\(\int \frac {(d+e e^{h+i x}) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx\) [573]

Optimal result

Integrand size = 44, antiderivative size = 599 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3}{3 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3}{3 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^2 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^2 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^3}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3} \]

[Out]

1/3*(g*x+f)^3*(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)^2*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))-2*g*(g*x+f)*polylog(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))+2*g^2*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))+1/3*(g
*x+f)^3*(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)^2*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))-2*g*(g*x+f)*polylog(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))+2*g^2*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))

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2297, 2215, 2221, 2611, 2320, 6724} \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=-\frac {2 g (f+g x) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2 \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 g (f+g x) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^2 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {(f+g x)^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{i \left (b-\sqrt {b^2-4 a c}\right )}-\frac {(f+g x)^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{i \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{3 g \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(f+g x)^3 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{3 g \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 g^2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^3 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 g^2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^3 \left (\sqrt {b^2-4 a c}+b\right )} \]

[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 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]

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)^2}{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)^2}{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)^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+i x} (f+g x)^2}{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)^2}{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)^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+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)^2 \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 (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+i x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b+\sqrt {b^2-4 a c}\right ) i}+\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+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)^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+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)^2 \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 {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+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\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+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\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+i x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\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+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)^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+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)^2 \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 {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+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\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+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {\left (2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2\right ) \text {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+i x}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3}+\frac {\left (2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2\right ) \text {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+i x}\right )}{\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)^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+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)^2 \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 {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+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\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+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\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+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^3}+\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+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1419\) vs. \(2(599)=1198\).

Time = 1.46 (sec) , antiderivative size = 1419, normalized size of antiderivative = 2.37 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=-\frac {-6 \sqrt {-\left (b^2-4 a c\right )^2} d f g i^3 x^2-2 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 i^3 x^3+6 b \sqrt {b^2-4 a c} d f^2 i^2 \arctan \left (\frac {b+2 c e^{h+i x}}{\sqrt {-b^2+4 a c}}\right )-12 a \sqrt {b^2-4 a c} e f^2 i^2 \arctan \left (\frac {b+2 c e^{h+i x}}{\sqrt {-b^2+4 a c}}\right )-6 \sqrt {-\left (b^2-4 a c\right )^2} d f^2 i^2 \log \left (e^{h+i x}\right )+6 \sqrt {-\left (b^2-4 a c\right )^2} d f g i^2 x \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )+6 b \sqrt {-b^2+4 a c} d f g i^2 x \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )-12 a \sqrt {-b^2+4 a c} e f g i^2 x \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )+3 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 i^2 x^2 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )+3 b \sqrt {-b^2+4 a c} d g^2 i^2 x^2 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )-6 a \sqrt {-b^2+4 a c} e g^2 i^2 x^2 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )+6 \sqrt {-\left (b^2-4 a c\right )^2} d f g i^2 x \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )-6 b \sqrt {-b^2+4 a c} d f g i^2 x \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )+12 a \sqrt {-b^2+4 a c} e f g i^2 x \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )+3 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 i^2 x^2 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )-3 b \sqrt {-b^2+4 a c} d g^2 i^2 x^2 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )+6 a \sqrt {-b^2+4 a c} e g^2 i^2 x^2 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )+3 \sqrt {-\left (b^2-4 a c\right )^2} d f^2 i^2 \log \left (a+e^{h+i x} \left (b+c e^{h+i x}\right )\right )+6 \left (\sqrt {-\left (b^2-4 a c\right )^2} d+b \sqrt {-b^2+4 a c} d-2 a \sqrt {-b^2+4 a c} e\right ) g i (f+g x) \operatorname {PolyLog}\left (2,\frac {2 c e^{h+i x}}{-b+\sqrt {b^2-4 a c}}\right )+6 \left (\sqrt {-\left (b^2-4 a c\right )^2} d-b \sqrt {-b^2+4 a c} d+2 a \sqrt {-b^2+4 a c} e\right ) g i (f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )-6 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 \operatorname {PolyLog}\left (3,\frac {2 c e^{h+i x}}{-b+\sqrt {b^2-4 a c}}\right )-6 b \sqrt {-b^2+4 a c} d g^2 \operatorname {PolyLog}\left (3,\frac {2 c e^{h+i x}}{-b+\sqrt {b^2-4 a c}}\right )+12 a \sqrt {-b^2+4 a c} e g^2 \operatorname {PolyLog}\left (3,\frac {2 c e^{h+i x}}{-b+\sqrt {b^2-4 a c}}\right )-6 \sqrt {-\left (b^2-4 a c\right )^2} d g^2 \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )+6 b \sqrt {-b^2+4 a c} d g^2 \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )-12 a \sqrt {-b^2+4 a c} e g^2 \operatorname {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} \]

[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]

-1/6*(-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*ArcTan[(b + 2*c
*E^(h + i*x))/Sqrt[-b^2 + 4*a*c]] - 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f^2*i^2*Log[E^(h + i*x)] + 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 - 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 + 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])] + 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*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*PolyLo
g[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])])/(a*Sqrt[-(b^2 - 4*a
*c)^2]*i^3)

Maple [F]

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

[In]

int((d+e*exp(i*x+h))*(g*x+f)^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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (543) = 1086\).

Time = 0.35 (sec) , antiderivative size = 1193, normalized size of antiderivative = 1.99 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Too large to display} \]

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)^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)

Sympy [F]

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

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)**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)

Maxima [F(-2)]

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

[In]

integrate((d+e*exp(i*x+h))*(g*x+f)^2/(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]

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

[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)

Mupad [F(-1)]

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

[In]

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