Optimal. Leaf size=57 \[ \frac{b c \log (F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{e^2}-\frac{F^{c (a+b x)}}{e (d+e x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0775856, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{b c \log (F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{e^2}-\frac{F^{c (a+b x)}}{e (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[F^(c*(a + b*x))/(d + e*x)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.91969, size = 51, normalized size = 0.89 \[ - \frac{F^{c \left (a + b x\right )}}{e \left (d + e x\right )} + \frac{F^{\frac{c \left (a e - b d\right )}{e}} b c \log{\left (F \right )} \operatorname{Ei}{\left (\frac{b c \left (d + e x\right ) \log{\left (F \right )}}{e} \right )}}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(c*(b*x+a))/(e*x+d)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.118037, size = 55, normalized size = 0.96 \[ \frac{F^{a c} \left (b c \log (F) F^{-\frac{b c d}{e}} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )-\frac{e F^{b c x}}{d+e x}\right )}{e^2} \]
Antiderivative was successfully verified.
[In] Integrate[F^(c*(a + b*x))/(d + e*x)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.034, size = 97, normalized size = 1.7 \[ -{\frac{{F}^{c \left ( bx+a \right ) }cb\ln \left ( F \right ) }{{e}^{2}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-1}}-{\frac{cb\ln \left ( F \right ) }{{e}^{2}}{F}^{{\frac{c \left ( ea-bd \right ) }{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -\ln \left ( F \right ) ac-{\frac{-eac\ln \left ( F \right ) +\ln \left ( F \right ) bcd}{e}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(c*(b*x+a))/(e*x+d)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.786934, size = 59, normalized size = 1.04 \[ -\frac{F^{a c} exp_integral_e\left (2, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{{\left (e x + d\right )} F^{\frac{b c d}{e}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e*x + d)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.245465, size = 104, normalized size = 1.82 \[ -\frac{F^{b c x + a c} e - \frac{{\left (b c e x + b c d\right )}{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right ) \log \left (F\right )}{F^{\frac{b c d - a c e}{e}}}}{e^{3} x + d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e*x + d)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{c \left (a + b x\right )}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(c*(b*x+a))/(e*x+d)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e*x + d)^2,x, algorithm="giac")
[Out]