\(\int \frac {(d+e e^{h+i x}) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx\) [572]

Optimal result

Integrand size = 44, antiderivative size = 770 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {3 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^3}+\frac {6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3}-\frac {6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3 \operatorname {PolyLog}\left (4,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^4}-\frac {6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3 \operatorname {PolyLog}\left (4,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^4} \]

[Out]

1/4*(g*x+f)^4*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/g/(b-(-4*a*c+b^2)^(1/2))-(g*x+f)^3*ln(1+2*c*exp(i*x+h)/(b-(-
4*a*c+b^2)^(1/2)))*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/i/(b-(-4*a*c+b^2)^(1/2))-3*g*(g*x+f)^2*polylog(2,-2*c*e
xp(i*x+h)/(b-(-4*a*c+b^2)^(1/2)))*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/i^2/(b-(-4*a*c+b^2)^(1/2))+6*g^2*(g*x+f)
*polylog(3,-2*c*exp(i*x+h)/(b-(-4*a*c+b^2)^(1/2)))*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/i^3/(b-(-4*a*c+b^2)^(1/
2))-6*g^3*polylog(4,-2*c*exp(i*x+h)/(b-(-4*a*c+b^2)^(1/2)))*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))/i^4/(b-(-4*a*c
+b^2)^(1/2))+1/4*(g*x+f)^4*(e+(b*e-2*c*d)/(-4*a*c+b^2)^(1/2))/g/(b+(-4*a*c+b^2)^(1/2))-(g*x+f)^3*ln(1+2*c*exp(
i*x+h)/(b+(-4*a*c+b^2)^(1/2)))*(e+(b*e-2*c*d)/(-4*a*c+b^2)^(1/2))/i/(b+(-4*a*c+b^2)^(1/2))-3*g*(g*x+f)^2*polyl
og(2,-2*c*exp(i*x+h)/(b+(-4*a*c+b^2)^(1/2)))*(e+(b*e-2*c*d)/(-4*a*c+b^2)^(1/2))/i^2/(b+(-4*a*c+b^2)^(1/2))+6*g
^2*(g*x+f)*polylog(3,-2*c*exp(i*x+h)/(b+(-4*a*c+b^2)^(1/2)))*(e+(b*e-2*c*d)/(-4*a*c+b^2)^(1/2))/i^3/(b+(-4*a*c
+b^2)^(1/2))-6*g^3*polylog(4,-2*c*exp(i*x+h)/(b+(-4*a*c+b^2)^(1/2)))*(e+(b*e-2*c*d)/(-4*a*c+b^2)^(1/2))/i^4/(b
+(-4*a*c+b^2)^(1/2))

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 770, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {2297, 2215, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\frac {6 g^2 (f+g x) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^3 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {6 g^2 (f+g x) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^3 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {3 g (f+g x)^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {3 g (f+g x)^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^2 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {(f+g x)^3 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{i \left (b-\sqrt {b^2-4 a c}\right )}-\frac {(f+g x)^3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{i \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^4 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{4 g \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^4 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{4 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {6 g^3 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {PolyLog}\left (4,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^4 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {6 g^3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (4,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^4 \left (\sqrt {b^2-4 a c}+b\right )} \]

[In]

Int[((d + e*E^(h + i*x))*(f + g*x)^3)/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)),x]

[Out]

((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^4)/(4*(b + Sqrt[b^2 - 4*a*c])*g) + ((e + (2*c*d - b*e)/Sqrt[b
^2 - 4*a*c])*(f + g*x)^4)/(4*(b - Sqrt[b^2 - 4*a*c])*g) - ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^3*L
og[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*i) - ((e - (2*c*d - b*e)/Sqrt[b^2
- 4*a*c])*(f + g*x)^3*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*i) - (3*(e
+ (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*(f + g*x)^2*PolyLog[2, (-2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b -
Sqrt[b^2 - 4*a*c])*i^2) - (3*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*(f + g*x)^2*PolyLog[2, (-2*c*E^(h + i*x)
)/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*i^2) + (6*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^2*(f +
g*x)*PolyLog[3, (-2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*i^3) + (6*(e - (2*c*d -
b*e)/Sqrt[b^2 - 4*a*c])*g^2*(f + g*x)*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 -
4*a*c])*i^3) - (6*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^3*PolyLog[4, (-2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a
*c])])/((b - Sqrt[b^2 - 4*a*c])*i^4) - (6*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^3*PolyLog[4, (-2*c*E^(h + i*
x))/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*i^4)

Rule 2215

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

Rule 2221

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

Rule 2297

Int[(((i_.)*(F_)^(u_) + (h_))*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[Simplify[(2*c*h - b*i)/q] + i, Int[(f + g*x)^m/(b - q + 2*c*F^u), x]
, x] - Dist[Simplify[(2*c*h - b*i)/q] - i, Int[(f + g*x)^m/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f,
g, h, i}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = -\left (\left (-e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {(f+g x)^3}{b+\sqrt {b^2-4 a c}+2 c e^{h+i x}} \, dx\right )+\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {(f+g x)^3}{b-\sqrt {b^2-4 a c}+2 c e^{h+i x}} \, dx \\ & = \frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (2 c \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {e^{h+i x} (f+g x)^3}{b+\sqrt {b^2-4 a c}+2 c e^{h+i x}} \, dx}{b+\sqrt {b^2-4 a c}}-\frac {\left (2 c \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {e^{h+i x} (f+g x)^3}{b-\sqrt {b^2-4 a c}+2 c e^{h+i x}} \, dx}{b-\sqrt {b^2-4 a c}} \\ & = \frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}+\frac {\left (3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \int (f+g x)^2 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b+\sqrt {b^2-4 a c}\right ) i}+\frac {\left (3 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g\right ) \int (f+g x)^2 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b-\sqrt {b^2-4 a c}\right ) i} \\ & = \frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {3 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {\left (6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2\right ) \int (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {\left (6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2\right ) \int (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b-\sqrt {b^2-4 a c}\right ) i^2} \\ & = \frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {3 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^3}+\frac {6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3}-\frac {\left (6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3\right ) \int \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b+\sqrt {b^2-4 a c}\right ) i^3}-\frac {\left (6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3\right ) \int \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b-\sqrt {b^2-4 a c}\right ) i^3} \\ & = \frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {3 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^3}+\frac {6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3}-\frac {\left (6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^{h+i x}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^4}-\frac {\left (6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,e^{h+i x}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^4} \\ & = \frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {3 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^3}+\frac {6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3}-\frac {6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3 \text {Li}_4\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^4}-\frac {6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3 \text {Li}_4\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^4} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2448\) vs. \(2(770)=1540\).

Time = 2.62 (sec) , antiderivative size = 2448, normalized size of antiderivative = 3.18 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Result too large to show} \]

[In]

Integrate[((d + e*E^(h + i*x))*(f + g*x)^3)/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)),x]

[Out]

-1/4*(-6*Sqrt[-(b^2 - 4*a*c)^2]*d*f^2*g*i^4*x^2 - 4*Sqrt[-(b^2 - 4*a*c)^2]*d*f*g^2*i^4*x^3 - Sqrt[-(b^2 - 4*a*
c)^2]*d*g^3*i^4*x^4 + 4*b*Sqrt[b^2 - 4*a*c]*d*f^3*i^3*ArcTan[(b + 2*c*E^(h + i*x))/Sqrt[-b^2 + 4*a*c]] - 8*a*S
qrt[b^2 - 4*a*c]*e*f^3*i^3*ArcTan[(b + 2*c*E^(h + i*x))/Sqrt[-b^2 + 4*a*c]] - 4*Sqrt[-(b^2 - 4*a*c)^2]*d*f^3*i
^3*Log[E^(h + i*x)] + 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])
] + 6*b*Sqrt[-b^2 + 4*a*c]*d*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] - 12*a*Sqrt[-b^2 +
4*a*c]*e*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f*g^2*i^
3*x^2*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 6*b*Sqrt[-b^2 + 4*a*c]*d*f*g^2*i^3*x^2*Log[1 + (2*c
*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] - 12*a*Sqrt[-b^2 + 4*a*c]*e*f*g^2*i^3*x^2*Log[1 + (2*c*E^(h + i*x))/(b
- Sqrt[b^2 - 4*a*c])] + 2*Sqrt[-(b^2 - 4*a*c)^2]*d*g^3*i^3*x^3*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c
])] + 2*b*Sqrt[-b^2 + 4*a*c]*d*g^3*i^3*x^3*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] - 4*a*Sqrt[-b^2
+ 4*a*c]*e*g^3*i^3*x^3*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f^2*g*i
^3*x*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 6*b*Sqrt[-b^2 + 4*a*c]*d*f^2*g*i^3*x*Log[1 + (2*c*E^
(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 12*a*Sqrt[-b^2 + 4*a*c]*e*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/(b + Sqr
t[b^2 - 4*a*c])] + 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f*g^2*i^3*x^2*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])]
- 6*b*Sqrt[-b^2 + 4*a*c]*d*f*g^2*i^3*x^2*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 12*a*Sqrt[-b^2
+ 4*a*c]*e*f*g^2*i^3*x^2*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 2*Sqrt[-(b^2 - 4*a*c)^2]*d*g^3*i
^3*x^3*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 2*b*Sqrt[-b^2 + 4*a*c]*d*g^3*i^3*x^3*Log[1 + (2*c*
E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 4*a*Sqrt[-b^2 + 4*a*c]*e*g^3*i^3*x^3*Log[1 + (2*c*E^(h + i*x))/(b + Sq
rt[b^2 - 4*a*c])] + 2*Sqrt[-(b^2 - 4*a*c)^2]*d*f^3*i^3*Log[a + E^(h + i*x)*(b + c*E^(h + i*x))] + 6*(Sqrt[-(b^
2 - 4*a*c)^2]*d + b*Sqrt[-b^2 + 4*a*c]*d - 2*a*Sqrt[-b^2 + 4*a*c]*e)*g*i^2*(f + g*x)^2*PolyLog[2, (2*c*E^(h +
i*x))/(-b + Sqrt[b^2 - 4*a*c])] + 6*(Sqrt[-(b^2 - 4*a*c)^2]*d - b*Sqrt[-b^2 + 4*a*c]*d + 2*a*Sqrt[-b^2 + 4*a*c
]*e)*g*i^2*(f + g*x)^2*PolyLog[2, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 12*Sqrt[-(b^2 - 4*a*c)^2]*d*f*
g^2*i*PolyLog[3, (2*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a*c])] - 12*b*Sqrt[-b^2 + 4*a*c]*d*f*g^2*i*PolyLog[3, (2
*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a*c])] + 24*a*Sqrt[-b^2 + 4*a*c]*e*f*g^2*i*PolyLog[3, (2*c*E^(h + i*x))/(-b
+ Sqrt[b^2 - 4*a*c])] - 12*Sqrt[-(b^2 - 4*a*c)^2]*d*g^3*i*x*PolyLog[3, (2*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a
*c])] - 12*b*Sqrt[-b^2 + 4*a*c]*d*g^3*i*x*PolyLog[3, (2*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a*c])] + 24*a*Sqrt[-
b^2 + 4*a*c]*e*g^3*i*x*PolyLog[3, (2*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a*c])] - 12*Sqrt[-(b^2 - 4*a*c)^2]*d*f*
g^2*i*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 12*b*Sqrt[-b^2 + 4*a*c]*d*f*g^2*i*PolyLog[3, (-
2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 24*a*Sqrt[-b^2 + 4*a*c]*e*f*g^2*i*PolyLog[3, (-2*c*E^(h + i*x))/(b
+ Sqrt[b^2 - 4*a*c])] - 12*Sqrt[-(b^2 - 4*a*c)^2]*d*g^3*i*x*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a
*c])] + 12*b*Sqrt[-b^2 + 4*a*c]*d*g^3*i*x*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 24*a*Sqrt[-
b^2 + 4*a*c]*e*g^3*i*x*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 12*Sqrt[-(b^2 - 4*a*c)^2]*d*g^
3*PolyLog[4, (2*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a*c])] + 12*b*Sqrt[-b^2 + 4*a*c]*d*g^3*PolyLog[4, (2*c*E^(h
+ i*x))/(-b + Sqrt[b^2 - 4*a*c])] - 24*a*Sqrt[-b^2 + 4*a*c]*e*g^3*PolyLog[4, (2*c*E^(h + i*x))/(-b + Sqrt[b^2
- 4*a*c])] + 12*Sqrt[-(b^2 - 4*a*c)^2]*d*g^3*PolyLog[4, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 12*b*Sqr
t[-b^2 + 4*a*c]*d*g^3*PolyLog[4, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 24*a*Sqrt[-b^2 + 4*a*c]*e*g^3*P
olyLog[4, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/(a*Sqrt[-(b^2 - 4*a*c)^2]*i^4)

Maple [F]

\[\int \frac {\left (d +e \,{\mathrm e}^{i x +h}\right ) \left (g x +f \right )^{3}}{a +b \,{\mathrm e}^{i x +h}+c \,{\mathrm e}^{2 i x +2 h}}d x\]

[In]

int((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)

[Out]

int((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1859 vs. \(2 (700) = 1400\).

Time = 0.41 (sec) , antiderivative size = 1859, normalized size of antiderivative = 2.41 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Too large to display} \]

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="fricas")

[Out]

1/4*((b^2 - 4*a*c)*d*g^3*i^4*x^4 + 4*(b^2 - 4*a*c)*d*f*g^2*i^4*x^3 + 6*(b^2 - 4*a*c)*d*f^2*g*i^4*x^2 + 4*(b^2
- 4*a*c)*d*f^3*i^4*x - 6*((b^2 - 4*a*c)*d*g^3*i^2*x^2 + 2*(b^2 - 4*a*c)*d*f*g^2*i^2*x + (b^2 - 4*a*c)*d*f^2*g*
i^2 + ((a*b*d - 2*a^2*e)*g^3*i^2*x^2 + 2*(a*b*d - 2*a^2*e)*f*g^2*i^2*x + (a*b*d - 2*a^2*e)*f^2*g*i^2)*sqrt((b^
2 - 4*a*c)/a^2))*dilog(-1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) + b*e^(i*x + h) + 2*a)/a + 1) - 6*((b^2 - 4
*a*c)*d*g^3*i^2*x^2 + 2*(b^2 - 4*a*c)*d*f*g^2*i^2*x + (b^2 - 4*a*c)*d*f^2*g*i^2 - ((a*b*d - 2*a^2*e)*g^3*i^2*x
^2 + 2*(a*b*d - 2*a^2*e)*f*g^2*i^2*x + (a*b*d - 2*a^2*e)*f^2*g*i^2)*sqrt((b^2 - 4*a*c)/a^2))*dilog(1/2*(a*sqrt
((b^2 - 4*a*c)/a^2)*e^(i*x + h) - b*e^(i*x + h) - 2*a)/a + 1) + 2*((b^2 - 4*a*c)*d*g^3*h^3 - 3*(b^2 - 4*a*c)*d
*f*g^2*h^2*i + 3*(b^2 - 4*a*c)*d*f^2*g*h*i^2 - (b^2 - 4*a*c)*d*f^3*i^3 - ((a*b*d - 2*a^2*e)*g^3*h^3 - 3*(a*b*d
- 2*a^2*e)*f*g^2*h^2*i + 3*(a*b*d - 2*a^2*e)*f^2*g*h*i^2 - (a*b*d - 2*a^2*e)*f^3*i^3)*sqrt((b^2 - 4*a*c)/a^2)
)*log(2*c*e^(i*x + h) + a*sqrt((b^2 - 4*a*c)/a^2) + b) + 2*((b^2 - 4*a*c)*d*g^3*h^3 - 3*(b^2 - 4*a*c)*d*f*g^2*
h^2*i + 3*(b^2 - 4*a*c)*d*f^2*g*h*i^2 - (b^2 - 4*a*c)*d*f^3*i^3 + ((a*b*d - 2*a^2*e)*g^3*h^3 - 3*(a*b*d - 2*a^
2*e)*f*g^2*h^2*i + 3*(a*b*d - 2*a^2*e)*f^2*g*h*i^2 - (a*b*d - 2*a^2*e)*f^3*i^3)*sqrt((b^2 - 4*a*c)/a^2))*log(2
*c*e^(i*x + h) - a*sqrt((b^2 - 4*a*c)/a^2) + b) - 2*((b^2 - 4*a*c)*d*g^3*i^3*x^3 + 3*(b^2 - 4*a*c)*d*f*g^2*i^3
*x^2 + 3*(b^2 - 4*a*c)*d*f^2*g*i^3*x + (b^2 - 4*a*c)*d*g^3*h^3 - 3*(b^2 - 4*a*c)*d*f*g^2*h^2*i + 3*(b^2 - 4*a*
c)*d*f^2*g*h*i^2 + ((a*b*d - 2*a^2*e)*g^3*i^3*x^3 + 3*(a*b*d - 2*a^2*e)*f*g^2*i^3*x^2 + 3*(a*b*d - 2*a^2*e)*f^
2*g*i^3*x + (a*b*d - 2*a^2*e)*g^3*h^3 - 3*(a*b*d - 2*a^2*e)*f*g^2*h^2*i + 3*(a*b*d - 2*a^2*e)*f^2*g*h*i^2)*sqr
t((b^2 - 4*a*c)/a^2))*log(1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) + b*e^(i*x + h) + 2*a)/a) - 2*((b^2 - 4*a
*c)*d*g^3*i^3*x^3 + 3*(b^2 - 4*a*c)*d*f*g^2*i^3*x^2 + 3*(b^2 - 4*a*c)*d*f^2*g*i^3*x + (b^2 - 4*a*c)*d*g^3*h^3
- 3*(b^2 - 4*a*c)*d*f*g^2*h^2*i + 3*(b^2 - 4*a*c)*d*f^2*g*h*i^2 - ((a*b*d - 2*a^2*e)*g^3*i^3*x^3 + 3*(a*b*d -
2*a^2*e)*f*g^2*i^3*x^2 + 3*(a*b*d - 2*a^2*e)*f^2*g*i^3*x + (a*b*d - 2*a^2*e)*g^3*h^3 - 3*(a*b*d - 2*a^2*e)*f*g
^2*h^2*i + 3*(a*b*d - 2*a^2*e)*f^2*g*h*i^2)*sqrt((b^2 - 4*a*c)/a^2))*log(-1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*
x + h) - b*e^(i*x + h) - 2*a)/a) - 12*((b^2 - 4*a*c)*d*g^3 + (a*b*d - 2*a^2*e)*g^3*sqrt((b^2 - 4*a*c)/a^2))*po
lylog(4, -1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) + b*e^(i*x + h))/a) - 12*((b^2 - 4*a*c)*d*g^3 - (a*b*d -
2*a^2*e)*g^3*sqrt((b^2 - 4*a*c)/a^2))*polylog(4, 1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) - b*e^(i*x + h))/a
) + 12*((b^2 - 4*a*c)*d*g^3*i*x + (b^2 - 4*a*c)*d*f*g^2*i + ((a*b*d - 2*a^2*e)*g^3*i*x + (a*b*d - 2*a^2*e)*f*g
^2*i)*sqrt((b^2 - 4*a*c)/a^2))*polylog(3, -1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) + b*e^(i*x + h))/a) + 12
*((b^2 - 4*a*c)*d*g^3*i*x + (b^2 - 4*a*c)*d*f*g^2*i - ((a*b*d - 2*a^2*e)*g^3*i*x + (a*b*d - 2*a^2*e)*f*g^2*i)*
sqrt((b^2 - 4*a*c)/a^2))*polylog(3, 1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) - b*e^(i*x + h))/a))/((a*b^2 -
4*a^2*c)*i^4)

Sympy [F]

\[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int \frac {\left (d + e e^{h} e^{i x}\right ) \left (f + g x\right )^{3}}{a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}}\, dx \]

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)**3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)

[Out]

Integral((d + e*exp(h)*exp(i*x))*(f + g*x)**3/(a + b*exp(h)*exp(i*x) + c*exp(2*h)*exp(2*i*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [F]

\[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (e e^{\left (i x + h\right )} + d\right )}}{c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a} \,d x } \]

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="giac")

[Out]

integrate((g*x + f)^3*(e*e^(i*x + h) + d)/(c*e^(2*i*x + 2*h) + b*e^(i*x + h) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\left (d+e\,{\mathrm {e}}^{h+i\,x}\right )}{a+b\,{\mathrm {e}}^{h+i\,x}+c\,{\mathrm {e}}^{2\,h+2\,i\,x}} \,d x \]

[In]

int(((f + g*x)^3*(d + e*exp(h + i*x)))/(a + b*exp(h + i*x) + c*exp(2*h + 2*i*x)),x)

[Out]

int(((f + g*x)^3*(d + e*exp(h + i*x)))/(a + b*exp(h + i*x) + c*exp(2*h + 2*i*x)), x)