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

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

[Out]

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

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Rubi [A]  time = 0.0505333, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(a+b x) f^{\frac{c}{a+b x}}}{b}-\frac{c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[f^(c/(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.45494, size = 32, normalized size = 0.78 \[ - \frac{c \log{\left (f \right )} \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{a + b x} \right )}}{b} + \frac{f^{\frac{c}{a + b x}} \left (a + b x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0152264, size = 41, normalized size = 1. \[ \frac{(a+b x) f^{\frac{c}{a+b x}}}{b}-\frac{c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(c/(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.021, size = 52, normalized size = 1.3 \[{f}^{{\frac{c}{bx+a}}}x+{\frac{a}{b}{f}^{{\frac{c}{bx+a}}}}+{\frac{c\ln \left ( f \right ) }{b}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c/(b*x+a)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*c*integrate(f^(c/(b*x + a))*x/(b^2*x^2 + 2*a*b*x + a^2), x)*log(f) + f^(c/(b*x
+ a))*x

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Fricas [A]  time = 0.258401, size = 54, normalized size = 1.32 \[ -\frac{c{\rm Ei}\left (\frac{c \log \left (f\right )}{b x + a}\right ) \log \left (f\right ) -{\left (b x + a\right )} f^{\frac{c}{b x + a}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(c/(b*x+a)),x)

[Out]

Integral(f**(c/(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(c/(b*x + a)), x)