\(\int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx\) [424]

Optimal result

Integrand size = 26, antiderivative size = 366 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\frac {d^2 F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{2 h (d g-c h)^2}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{2 h (g+h x)^2}+\frac {d (b c-a d) f F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}} \log (F)}{2 (d g-c h)^3}-\frac {(b c-a d) f F^{e+\frac {f (a+b x)}{c+d x}} \log (F)}{2 (d g-c h)^2 (g+h x)}+\frac {d (b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^3}+\frac {(b c-a d)^2 f^2 F^{e+\frac {f (b g-a h)}{d g-c h}} h \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{2 (d g-c h)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 2.82 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2264, 6874, 2262, 2240, 2241, 2263, 2265, 2209} \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\frac {d^2 F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{2 h (d g-c h)^2}+\frac {f^2 h \log ^2(F) (b c-a d)^2 F^{\frac {f (b g-a h)}{d g-c h}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{2 (d g-c h)^4}+\frac {d f \log (F) (b c-a d) F^{\frac {f (b g-a h)}{d g-c h}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^3}-\frac {F^{\frac {f (a+b x)}{c+d x}+e}}{2 h (g+h x)^2}+\frac {d f \log (F) (b c-a d) F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{2 (d g-c h)^3}-\frac {f \log (F) (b c-a d) F^{\frac {f (a+b x)}{c+d x}+e}}{2 (g+h x) (d g-c h)^2} \]

[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)

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

Mathematica [F]

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

[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]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2018\) vs. \(2(356)=712\).

Time = 0.73 (sec) , antiderivative size = 2019, normalized size of antiderivative = 5.52

method	result	size
risch	\(\text {Expression too large to display}\)	\(2019\)

[In]

int(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (356) = 712\).

Time = 0.37 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.06 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\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 )}} \]

[In]

integrate(F^(e+f*(b*x+a)/(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)

Sympy [F(-1)]

Timed out. \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\int { \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{3}} \,d x } \]

[In]

integrate(F^(e+f*(b*x+a)/(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)

Giac [F]

\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\int { \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{3}} \,d x } \]

[In]

integrate(F^(e+f*(b*x+a)/(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)

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx=\int \frac {F^{e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}}}{{\left (g+h\,x\right )}^3} \,d x \]

[In]

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

[Out]

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