3.5.25 \(\int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^4} \, dx\) [425]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 6.47, antiderivative size = 634, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2264, 6874, 2262, 2240, 2241, 2263, 2265, 2209} \begin {gather*} \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 {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\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 {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\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 {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\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} \end {gather*}

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)

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2262

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :>
Int[(g + h*x)^m*F^((d*e + b*f)/d - f*((b*c - a*d)/(d*(c + d*x)))), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m},
x] && NeQ[b*c - a*d, 0] && EqQ[d*g - c*h, 0]

Rule 2263

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/((g_.) + (h_.)*(x_)), x_Symbol] :> Dist[d
/h, Int[F^(e + f*((a + b*x)/(c + d*x)))/(c + d*x), x], x] - Dist[(d*g - c*h)/h, Int[F^(e + f*((a + b*x)/(c + d
*x)))/((c + d*x)*(g + h*x)), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g -
c*h, 0]

Rule 2264

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_), x_Symbol] :> S
imp[(g + h*x)^(m + 1)*(F^(e + f*((a + b*x)/(c + d*x)))/(h*(m + 1))), x] - Dist[f*(b*c - a*d)*(Log[F]/(h*(m + 1
))), Int[(g + h*x)^(m + 1)*(F^(e + f*((a + b*x)/(c + d*x)))/(c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f
, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0] && ILtQ[m, -1]

Rule 2265

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/(((g_.) + (h_.)*(x_))*((i_.) + (j_.)*(x_)
)), x_Symbol] :> Dist[-d/(h*(d*i - c*j)), Subst[Int[F^(e + f*((b*i - a*j)/(d*i - c*j)) - (b*c - a*d)*f*(x/(d*i
- c*j)))/x, x], x, (i + j*x)/(c + d*x)], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && EqQ[d*g - c*h, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^4} \, dx &=-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac {((b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)^3} \, dx}{3 h}\\ &=-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac {((b c-a d) f \log (F)) \int \left (\frac {d^3 F^{e+\frac {f (a+b x)}{c+d x}}}{(d g-c h)^3 (c+d x)^2}-\frac {3 d^3 F^{e+\frac {f (a+b x)}{c+d x}} h}{(d g-c h)^4 (c+d x)}+\frac {F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^2 (g+h x)^3}+\frac {2 d F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^3 (g+h x)^2}+\frac {3 d^2 F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^4 (g+h x)}\right ) \, dx}{3 h}\\ &=-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac {\left (d^3 (b c-a d) f \log (F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{(d g-c h)^4}+\frac {\left (d^2 (b c-a d) f h \log (F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx}{(d g-c h)^4}+\frac {\left (d^3 (b c-a d) f \log (F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{3 h (d g-c h)^3}+\frac {(2 d (b c-a d) f h \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx}{3 (d g-c h)^3}+\frac {((b c-a d) f h \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx}{3 (d g-c h)^2}\\ &=-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}-\frac {\left (d^3 (b c-a d) f \log (F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{(d g-c h)^4}+\frac {\left (d^3 (b c-a d) f \log (F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{(d g-c h)^4}-\frac {\left (d^2 (b c-a d) f \log (F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{(d g-c h)^3}+\frac {\left (d^3 (b c-a d) f \log (F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{3 h (d g-c h)^3}+\frac {\left (2 d (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)} \, dx}{3 (d g-c h)^3}+\frac {\left ((b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)^2} \, dx}{6 (d g-c h)^2}\\ &=\frac {d^3 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac {d^2 (b c-a d) f F^{e+\frac {b f}{d}} \text {Ei}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right ) \log (F)}{(d g-c h)^4}+\frac {\left (d^2 (b c-a d) f \log (F)\right ) \text {Subst}\left (\int \frac {F^{e+\frac {f (b g-a h)}{d g-c h}-\frac {(b c-a d) f x}{d g-c h}}}{x} \, dx,x,\frac {g+h x}{c+d x}\right )}{(d g-c h)^4}+\frac {\left (d^3 (b c-a d) f \log (F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{(d g-c h)^4}+\frac {\left (2 d (b c-a d)^2 f^2 \log ^2(F)\right ) \int \left (\frac {d F^{e+\frac {f (a+b x)}{c+d x}}}{(d g-c h) (c+d x)^2}-\frac {d F^{e+\frac {f (a+b x)}{c+d x}} h}{(d g-c h)^2 (c+d x)}+\frac {F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^2 (g+h x)}\right ) \, dx}{3 (d g-c h)^3}+\frac {\left ((b c-a d)^2 f^2 \log ^2(F)\right ) \int \left (\frac {d^2 F^{e+\frac {f (a+b x)}{c+d x}}}{(d g-c h)^2 (c+d x)^2}-\frac {2 d^2 F^{e+\frac {f (a+b x)}{c+d x}} h}{(d g-c h)^3 (c+d x)}+\frac {F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^2 (g+h x)^2}+\frac {2 d F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^3 (g+h x)}\right ) \, dx}{6 (d g-c h)^2}\\ &=\frac {d^3 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac {d^2 (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^4}-\frac {\left (d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{3 (d g-c h)^5}-\frac {\left (2 d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac {\left (d (b c-a d)^2 f^2 h^2 \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx}{3 (d g-c h)^5}+\frac {\left (2 d (b c-a d)^2 f^2 h^2 \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx}{3 (d g-c h)^5}+\frac {\left (d^2 (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{6 (d g-c h)^4}+\frac {\left (2 d^2 (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{3 (d g-c h)^4}+\frac {\left ((b c-a d)^2 f^2 h^2 \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx}{6 (d g-c h)^4}\\ &=\frac {d^3 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac {d^2 (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^4}-\frac {(b c-a d)^2 f^2 F^{e+\frac {f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}-\frac {\left (d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac {\left (d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{3 (d g-c h)^5}-\frac {\left (2 d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac {\left (2 d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac {\left (d^2 (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{6 (d g-c h)^4}+\frac {\left (2 d^2 (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{3 (d g-c h)^4}-\frac {\left (d (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{3 (d g-c h)^4}-\frac {\left (2 d (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{3 (d g-c h)^4}+\frac {\left ((b c-a d)^3 f^3 h \log ^3(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)} \, dx}{6 (d g-c h)^4}\\ &=\frac {d^3 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac {5 d^2 (b c-a d) f F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac {d^2 (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^4}-\frac {(b c-a d)^2 f^2 F^{e+\frac {f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac {d (b c-a d)^2 f^2 F^{e+\frac {b f}{d}} h \text {Ei}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}+\frac {\left (d (b c-a d)^2 f^2 h \log ^2(F)\right ) \text {Subst}\left (\int \frac {F^{e+\frac {f (b g-a h)}{d g-c h}-\frac {(b c-a d) f x}{d g-c h}}}{x} \, dx,x,\frac {g+h x}{c+d x}\right )}{3 (d g-c h)^5}+\frac {\left (2 d (b c-a d)^2 f^2 h \log ^2(F)\right ) \text {Subst}\left (\int \frac {F^{e+\frac {f (b g-a h)}{d g-c h}-\frac {(b c-a d) f x}{d g-c h}}}{x} \, dx,x,\frac {g+h x}{c+d x}\right )}{3 (d g-c h)^5}+\frac {\left (d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac {\left (2 d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac {\left ((b c-a d)^3 f^3 h \log ^3(F)\right ) \int \left (\frac {d F^{e+\frac {f (a+b x)}{c+d x}}}{(d g-c h) (c+d x)^2}-\frac {d F^{e+\frac {f (a+b x)}{c+d x}} h}{(d g-c h)^2 (c+d x)}+\frac {F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^2 (g+h x)}\right ) \, dx}{6 (d g-c h)^4}\\ &=\frac {d^3 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac {5 d^2 (b c-a d) f F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac {d^2 (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^4}-\frac {(b c-a d)^2 f^2 F^{e+\frac {f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac {d (b c-a d)^2 f^2 F^{e+\frac {f (b g-a h)}{d g-c h}} h \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}-\frac {\left (d (b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{6 (d g-c h)^6}+\frac {\left ((b c-a d)^3 f^3 h^3 \log ^3(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx}{6 (d g-c h)^6}+\frac {\left (d (b c-a d)^3 f^3 h \log ^3(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{6 (d g-c h)^5}\\ &=\frac {d^3 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac {5 d^2 (b c-a d) f F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac {d^2 (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^4}-\frac {(b c-a d)^2 f^2 F^{e+\frac {f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac {d (b c-a d)^2 f^2 F^{e+\frac {f (b g-a h)}{d g-c h}} h \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}-\frac {\left (d (b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{6 (d g-c h)^6}+\frac {\left (d (b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{6 (d g-c h)^6}+\frac {\left (d (b c-a d)^3 f^3 h \log ^3(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{6 (d g-c h)^5}-\frac {\left ((b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{6 (d g-c h)^5}\\ &=\frac {d^3 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac {5 d^2 (b c-a d) f F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac {d^2 (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^4}+\frac {d (b c-a d)^2 f^2 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} h \log ^2(F)}{6 (d g-c h)^5}-\frac {(b c-a d)^2 f^2 F^{e+\frac {f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac {d (b c-a d)^2 f^2 F^{e+\frac {f (b g-a h)}{d g-c h}} h \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}+\frac {(b c-a d)^3 f^3 F^{e+\frac {b f}{d}} h^2 \text {Ei}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right ) \log ^3(F)}{6 (d g-c h)^6}+\frac {\left ((b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \text {Subst}\left (\int \frac {F^{e+\frac {f (b g-a h)}{d g-c h}-\frac {(b c-a d) f x}{d g-c h}}}{x} \, dx,x,\frac {g+h x}{c+d x}\right )}{6 (d g-c h)^6}+\frac {\left (d (b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{6 (d g-c h)^6}\\ &=\frac {d^3 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac {5 d^2 (b c-a d) f F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac {2 d (b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac {d^2 (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^4}+\frac {d (b c-a d)^2 f^2 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} h \log ^2(F)}{6 (d g-c h)^5}-\frac {(b c-a d)^2 f^2 F^{e+\frac {f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac {d (b c-a d)^2 f^2 F^{e+\frac {f (b g-a h)}{d g-c h}} h \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}+\frac {(b c-a d)^3 f^3 F^{e+\frac {f (b g-a h)}{d g-c h}} h^2 \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^3(F)}{6 (d g-c h)^6}\\ \end {align*}

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Mathematica [F]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^4} \, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(4670\) vs. \(2(618)=1236\).
time = 0.16, size = 4671, normalized size = 7.37

method	result	size
risch	\(\text {Expression too large to display}\)	\(4671\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2251 vs. \(2 (628) = 1256\).
time = 0.44, size = 2251, normalized size = 3.55 \begin {gather*} \text {Too large to display} \end {gather*}

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, 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^((b*f*g - a*f*h + (d*g - c*h)*e)/(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)*log(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^((b*f*x + a*f + (d*x + c)
*e)/(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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {F^{e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}}}{{\left (g+h\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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