Rubi [A]  time = 0.636027, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{b c \log (f) f^{\frac{c}{a}} \text{ExpIntegralEi}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )}{a^2}-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x} \]

Antiderivative was successfully verified.

[In]  Int[f^(c/(a + b*x))/x^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - b c \log{\left (f \right )} \int \frac{f^{\frac{c}{a + b x}}}{x \left (a + b x\right )^{2}}\, dx - \frac{f^{\frac{c}{a + b x}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(c/(b*x+a))/x**2,x)

[Out]

-b*c*log(f)*Integral(f**(c/(a + b*x))/(x*(a + b*x)**2), x) - f**(c/(a + b*x))/x

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Mathematica [A]  time = 0.0897092, size = 68, normalized size = 1. \[ -\frac{b c \log (f) f^{\frac{c}{a}} \text{ExpIntegralEi}\left (-\frac{b c x \log (f)}{a^2+a b x}\right )}{a^2}-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(c/(a + b*x))/x^2,x]

[Out]

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

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Maple [A]  time = 0.033, size = 80, normalized size = 1.2 \[{\frac{c\ln \left ( f \right ) b}{{a}^{2}}{f}^{{\frac{c}{bx+a}}} \left ({\frac{c\ln \left ( f \right ) }{bx+a}}-{\frac{c\ln \left ( f \right ) }{a}} \right ) ^{-1}}+{\frac{c\ln \left ( f \right ) b}{{a}^{2}}{f}^{{\frac{c}{a}}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}}+{\frac{c\ln \left ( f \right ) }{a}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c/(b*x+a))/x^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{\frac{c}{b x + a}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c/(b*x + a))/x^2,x, algorithm="maxima")

[Out]

integrate(f^(c/(b*x + a))/x^2, x)

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Fricas [A]  time = 0.253831, size = 81, normalized size = 1.19 \[ -\frac{b c f^{\frac{c}{a}} x{\rm Ei}\left (-\frac{b c x \log \left (f\right )}{a b x + a^{2}}\right ) \log \left (f\right ) +{\left (a b x + a^{2}\right )} f^{\frac{c}{b x + a}}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c/(b*x + a))/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{\frac{c}{a + b x}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c/(b*x+a))/x**2,x)

[Out]

Integral(f**(c/(a + b*x))/x**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{\frac{c}{b x + a}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(c/(b*x + a))/x^2,x, algorithm="giac")

[Out]

integrate(f^(c/(b*x + a))/x^2, x)