Optimal. Leaf size=104 \[ \frac{F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{h}-\frac{F^{\frac{b f}{d}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (b c-a d)}{d (c+d x)}\right )}{h} \]
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Rubi [A] time = 1.60172, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{h}-\frac{F^{\frac{b f}{d}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (b c-a d)}{d (c+d x)}\right )}{h} \]
Antiderivative was successfully verified.
[In] Int[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x),x]
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Rubi in Sympy [A] time = 39.3447, size = 90, normalized size = 0.87 \[ - \frac{F^{\frac{b f + d e}{d}} \operatorname{Ei}{\left (\frac{f \left (a d - b c\right ) \log{\left (F \right )}}{d \left (c + d x\right )} \right )}}{h} + \frac{F^{\frac{e \left (c h - d g\right ) + f \left (a h - b g\right )}{c h - d g}} \operatorname{Ei}{\left (- \frac{f \left (g + h x\right ) \left (a d - b c\right ) \log{\left (F \right )}}{\left (c + d x\right ) \left (c h - d g\right )} \right )}}{h} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(e+f*(b*x+a)/(d*x+c))/(h*x+g),x)
[Out]
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Mathematica [A] time = 0.223699, size = 103, normalized size = 0.99 \[ \frac{F^{\frac{b f}{d}+e} \left (F^{\frac{f h (b c-a d)}{d (d g-c h)}} \text{ExpIntegralEi}\left (\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (c h-d g)}\right )-\text{ExpIntegralEi}\left (\frac{\log (F) (a d f-b c f)}{d (c+d x)}\right )\right )}{h} \]
Antiderivative was successfully verified.
[In] Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x),x]
[Out]
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Maple [B] time = 0.055, size = 432, normalized size = 4.2 \[{\frac{ad}{h \left ( ad-cb \right ) }{F}^{{\frac{bf+ed}{d}}}{\it Ei} \left ( 1,-{\frac{f \left ( ad-cb \right ) \ln \left ( F \right ) }{ \left ( dx+c \right ) d}}-{\frac{ \left ( bf+ed \right ) \ln \left ( F \right ) }{d}}-{\frac{-\ln \left ( F \right ) bf-de\ln \left ( F \right ) }{d}} \right ) }-{\frac{cb}{h \left ( ad-cb \right ) }{F}^{{\frac{bf+ed}{d}}}{\it Ei} \left ( 1,-{\frac{f \left ( ad-cb \right ) \ln \left ( F \right ) }{ \left ( dx+c \right ) d}}-{\frac{ \left ( bf+ed \right ) \ln \left ( F \right ) }{d}}-{\frac{-\ln \left ( F \right ) bf-de\ln \left ( F \right ) }{d}} \right ) }-{\frac{ad}{h \left ( ad-cb \right ) }{F}^{{\frac{afh-bfg+che-deg}{ch-dg}}}{\it Ei} \left ( 1,-{\frac{f \left ( ad-cb \right ) \ln \left ( F \right ) }{ \left ( dx+c \right ) d}}-{\frac{ \left ( bf+ed \right ) \ln \left ( F \right ) }{d}}-{\frac{-\ln \left ( F \right ) afh+\ln \left ( F \right ) bfg-\ln \left ( F \right ) ceh+\ln \left ( F \right ) deg}{ch-dg}} \right ) }+{\frac{cb}{h \left ( ad-cb \right ) }{F}^{{\frac{afh-bfg+che-deg}{ch-dg}}}{\it Ei} \left ( 1,-{\frac{f \left ( ad-cb \right ) \ln \left ( F \right ) }{ \left ( dx+c \right ) d}}-{\frac{ \left ( bf+ed \right ) \ln \left ( F \right ) }{d}}-{\frac{-\ln \left ( F \right ) afh+\ln \left ( F \right ) bfg-\ln \left ( F \right ) ceh+\ln \left ( F \right ) deg}{ch-dg}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{e + \frac{{\left (b x + a\right )} f}{d x + c}}}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g),x, algorithm="maxima")
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Fricas [A] time = 0.306801, size = 182, normalized size = 1.75 \[ -\frac{F^{\frac{d e + b f}{d}}{\rm Ei}\left (-\frac{{\left (b c - a d\right )} f \log \left (F\right )}{d^{2} x + c d}\right ) - F^{\frac{{\left (d e + b f\right )} g -{\left (c e + a f\right )} h}{d g - c h}}{\rm Ei}\left (-\frac{{\left ({\left (b c - a d\right )} f h x +{\left (b c - a d\right )} f g\right )} \log \left (F\right )}{c d g - c^{2} h +{\left (d^{2} g - c d h\right )} x}\right )}{h} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(e+f*(b*x+a)/(d*x+c))/(h*x+g),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{e + \frac{{\left (b x + a\right )} f}{d x + c}}}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g),x, algorithm="giac")
[Out]