3.424 \(\int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx\)

Optimal. Leaf size=366 \[ \frac{d^2 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{2 h (d g-c h)^2}+\frac{f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{2 (d g-c h)^4}+\frac{d f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{(d g-c h)^3}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{2 h (g+h x)^2}+\frac{d f \log (F) (b c-a d) F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{2 (d g-c h)^3}-\frac{f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{2 (g+h x) (d g-c h)^2} \]

[Out]

(d^2*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(2*h*(d*g - c*h)^2) - F^(e
+ (f*(a + b*x))/(c + d*x))/(2*h*(g + h*x)^2) + (d*(b*c - a*d)*f*F^(e + (b*f)/d
- ((b*c - a*d)*f)/(d*(c + d*x)))*Log[F])/(2*(d*g - c*h)^3) - ((b*c - a*d)*f*F^(e
+ (f*(a + b*x))/(c + d*x))*Log[F])/(2*(d*g - c*h)^2*(g + h*x)) + (d*(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)^3 + ((b*c - a*d)^2*f^2*F^(
e + (f*(b*g - a*h))/(d*g - c*h))*h*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[
F])/((d*g - c*h)*(c + d*x)))]*Log[F]^2)/(2*(d*g - c*h)^4)

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Rubi [A]  time = 7.40363, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{d^2 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{2 h (d g-c h)^2}+\frac{f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{2 (d g-c h)^4}+\frac{d f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{(d g-c h)^3}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{2 h (g+h x)^2}+\frac{d f \log (F) (b c-a d) F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{2 (d g-c h)^3}-\frac{f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{2 (g+h x) (d g-c h)^2} \]

Antiderivative was successfully verified.

[In]  Int[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^3,x]

[Out]

(d^2*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(2*h*(d*g - c*h)^2) - F^(e
+ (f*(a + b*x))/(c + d*x))/(2*h*(g + h*x)^2) + (d*(b*c - a*d)*f*F^(e + (b*f)/d
- ((b*c - a*d)*f)/(d*(c + d*x)))*Log[F])/(2*(d*g - c*h)^3) - ((b*c - a*d)*f*F^(e
+ (f*(a + b*x))/(c + d*x))*Log[F])/(2*(d*g - c*h)^2*(g + h*x)) + (d*(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)^3 + ((b*c - a*d)^2*f^2*F^(
e + (f*(b*g - a*h))/(d*g - c*h))*h*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[
F])/((d*g - c*h)*(c + d*x)))]*Log[F]^2)/(2*(d*g - c*h)^4)

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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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.315812, size = 0, normalized size = 0. \[ \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^3,x]

[Out]

Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^3, x]

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Maple [B]  time = 0.095, size = 1934, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^3,x)

[Out]

-ln(F)*f*d^2/(c*h-d*g)^3*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+ln(F)*f*d/(c*h-d*g)^3
*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-ln(F)*f*d^2/(c*h-d*g)^3*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+ln(F)*f*d/(c*h-d*g)^3*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-1/2
*ln(F)^2*f^2*d^2*h/(c*h-d*g)^4*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)^2*a^2+ln(F)^2*f^2
*d*h/(c*h-d*g)^4*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)^2*a*c*b-1/2*ln(F)^2*f^2*h/(c*h-
d*g)^4*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)^2*c^2*b^2-1/2*ln(F)^2*f^2*d^2*h/(c*h-d*g)
^4*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^2+ln(F)^2*f^2*d*h/(c*h-d*g)^4*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*c*b-1/2*ln(F)^2*f^2*h/(c*h-d*g)^4*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^2*b^2-1/2*ln(F)^2*f^2*d^2*h/(c*h-d*g)^4*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^2+ln(F)^2*f^2*d*h/(c*h-d*g)^4*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)*l
n(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*c*b-1/2*l
n(F)^2*f^2*h/(c*h-d*g)^4*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^2*b^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^3,x, algorithm="maxima")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^3, x)

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Fricas [A]  time = 0.336944, size = 1019, normalized size = 2.78 \[ \frac{{\left ({\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} h^{3} x^{2} + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} g h^{2} x +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{2} h\right )} \log \left (F\right )^{2} + 2 \,{\left ({\left (b c d^{2} - a d^{3}\right )} f g^{3} -{\left (b c^{2} d - a c d^{2}\right )} f g^{2} h +{\left ({\left (b c d^{2} - a d^{3}\right )} f g h^{2} -{\left (b c^{2} d - a c d^{2}\right )} f h^{3}\right )} x^{2} + 2 \,{\left ({\left (b c d^{2} - a d^{3}\right )} f g^{2} h -{\left (b c^{2} d - a c d^{2}\right )} f g h^{2}\right )} x\right )} \log \left (F\right )\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 ) +{\left (2 \, c d^{3} g^{3} - 5 \, c^{2} d^{2} g^{2} h + 4 \, c^{3} d g h^{2} - c^{4} h^{3} +{\left (d^{4} g^{2} h - 2 \, c d^{3} g h^{2} + c^{2} d^{2} h^{3}\right )} x^{2} + 2 \,{\left (d^{4} g^{3} - 2 \, c d^{3} g^{2} h + c^{2} d^{2} g h^{2}\right )} x +{\left ({\left (b c^{2} d - a c d^{2}\right )} f g^{2} h -{\left (b c^{3} - a c^{2} d\right )} f g h^{2} +{\left ({\left (b c d^{2} - a d^{3}\right )} f g h^{2} -{\left (b c^{2} d - a c d^{2}\right )} f h^{3}\right )} x^{2} +{\left ({\left (b c d^{2} - a d^{3}\right )} f g^{2} h -{\left (b c^{3} - a c^{2} d\right )} f h^{3}\right )} x\right )} \log \left (F\right )\right )} F^{\frac{c e + a f +{\left (d e + b f\right )} x}{d x + c}}}{2 \,{\left (d^{4} g^{6} - 4 \, c d^{3} g^{5} h + 6 \, c^{2} d^{2} g^{4} h^{2} - 4 \, c^{3} d g^{3} h^{3} + c^{4} g^{2} h^{4} +{\left (d^{4} g^{4} h^{2} - 4 \, c d^{3} g^{3} h^{3} + 6 \, c^{2} d^{2} g^{2} h^{4} - 4 \, c^{3} d g h^{5} + c^{4} h^{6}\right )} x^{2} + 2 \,{\left (d^{4} g^{5} h - 4 \, c d^{3} g^{4} h^{2} + 6 \, c^{2} d^{2} g^{3} h^{3} - 4 \, c^{3} d g^{2} h^{4} + c^{4} g h^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^3,x, algorithm="fricas")

[Out]

1/2*((((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*h^3*x^2 + 2*(b^2*c^2 - 2*a*b*c*d + a^
2*d^2)*f^2*g*h^2*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*g^2*h)*log(F)^2 + 2*((b
*c*d^2 - a*d^3)*f*g^3 - (b*c^2*d - a*c*d^2)*f*g^2*h + ((b*c*d^2 - a*d^3)*f*g*h^2
- (b*c^2*d - a*c*d^2)*f*h^3)*x^2 + 2*((b*c*d^2 - a*d^3)*f*g^2*h - (b*c^2*d - a*
c*d^2)*f*g*h^2)*x)*log(F))*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))
+ (2*c*d^3*g^3 - 5*c^2*d^2*g^2*h + 4*c^3*d*g*h^2 - c^4*h^3 + (d^4*g^2*h - 2*c*d
^3*g*h^2 + c^2*d^2*h^3)*x^2 + 2*(d^4*g^3 - 2*c*d^3*g^2*h + c^2*d^2*g*h^2)*x + ((
b*c^2*d - a*c*d^2)*f*g^2*h - (b*c^3 - a*c^2*d)*f*g*h^2 + ((b*c*d^2 - a*d^3)*f*g*
h^2 - (b*c^2*d - a*c*d^2)*f*h^3)*x^2 + ((b*c*d^2 - a*d^3)*f*g^2*h - (b*c^3 - a*c
^2*d)*f*h^3)*x)*log(F))*F^((c*e + a*f + (d*e + b*f)*x)/(d*x + c)))/(d^4*g^6 - 4*
c*d^3*g^5*h + 6*c^2*d^2*g^4*h^2 - 4*c^3*d*g^3*h^3 + c^4*g^2*h^4 + (d^4*g^4*h^2 -
4*c*d^3*g^3*h^3 + 6*c^2*d^2*g^2*h^4 - 4*c^3*d*g*h^5 + c^4*h^6)*x^2 + 2*(d^4*g^5
*h - 4*c*d^3*g^4*h^2 + 6*c^2*d^2*g^3*h^3 - 4*c^3*d*g^2*h^4 + c^4*g*h^5)*x)

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{e + \frac{{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^3,x, algorithm="giac")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^3, x)