3.422 \(\int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{g+h x} \, dx\)

Optimal. Leaf size=104 \[ \frac{F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{h}-\frac{F^{\frac{b f}{d}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (b c-a d)}{d (c+d x)}\right )}{h} \]

[Out]

-((F^(e + (b*f)/d)*ExpIntegralEi[-(((b*c - a*d)*f*Log[F])/(d*(c + d*x)))])/h) +
(F^(e + (f*(b*g - a*h))/(d*g - c*h))*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Lo
g[F])/((d*g - c*h)*(c + d*x)))])/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]

[Out]

-((F^(e + (b*f)/d)*ExpIntegralEi[-(((b*c - a*d)*f*Log[F])/(d*(c + d*x)))])/h) +
(F^(e + (f*(b*g - a*h))/(d*g - c*h))*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Lo
g[F])/((d*g - c*h)*(c + d*x)))])/h

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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]

-F**((b*f + d*e)/d)*Ei(f*(a*d - b*c)*log(F)/(d*(c + d*x)))/h + F**((e*(c*h - d*g
) + f*(a*h - b*g))/(c*h - d*g))*Ei(-f*(g + h*x)*(a*d - b*c)*log(F)/((c + d*x)*(c
*h - d*g)))/h

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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]

(F^(e + (b*f)/d)*(-ExpIntegralEi[((-(b*c*f) + a*d*f)*Log[F])/(d*(c + d*x))] + F^
(((b*c - a*d)*f*h)/(d*(d*g - c*h)))*ExpIntegralEi[((b*c - a*d)*f*(g + h*x)*Log[F
])/((-(d*g) + c*h)*(c + d*x))]))/h

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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)

[Out]

d/h/(a*d-b*c)*F^((b*f+d*e)/d)*Ei(1,-f*(a*d-b*c)*ln(F)/d/(d*x+c)-(b*f+d*e)*ln(F)/
d-(-ln(F)*b*f-d*e*ln(F))/d)*a-1/h/(a*d-b*c)*F^((b*f+d*e)/d)*Ei(1,-f*(a*d-b*c)*ln
(F)/d/(d*x+c)-(b*f+d*e)*ln(F)/d-(-ln(F)*b*f-d*e*ln(F))/d)*c*b-d/h/(a*d-b*c)*F^((
a*f*h-b*f*g+c*e*h-d*e*g)/(c*h-d*g))*Ei(1,-f*(a*d-b*c)*ln(F)/d/(d*x+c)-(b*f+d*e)*
ln(F)/d-(-ln(F)*a*f*h+ln(F)*b*f*g-ln(F)*c*e*h+ln(F)*d*e*g)/(c*h-d*g))*a+1/h/(a*d
-b*c)*F^((a*f*h-b*f*g+c*e*h-d*e*g)/(c*h-d*g))*Ei(1,-f*(a*d-b*c)*ln(F)/d/(d*x+c)-
(b*f+d*e)*ln(F)/d-(-ln(F)*a*f*h+ln(F)*b*f*g-ln(F)*c*e*h+ln(F)*d*e*g)/(c*h-d*g))*
c*b

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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")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g), x)

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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]

-(F^((d*e + b*f)/d)*Ei(-(b*c - a*d)*f*log(F)/(d^2*x + c*d)) - F^(((d*e + b*f)*g
- (c*e + a*f)*h)/(d*g - c*h))*Ei(-((b*c - a*d)*f*h*x + (b*c - a*d)*f*g)*log(F)/(
c*d*g - c^2*h + (d^2*g - c*d*h)*x)))/h

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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]

Timed 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]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g), x)