3.423 \(\int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx\)

Optimal. Leaf size=159 \[ \frac{f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{(d g-c h)^2}+\frac{d F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{h (d g-c h)}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{h (g+h x)} \]

[Out]

(d*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(h*(d*g - c*h)) - F^(e + (f*
(a + b*x))/(c + d*x))/(h*(g + h*x)) + ((b*c - a*d)*f*F^(e + (f*(b*g - a*h))/(d*g
- c*h))*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)
))]*Log[F])/(d*g - c*h)^2

_______________________________________________________________________________________

Rubi [A]  time = 3.63333, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{(d g-c h)^2}+\frac{d F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{h (d g-c h)}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{h (g+h x)} \]

Antiderivative was successfully verified.

[In]  Int[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^2,x]

[Out]

(d*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(h*(d*g - c*h)) - F^(e + (f*
(a + b*x))/(c + d*x))/(h*(g + h*x)) + ((b*c - a*d)*f*F^(e + (f*(b*g - a*h))/(d*g
- c*h))*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)
))]*Log[F])/(d*g - c*h)^2

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.57565, size = 0, normalized size = 0. \[ \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^2,x]

[Out]

Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^2, x]

_______________________________________________________________________________________

Maple [B]  time = 0.05, size = 560, normalized size = 3.5 \[{\frac{\ln \left ( F \right ) adf}{ \left ( ch-dg \right ) ^{2}}{F}^{{\frac{bfx+dex+af+ce}{dx+c}}} \left ({\frac{f\ln \left ( F \right ) a}{dx+c}}-{\frac{cb\ln \left ( F \right ) f}{ \left ( dx+c \right ) d}}+{\frac{\ln \left ( F \right ) bf}{d}}+\ln \left ( F \right ) e-{\frac{\ln \left ( F \right ) afh}{ch-dg}}+{\frac{\ln \left ( F \right ) bfg}{ch-dg}}-{\frac{\ln \left ( F \right ) ceh}{ch-dg}}+{\frac{\ln \left ( F \right ) deg}{ch-dg}} \right ) ^{-1}}-{\frac{cb\ln \left ( F \right ) f}{ \left ( ch-dg \right ) ^{2}}{F}^{{\frac{bfx+dex+af+ce}{dx+c}}} \left ({\frac{f\ln \left ( F \right ) a}{dx+c}}-{\frac{cb\ln \left ( F \right ) f}{ \left ( dx+c \right ) d}}+{\frac{\ln \left ( F \right ) bf}{d}}+\ln \left ( F \right ) e-{\frac{\ln \left ( F \right ) afh}{ch-dg}}+{\frac{\ln \left ( F \right ) bfg}{ch-dg}}-{\frac{\ln \left ( F \right ) ceh}{ch-dg}}+{\frac{\ln \left ( F \right ) deg}{ch-dg}} \right ) ^{-1}}+{\frac{\ln \left ( F \right ) adf}{ \left ( ch-dg \right ) ^{2}}{F}^{{\frac{afh-bfg+che-deg}{ch-dg}}}{\it Ei} \left ( 1,-{\frac{\ln \left ( F \right ) f \left ( ad-cb \right ) }{ \left ( dx+c \right ) d}}-{\frac{ \left ( bf+ed \right ) \ln \left ( F \right ) }{d}}-{\frac{-\ln \left ( F \right ) afh+\ln \left ( F \right ) bfg-\ln \left ( F \right ) ceh+\ln \left ( F \right ) deg}{ch-dg}} \right ) }-{\frac{cb\ln \left ( F \right ) f}{ \left ( ch-dg \right ) ^{2}}{F}^{{\frac{afh-bfg+che-deg}{ch-dg}}}{\it Ei} \left ( 1,-{\frac{\ln \left ( F \right ) f \left ( ad-cb \right ) }{ \left ( dx+c \right ) d}}-{\frac{ \left ( bf+ed \right ) \ln \left ( F \right ) }{d}}-{\frac{-\ln \left ( F \right ) afh+\ln \left ( F \right ) bfg-\ln \left ( F \right ) ceh+\ln \left ( F \right ) deg}{ch-dg}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^2,x)

[Out]

f*ln(F)/(c*h-d*g)^2*F^((b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(f*ln(F)/(d*x+c)*a-f*ln(F)
/d/(d*x+c)*c*b+ln(F)/d*b*f+ln(F)*e-1/(c*h-d*g)*ln(F)*a*f*h+1/(c*h-d*g)*ln(F)*b*f
*g-1/(c*h-d*g)*ln(F)*c*e*h+1/(c*h-d*g)*ln(F)*d*e*g)*a*d-f*ln(F)/(c*h-d*g)^2*F^((
b*f*x+d*e*x+a*f+c*e)/(d*x+c))/(f*ln(F)/(d*x+c)*a-f*ln(F)/d/(d*x+c)*c*b+ln(F)/d*b
*f+ln(F)*e-1/(c*h-d*g)*ln(F)*a*f*h+1/(c*h-d*g)*ln(F)*b*f*g-1/(c*h-d*g)*ln(F)*c*e
*h+1/(c*h-d*g)*ln(F)*d*e*g)*c*b+f*ln(F)/(c*h-d*g)^2*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*d-f*ln(F)/(c*h-d*g)^2*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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^2,x, algorithm="maxima")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^2, x)

_______________________________________________________________________________________

Fricas [A]  time = 0.264283, size = 297, normalized size = 1.87 \[ \frac{{\left ({\left (b c - a d\right )} f h x +{\left (b c - a d\right )} f g\right )} F^{\frac{{\left (d e + b f\right )} g -{\left (c e + a f\right )} h}{d g - c h}}{\rm Ei}\left (-\frac{{\left ({\left (b c - a d\right )} f h x +{\left (b c - a d\right )} f g\right )} \log \left (F\right )}{c d g - c^{2} h +{\left (d^{2} g - c d h\right )} x}\right ) \log \left (F\right ) +{\left (c d g - c^{2} h +{\left (d^{2} g - c d h\right )} x\right )} F^{\frac{c e + a f +{\left (d e + b f\right )} x}{d x + c}}}{d^{2} g^{3} - 2 \, c d g^{2} h + c^{2} g h^{2} +{\left (d^{2} g^{2} h - 2 \, c d g h^{2} + c^{2} h^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^2,x, algorithm="fricas")

[Out]

(((b*c - a*d)*f*h*x + (b*c - a*d)*f*g)*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))*log(F) + (c*d*g - c^2*h + (d^2*g - c*d*h)*x)*F^((c*e + a*f + (d*e +
b*f)*x)/(d*x + c)))/(d^2*g^3 - 2*c*d*g^2*h + c^2*g*h^2 + (d^2*g^2*h - 2*c*d*g*h
^2 + c^2*h^3)*x)

_______________________________________________________________________________________

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{e + \frac{{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^2,x, algorithm="giac")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^2, x)