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

Optimal. Leaf size=599 \[ -\frac{2 g (f+g x) \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^2 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 g (f+g x) \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{2 g^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^3 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{2 g^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^3 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(f+g x)^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )}{i \left (b-\sqrt{b^2-4 a c}\right )}-\frac{(f+g x)^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )}{i \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )}{3 g \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right )}{3 g \left (b-\sqrt{b^2-4 a c}\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.00222, antiderivative size = 599, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2265, 2184, 2190, 2531, 2282, 6589} \[ -\frac{2 g (f+g x) \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^2 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 g (f+g x) \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{2 g^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^3 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{2 g^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^3 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(f+g x)^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )}{i \left (b-\sqrt{b^2-4 a c}\right )}-\frac{(f+g x)^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )}{i \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )}{3 g \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right )}{3 g \left (b-\sqrt{b^2-4 a c}\right )} \]

Antiderivative was successfully verified.

[In]

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

Rule 2265

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 2184

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2531

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 2282

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 6589

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]

Rubi steps

\begin{align*} \int \frac{\left (d+e e^{h+573 x}\right ) (f+g x)^2}{a+b e^{h+573 x}+c e^{2 h+1146 x}} \, dx &=-\left (\left (-e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{(f+g x)^2}{b+\sqrt{b^2-4 a c}+2 c e^{h+573 x}} \, dx\right )+\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{(f+g x)^2}{b-\sqrt{b^2-4 a c}+2 c e^{h+573 x}} \, dx\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3}{3 \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}{3 \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+573 x} (f+g x)^2}{b+\sqrt{b^2-4 a c}+2 c e^{h+573 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+573 x} (f+g x)^2}{b-\sqrt{b^2-4 a c}+2 c e^{h+573 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)^3}{3 \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}{3 \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)^2 \log \left (1+\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right )}{573 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2 \log \left (1+\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right )}{573 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g\right ) \int (f+g x) \log \left (1+\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right ) \, dx}{573 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g\right ) \int (f+g x) \log \left (1+\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right ) \, dx}{573 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3}{3 \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}{3 \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)^2 \log \left (1+\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right )}{573 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2 \log \left (1+\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right )}{573 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x) \text{Li}_2\left (-\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right )}{328329 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x) \text{Li}_2\left (-\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right )}{328329 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2\right ) \int \text{Li}_2\left (-\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right ) \, dx}{328329 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2\right ) \int \text{Li}_2\left (-\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right ) \, dx}{328329 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3}{3 \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}{3 \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)^2 \log \left (1+\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right )}{573 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2 \log \left (1+\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right )}{573 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x) \text{Li}_2\left (-\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right )}{328329 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x) \text{Li}_2\left (-\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right )}{328329 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{x} \, dx,x,e^{h+573 x}\right )}{188132517 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{\left (2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{2 c x}{-b+\sqrt{b^2-4 a c}}\right )}{x} \, dx,x,e^{h+573 x}\right )}{188132517 \left (b-\sqrt{b^2-4 a c}\right )}\\ &=\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^3}{3 \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}{3 \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)^2 \log \left (1+\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right )}{573 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f+g x)^2 \log \left (1+\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right )}{573 \left (b+\sqrt{b^2-4 a c}\right )}-\frac{2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x) \text{Li}_2\left (-\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right )}{328329 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g (f+g x) \text{Li}_2\left (-\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right )}{328329 \left (b+\sqrt{b^2-4 a c}\right )}+\frac{2 \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2 \text{Li}_3\left (-\frac{2 c e^{h+573 x}}{b-\sqrt{b^2-4 a c}}\right )}{188132517 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) g^2 \text{Li}_3\left (-\frac{2 c e^{h+573 x}}{b+\sqrt{b^2-4 a c}}\right )}{188132517 \left (b+\sqrt{b^2-4 a c}\right )}\\ \end{align*}

Mathematica [B]  time = 2.52292, size = 1412, normalized size = 2.36 \[ -\frac{-2 \sqrt{-\left (b^2-4 a c\right )^2} d g^2 x^3 i^3-6 \sqrt{-\left (b^2-4 a c\right )^2} d f g x^2 i^3-6 \sqrt{-\left (b^2-4 a c\right )^2} d f^2 x i^3+6 b \sqrt{b^2-4 a c} d f^2 \tan ^{-1}\left (\frac{b+2 c e^{h+i x}}{\sqrt{4 a c-b^2}}\right ) i^2+12 a \sqrt{4 a c-b^2} e f^2 \tanh ^{-1}\left (\frac{b+2 c e^{h+i x}}{\sqrt{b^2-4 a c}}\right ) i^2+3 \sqrt{-\left (b^2-4 a c\right )^2} d g^2 x^2 \log \left (\frac{2 e^{h+i x} c}{b-\sqrt{b^2-4 a c}}+1\right ) i^2+3 b \sqrt{4 a c-b^2} d g^2 x^2 \log \left (\frac{2 e^{h+i x} c}{b-\sqrt{b^2-4 a c}}+1\right ) i^2-6 a \sqrt{4 a c-b^2} e g^2 x^2 \log \left (\frac{2 e^{h+i x} c}{b-\sqrt{b^2-4 a c}}+1\right ) i^2+6 \sqrt{-\left (b^2-4 a c\right )^2} d f g x \log \left (\frac{2 e^{h+i x} c}{b-\sqrt{b^2-4 a c}}+1\right ) i^2+6 b \sqrt{4 a c-b^2} d f g x \log \left (\frac{2 e^{h+i x} c}{b-\sqrt{b^2-4 a c}}+1\right ) i^2-12 a \sqrt{4 a c-b^2} e f g x \log \left (\frac{2 e^{h+i x} c}{b-\sqrt{b^2-4 a c}}+1\right ) i^2+3 \sqrt{-\left (b^2-4 a c\right )^2} d g^2 x^2 \log \left (\frac{2 e^{h+i x} c}{b+\sqrt{b^2-4 a c}}+1\right ) i^2-3 b \sqrt{4 a c-b^2} d g^2 x^2 \log \left (\frac{2 e^{h+i x} c}{b+\sqrt{b^2-4 a c}}+1\right ) i^2+6 a \sqrt{4 a c-b^2} e g^2 x^2 \log \left (\frac{2 e^{h+i x} c}{b+\sqrt{b^2-4 a c}}+1\right ) i^2+6 \sqrt{-\left (b^2-4 a c\right )^2} d f g x \log \left (\frac{2 e^{h+i x} c}{b+\sqrt{b^2-4 a c}}+1\right ) i^2-6 b \sqrt{4 a c-b^2} d f g x \log \left (\frac{2 e^{h+i x} c}{b+\sqrt{b^2-4 a c}}+1\right ) i^2+12 a \sqrt{4 a c-b^2} e f g x \log \left (\frac{2 e^{h+i x} c}{b+\sqrt{b^2-4 a c}}+1\right ) i^2+3 \sqrt{-\left (b^2-4 a c\right )^2} d f^2 \log \left (a+e^{h+i x} \left (b+c e^{h+i x}\right )\right ) i^2+6 \left (\sqrt{-\left (b^2-4 a c\right )^2} d+b \sqrt{4 a c-b^2} d-2 a \sqrt{4 a c-b^2} e\right ) g (f+g x) \text{PolyLog}\left (2,\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}-b}\right ) i+6 \left (\sqrt{-\left (b^2-4 a c\right )^2} d-b \sqrt{4 a c-b^2} d+2 a \sqrt{4 a c-b^2} e\right ) g (f+g x) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b+\sqrt{b^2-4 a c}}\right ) i-6 \sqrt{-\left (b^2-4 a c\right )^2} d g^2 \text{PolyLog}\left (3,\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}-b}\right )-6 b \sqrt{4 a c-b^2} d g^2 \text{PolyLog}\left (3,\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}-b}\right )+12 a \sqrt{4 a c-b^2} e g^2 \text{PolyLog}\left (3,\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}-b}\right )-6 \sqrt{-\left (b^2-4 a c\right )^2} d g^2 \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b+\sqrt{b^2-4 a c}}\right )+6 b \sqrt{4 a c-b^2} d g^2 \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b+\sqrt{b^2-4 a c}}\right )-12 a \sqrt{4 a c-b^2} e g^2 \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b+\sqrt{b^2-4 a c}}\right )}{6 a \sqrt{-\left (b^2-4 a c\right )^2} i^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-6*Sqrt[-(b^2 - 4*a*c)^2]*d*f^2*i^3*x - 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f*g*i^3*x^2 - 2*Sqrt[-(b^2 - 4*a*c)^2]*d*
g^2*i^3*x^3 + 6*b*Sqrt[b^2 - 4*a*c]*d*f^2*i^2*ArcTan[(b + 2*c*E^(h + i*x))/Sqrt[-b^2 + 4*a*c]] + 12*a*Sqrt[-b^
2 + 4*a*c]*e*f^2*i^2*ArcTanh[(b + 2*c*E^(h + i*x))/Sqrt[b^2 - 4*a*c]] + 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f*g*i^2*x*L
og[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*i^2*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*g*i^2*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*
a*c])] + 3*Sqrt[-(b^2 - 4*a*c)^2]*d*g^2*i^2*x^2*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 3*b*Sqrt[
-b^2 + 4*a*c]*d*g^2*i^2*x^2*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] - 6*a*Sqrt[-b^2 + 4*a*c]*e*g^2*
i^2*x^2*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*i^2*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*g*i^2*x*Log[1 + (2*c*E^(h + i*x))/(b + Sqr
t[b^2 - 4*a*c])] + 12*a*Sqrt[-b^2 + 4*a*c]*e*f*g*i^2*x*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 3*
Sqrt[-(b^2 - 4*a*c)^2]*d*g^2*i^2*x^2*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 3*b*Sqrt[-b^2 + 4*a*
c]*d*g^2*i^2*x^2*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 6*a*Sqrt[-b^2 + 4*a*c]*e*g^2*i^2*x^2*Log
[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 3*Sqrt[-(b^2 - 4*a*c)^2]*d*f^2*i^2*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*(f + g*
x)*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*(f + g*x)*PolyLog[2, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 6*Sqrt[-
(b^2 - 4*a*c)^2]*d*g^2*PolyLog[3, (2*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a*c])] - 6*b*Sqrt[-b^2 + 4*a*c]*d*g^2*P
olyLog[3, (2*c*E^(h + i*x))/(-b + Sqrt[b^2 - 4*a*c])] + 12*a*Sqrt[-b^2 + 4*a*c]*e*g^2*PolyLog[3, (2*c*E^(h + i
*x))/(-b + Sqrt[b^2 - 4*a*c])] - 6*Sqrt[-(b^2 - 4*a*c)^2]*d*g^2*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 -
4*a*c])] + 6*b*Sqrt[-b^2 + 4*a*c]*d*g^2*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] - 12*a*Sqrt[-b^
2 + 4*a*c]*e*g^2*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/(6*a*Sqrt[-(b^2 - 4*a*c)^2]*i^3)

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Maple [F]  time = 0.208, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{{\rm e}^{ix+h}} \right ) \left ( gx+f \right ) ^{2}}{a+b{{\rm e}^{ix+h}}+c{{\rm e}^{2\,ix+2\,h}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 1.5907, size = 2732, normalized size = 4.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e e^{h} e^{i x}\right ) \left (f + g x\right )^{2}}{a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{2}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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