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

Optimal. Leaf size=120 \[ -\frac{c^2 \log ^2(f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right )}{2 b^2}+\frac{a c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right )}{b^2}+\frac{(a+b x)^2 f^{\frac{c}{a+b x}}}{2 b^2}-\frac{a (a+b x) f^{\frac{c}{a+b x}}}{b^2}+\frac{c \log (f) (a+b x) f^{\frac{c}{a+b x}}}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.19117, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{c^2 \log ^2(f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right )}{2 b^2}+\frac{a c \log (f) \text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right )}{b^2}+\frac{(a+b x)^2 f^{\frac{c}{a+b x}}}{2 b^2}-\frac{a (a+b x) f^{\frac{c}{a+b x}}}{b^2}+\frac{c \log (f) (a+b x) f^{\frac{c}{a+b x}}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[f^(c/(a + b*x))*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.992, size = 109, normalized size = 0.91 \[ \frac{a c \log{\left (f \right )} \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{a + b x} \right )}}{b^{2}} - \frac{a f^{\frac{c}{a + b x}} \left (a + b x\right )}{b^{2}} - \frac{c^{2} \log{\left (f \right )}^{2} \operatorname{Ei}{\left (\frac{c \log{\left (f \right )}}{a + b x} \right )}}{2 b^{2}} + \frac{c f^{\frac{c}{a + b x}} \left (a + b x\right ) \log{\left (f \right )}}{2 b^{2}} + \frac{f^{\frac{c}{a + b x}} \left (a + b x\right )^{2}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0764587, size = 82, normalized size = 0.68 \[ \frac{c \log (f) (2 a-c \log (f)) \text{ExpIntegralEi}\left (\frac{c \log (f)}{a+b x}\right )+b x f^{\frac{c}{a+b x}} (b x+c \log (f))}{2 b^2}-\frac{a (a-c \log (f)) f^{\frac{c}{a+b x}}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(c/(a + b*x))*x,x]

[Out]

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

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Maple [A]  time = 0.03, size = 126, normalized size = 1.1 \[{\frac{{x}^{2}}{2}{f}^{{\frac{c}{bx+a}}}}-{\frac{{a}^{2}}{2\,{b}^{2}}{f}^{{\frac{c}{bx+a}}}}+{\frac{c\ln \left ( f \right ) x}{2\,b}{f}^{{\frac{c}{bx+a}}}}+{\frac{ac\ln \left ( f \right ) }{2\,{b}^{2}}{f}^{{\frac{c}{bx+a}}}}+{\frac{{c}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}}{2\,{b}^{2}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) }-{\frac{ac\ln \left ( f \right ) }{{b}^{2}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(c/(b*x+a))*x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (b x^{2} + c x \log \left (f\right )\right )} f^{\frac{c}{b x + a}}}{2 \, b} - \int \frac{{\left (a^{2} c \log \left (f\right ) -{\left (b c^{2} \log \left (f\right )^{2} - 2 \, a b c \log \left (f\right )\right )} x\right )} f^{\frac{c}{b x + a}}}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x^2 + c*x*log(f))*f^(c/(b*x + a))/b - integrate(1/2*(a^2*c*log(f) - (b*c^
2*log(f)^2 - 2*a*b*c*log(f))*x)*f^(c/(b*x + a))/(b^3*x^2 + 2*a*b^2*x + a^2*b), x
)

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Fricas [A]  time = 0.269991, size = 96, normalized size = 0.8 \[ \frac{{\left (b^{2} x^{2} - a^{2} +{\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac{c}{b x + a}} -{\left (c^{2} \log \left (f\right )^{2} - 2 \, a c \log \left (f\right )\right )}{\rm Ei}\left (\frac{c \log \left (f\right )}{b x + a}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)

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

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)