3.425 \(\int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^4} \, dx\)

Optimal. Leaf size=634 \[ \frac{d^3 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{3 h (d g-c h)^3}+\frac{d^2 f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{(d g-c h)^4}+\frac{5 d^2 f \log (F) (b c-a d) F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{6 (d g-c h)^4}+\frac{f^3 h^2 \log ^3(F) (b c-a d)^3 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{6 (d g-c h)^6}+\frac{d f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{(d g-c h)^5}+\frac{d f^2 h \log ^2(F) (b c-a d)^2 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{6 (d g-c h)^5}-\frac{f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (a+b x)}{c+d x}+e}}{6 (g+h x) (d g-c h)^4}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{3 h (g+h x)^3}-\frac{2 d f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{3 (g+h x) (d g-c h)^3}-\frac{f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{6 (g+h x)^2 (d g-c h)^2} \]

[Out]

(d^3*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(3*h*(d*g - c*h)^3) - F^(e
+ (f*(a + b*x))/(c + d*x))/(3*h*(g + h*x)^3) + (5*d^2*(b*c - a*d)*f*F^(e + (b*f
)/d - ((b*c - a*d)*f)/(d*(c + d*x)))*Log[F])/(6*(d*g - c*h)^4) - ((b*c - a*d)*f*
F^(e + (f*(a + b*x))/(c + d*x))*Log[F])/(6*(d*g - c*h)^2*(g + h*x)^2) - (2*d*(b*
c - a*d)*f*F^(e + (f*(a + b*x))/(c + d*x))*Log[F])/(3*(d*g - c*h)^3*(g + h*x)) +
(d^2*(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)^4 + (d*(b
*c - a*d)^2*f^2*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x)))*h*Log[F]^2)/(6*(
d*g - c*h)^5) - ((b*c - a*d)^2*f^2*F^(e + (f*(a + b*x))/(c + d*x))*h*Log[F]^2)/(
6*(d*g - c*h)^4*(g + h*x)) + (d*(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)/(d*g - c*h)^5 + ((b*c - a*d)^3*f^3*F^(e + (f*(b*g - a*h))/(d*g - c*
h))*h^2*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x))
)]*Log[F]^3)/(6*(d*g - c*h)^6)

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Rubi [A]  time = 14.4596, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 48, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{d^3 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{3 h (d g-c h)^3}+\frac{d^2 f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{(d g-c h)^4}+\frac{5 d^2 f \log (F) (b c-a d) F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{6 (d g-c h)^4}+\frac{f^3 h^2 \log ^3(F) (b c-a d)^3 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{6 (d g-c h)^6}+\frac{d f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{ExpIntegralEi}\left (-\frac{f \log (F) (g+h x) (b c-a d)}{(c+d x) (d g-c h)}\right )}{(d g-c h)^5}+\frac{d f^2 h \log ^2(F) (b c-a d)^2 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{6 (d g-c h)^5}-\frac{f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (a+b x)}{c+d x}+e}}{6 (g+h x) (d g-c h)^4}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{3 h (g+h x)^3}-\frac{2 d f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{3 (g+h x) (d g-c h)^3}-\frac{f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{6 (g+h x)^2 (d g-c h)^2} \]

Antiderivative was successfully verified.

[In]  Int[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^4,x]

[Out]

(d^3*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(3*h*(d*g - c*h)^3) - F^(e
+ (f*(a + b*x))/(c + d*x))/(3*h*(g + h*x)^3) + (5*d^2*(b*c - a*d)*f*F^(e + (b*f
)/d - ((b*c - a*d)*f)/(d*(c + d*x)))*Log[F])/(6*(d*g - c*h)^4) - ((b*c - a*d)*f*
F^(e + (f*(a + b*x))/(c + d*x))*Log[F])/(6*(d*g - c*h)^2*(g + h*x)^2) - (2*d*(b*
c - a*d)*f*F^(e + (f*(a + b*x))/(c + d*x))*Log[F])/(3*(d*g - c*h)^3*(g + h*x)) +
(d^2*(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)^4 + (d*(b
*c - a*d)^2*f^2*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x)))*h*Log[F]^2)/(6*(
d*g - c*h)^5) - ((b*c - a*d)^2*f^2*F^(e + (f*(a + b*x))/(c + d*x))*h*Log[F]^2)/(
6*(d*g - c*h)^4*(g + h*x)) + (d*(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)/(d*g - c*h)^5 + ((b*c - a*d)^3*f^3*F^(e + (f*(b*g - a*h))/(d*g - c*
h))*h^2*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x))
)]*Log[F]^3)/(6*(d*g - c*h)^6)

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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)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 0.606867, size = 0, normalized size = 0. \[ \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^4,x]

[Out]

Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^4, x]

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Maple [B]  time = 0.162, size = 4471, normalized size = 7.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^4,x)

[Out]

-1/3*ln(F)^3*f^3*h^2/(c*h-d*g)^6*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)^3*b^3*c^3-1/6*l
n(F)^3*f^3*h^2/(c*h-d*g)^6*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*b^3*c^3-1/6*ln(F)^3
*f^3*h^2/(c*h-d*g)^6*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)*b^3*c^3-1/6*ln(F)^3*f^3*h^2
/(c*h-d*g)^6*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))*b^3*c^3+1/6*ln(F)^3*f^3*d^3*h^2/(c*h-d*g)^6*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^3+1/6*ln(F)^3*f^3*d^3*h^2/(c*h-d*g)^6*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^3+ln(F)^2*f^2*d^3*h/(c*h-d*g)^5*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^3*h/(c*h-d*g)^5*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^3*h/(c*h-d*g)^5*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+1/3*ln(F)^3*f^3*d^3*h^2/(c*h-d*g)^6*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)^3*a^3+1/6*ln(F)^3*f^3*d^3*h^2/(c*h-d*g)^6*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^3-ln(F)*f*d^2/(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*b-ln(F)*f*d^2/(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*b+ln(F)^2*f^2*d*h/(c*h-d*g)^5*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+ln(F)^2*f^2*d*h/(c*h-d*g)^5*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+ln(F)^2*f^2*d*h/(c*h-d*g)^5*F^((a*f*h-b*f*g+c*e*h-d*e*g)/(c*h-d*g))*E
i(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+ln(F)^3*f^3*d*h^2/(c*h-d*g)^6*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)^3*a*b^2*c^2-1/2*ln(F)^3*f^3*d^2*h^2/(c*h-d*g)^6*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*c*b+1/2*ln(F)^3*f^3*d*h^2/(c*h-d*g)^6*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*b^2*c^2-1/2*ln(F)^3*f^3*d^2*h^2/(c*h-d*g)^6*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*c*b+1/2*ln(F)^3*f^3*d*h^2/(c*h-d*g)^6*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*b^2*c^2-1/2*ln(F)^3*f^3*d^2*h^2/(c*h-d*g)^6*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*c*b+1/2*ln(F)^3*f^3*d*h
^2/(c*h-d*g)^6*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*b^2*c^2-2*ln(F)^2*f^2*d^2*h/(c*h-d*g)^5*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-2*ln(F)^2*f^2*d^2*h/(c*h-d*g)^5*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
-2*ln(F)^2*f^2*d^2*h/(c*h-d*g)^5*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*c*b-ln(F)^3*f^3*d^2*h^2/(c*h-d*g)^6*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)^3*a^2*c*b+ln(F)*f*d^3/(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+ln(F)*f*d^3/(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+l
n(F)*d*e*g)/(c*h-d*g))*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{e + \frac{{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^4,x, algorithm="maxima")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^4, x)

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Fricas [A]  time = 0.281614, size = 3038, normalized size = 4.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^4,x, algorithm="fricas")

[Out]

1/6*((((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*h^5*x^3 + 3*(b^3*
c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*g*h^4*x^2 + 3*(b^3*c^3 - 3*a*
b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*g^2*h^3*x + (b^3*c^3 - 3*a*b^2*c^2*d +
3*a^2*b*c*d^2 - a^3*d^3)*f^3*g^3*h^2)*log(F)^3 + 6*((b^2*c^2*d^2 - 2*a*b*c*d^3 +
a^2*d^4)*f^2*g^4*h - (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^2*g^3*h^2 + ((b^
2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^2*g*h^4 - (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*
c*d^3)*f^2*h^5)*x^3 + 3*((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^2*g^2*h^3 - (b^
2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^2*g*h^4)*x^2 + 3*((b^2*c^2*d^2 - 2*a*b*c*
d^3 + a^2*d^4)*f^2*g^3*h^2 - (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^2*g^2*h^3
)*x)*log(F)^2 + 6*((b*c*d^4 - a*d^5)*f*g^5 - 2*(b*c^2*d^3 - a*c*d^4)*f*g^4*h + (
b*c^3*d^2 - a*c^2*d^3)*f*g^3*h^2 + ((b*c*d^4 - a*d^5)*f*g^2*h^3 - 2*(b*c^2*d^3 -
a*c*d^4)*f*g*h^4 + (b*c^3*d^2 - a*c^2*d^3)*f*h^5)*x^3 + 3*((b*c*d^4 - a*d^5)*f*
g^3*h^2 - 2*(b*c^2*d^3 - a*c*d^4)*f*g^2*h^3 + (b*c^3*d^2 - a*c^2*d^3)*f*g*h^4)*x
^2 + 3*((b*c*d^4 - a*d^5)*f*g^4*h - 2*(b*c^2*d^3 - a*c*d^4)*f*g^3*h^2 + (b*c^3*d
^2 - a*c^2*d^3)*f*g^2*h^3)*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)) + (6*c*d^5*g^5 - 24*c^2*d^4*g^4*h + 38*c^3*d^3*g^3*h^2 - 30*c^4*d^2*
g^2*h^3 + 12*c^5*d*g*h^4 - 2*c^6*h^5 + 2*(d^6*g^3*h^2 - 3*c*d^5*g^2*h^3 + 3*c^2*
d^4*g*h^4 - c^3*d^3*h^5)*x^3 + 6*(d^6*g^4*h - 3*c*d^5*g^3*h^2 + 3*c^2*d^4*g^2*h^
3 - c^3*d^3*g*h^4)*x^2 + ((b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^2*g^3*h^2 -
(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^2*g^2*h^3 + ((b^2*c^2*d^2 - 2*a*b*c*d^3
+ a^2*d^4)*f^2*g*h^4 - (b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^2*h^5)*x^3 + (2
*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^2*g^2*h^3 - (b^2*c^3*d - 2*a*b*c^2*d^2
+ a^2*c*d^3)*f^2*g*h^4 - (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^2*h^5)*x^2 + ((
b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^2*g^3*h^2 + (b^2*c^3*d - 2*a*b*c^2*d^2 +
a^2*c*d^3)*f^2*g^2*h^3 - 2*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^2*g*h^4)*x)*l
og(F)^2 + 6*(d^6*g^5 - 3*c*d^5*g^4*h + 3*c^2*d^4*g^3*h^2 - c^3*d^3*g^2*h^3)*x +
(6*(b*c^2*d^3 - a*c*d^4)*f*g^4*h - 13*(b*c^3*d^2 - a*c^2*d^3)*f*g^3*h^2 + 8*(b*c
^4*d - a*c^3*d^2)*f*g^2*h^3 - (b*c^5 - a*c^4*d)*f*g*h^4 + 5*((b*c*d^4 - a*d^5)*f
*g^2*h^3 - 2*(b*c^2*d^3 - a*c*d^4)*f*g*h^4 + (b*c^3*d^2 - a*c^2*d^3)*f*h^5)*x^3
+ (11*(b*c*d^4 - a*d^5)*f*g^3*h^2 - 18*(b*c^2*d^3 - a*c*d^4)*f*g^2*h^3 + 3*(b*c^
3*d^2 - a*c^2*d^3)*f*g*h^4 + 4*(b*c^4*d - a*c^3*d^2)*f*h^5)*x^2 + (6*(b*c*d^4 -
a*d^5)*f*g^4*h - 2*(b*c^2*d^3 - a*c*d^4)*f*g^3*h^2 - 15*(b*c^3*d^2 - a*c^2*d^3)*
f*g^2*h^3 + 12*(b*c^4*d - a*c^3*d^2)*f*g*h^4 - (b*c^5 - a*c^4*d)*f*h^5)*x)*log(F
))*F^((c*e + a*f + (d*e + b*f)*x)/(d*x + c)))/(d^6*g^9 - 6*c*d^5*g^8*h + 15*c^2*
d^4*g^7*h^2 - 20*c^3*d^3*g^6*h^3 + 15*c^4*d^2*g^5*h^4 - 6*c^5*d*g^4*h^5 + c^6*g^
3*h^6 + (d^6*g^6*h^3 - 6*c*d^5*g^5*h^4 + 15*c^2*d^4*g^4*h^5 - 20*c^3*d^3*g^3*h^6
+ 15*c^4*d^2*g^2*h^7 - 6*c^5*d*g*h^8 + c^6*h^9)*x^3 + 3*(d^6*g^7*h^2 - 6*c*d^5*
g^6*h^3 + 15*c^2*d^4*g^5*h^4 - 20*c^3*d^3*g^4*h^5 + 15*c^4*d^2*g^3*h^6 - 6*c^5*d
*g^2*h^7 + c^6*g*h^8)*x^2 + 3*(d^6*g^8*h - 6*c*d^5*g^7*h^2 + 15*c^2*d^4*g^6*h^3
- 20*c^3*d^3*g^5*h^4 + 15*c^4*d^2*g^4*h^5 - 6*c^5*d*g^3*h^6 + c^6*g^2*h^7)*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)**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{e + \frac{{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^4,x, algorithm="giac")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^4, x)