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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 4.75749, 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, Rules used = {2232, 6742, 2230, 2209, 2210, 2231, 2233, 2178} \[ \frac{d^2 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{2 h (d g-c h)^2}+\frac{f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{2 (d g-c h)^4}+\frac{d f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^3}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{2 h (g+h x)^2}+\frac{d f \log (F) (b c-a d) F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{2 (d g-c h)^3}-\frac{f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{2 (g+h x) (d g-c h)^2} \]

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)

Rule 2232

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 6742

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

Rule 2230

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 2209

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 2210

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 2231

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 2233

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 2178

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

Rubi steps

\begin{align*} \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx &=-\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)) \operatorname{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 ) \operatorname{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]  time = 0.503373, 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]

________________________________________________________________________________________

Maple [B]  time = 0.213, size = 2014, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

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]

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

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

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, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.73227, size = 1507, normalized size = 4.12 \begin{align*} \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 )}} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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

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, algorithm="giac")

[Out]

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