\(\int \frac {(d+e e^{h+i x}) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx\) [495]

Optimal result

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

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

Mathematica [B] (verified)

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

Time = 2.73 (sec) , antiderivative size = 1419, normalized size of antiderivative = 2.37 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx =\text {Too large to display} \] Input:

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]
 

Output:

-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 - Sqr 
t[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*Lo 
g[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...
 

Rubi [A] (verified)

Time = 2.72 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.70, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2695, 2615, 2620, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2695

\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \int \frac {(f+g x)^2}{b+2 c e^{h+i x}-\sqrt {b^2-4 a c}}dx+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {(f+g x)^2}{b+2 c e^{h+i x}+\sqrt {b^2-4 a c}}dx\)

\(\Big \downarrow \) 2615

\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^3}{3 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \int \frac {e^{h+i x} (f+g x)^2}{b+2 c e^{h+i x}-\sqrt {b^2-4 a c}}dx}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^3}{3 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \int \frac {e^{h+i x} (f+g x)^2}{b+2 c e^{h+i x}+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}+b}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^3}{3 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^2 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {g \int (f+g x) \log \left (\frac {2 e^{h+i x} c}{b-\sqrt {b^2-4 a c}}+1\right )dx}{c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^3}{3 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^2 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {g \int (f+g x) \log \left (\frac {2 e^{h+i x} c}{b+\sqrt {b^2-4 a c}}+1\right )dx}{c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^3}{3 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^2 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {g \left (\frac {g \int \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )dx}{i}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}\right )}{c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^3}{3 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^2 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {g \left (\frac {g \int \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )dx}{i}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}\right )}{c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^3}{3 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^2 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {g \left (\frac {g \int e^{-h-i x} \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )de^{h+i x}}{i^2}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}\right )}{c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^3}{3 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^2 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {g \left (\frac {g \int e^{-h-i x} \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )de^{h+i x}}{i^2}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}\right )}{c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^3}{3 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^2 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {g \left (\frac {g \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}\right )}{c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^3}{3 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^2 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {g \left (\frac {g \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^2}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}\right )}{c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\)

Input:

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]
 

Output:

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

Defintions of rubi rules used

rule 2615

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] - Simp[ 
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 2620

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] - Si 
mp[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 2695

Int[(((i_.)*(F_)^(u_) + (h_))*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F 
_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(Simplify[(2*c*h - b*i)/q] + i)   Int[(f + g*x)^m/(b - q + 2*c*F^u), x], 
x] - Simp[(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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [F]

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

Input:

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)
 

Output:

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 = 0.12 (sec) , antiderivative size = 1193, normalized size of antiderivative = 1.99 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Too large to display} \] Input:

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")
 

Output:

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...
 

Sympy [F]

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

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)
 

Output:

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

Maxima [F(-2)]

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

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")
 

Output:

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

Giac [F]

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

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")
 

Output:

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

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)
 

Output:

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)
 

Reduce [F]

\[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^2}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\frac {-4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 e^{i x +h} c +b}{\sqrt {4 a c -b^{2}}}\right ) a e \,f^{2} i +2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 e^{i x +h} c +b}{\sqrt {4 a c -b^{2}}}\right ) b d \,f^{2} i +8 e^{h} \left (\int \frac {e^{i x} x^{2}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c e \,g^{2}-2 e^{h} \left (\int \frac {e^{i x} x^{2}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} e \,g^{2}+16 e^{h} \left (\int \frac {e^{i x} x}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c e f g -4 e^{h} \left (\int \frac {e^{i x} x}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} e f g +8 \left (\int \frac {x^{2}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c d \,g^{2}-2 \left (\int \frac {x^{2}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} d \,g^{2}+16 \left (\int \frac {x}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c d f g -4 \left (\int \frac {x}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} d f g +4 \,\mathrm {log}\left (e^{2 i x +2 h} c +e^{i x +h} b +a \right ) a c d \,f^{2} i -\mathrm {log}\left (e^{2 i x +2 h} c +e^{i x +h} b +a \right ) b^{2} d \,f^{2} i +8 a c d \,f^{2} x -2 b^{2} d \,f^{2} x}{2 a \left (4 a c -b^{2}\right )} \] Input:

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)
 

Output:

( - 4*sqrt(4*a*c - b**2)*atan((2*e**(h + i*x)*c + b)/sqrt(4*a*c - b**2))*a 
*e*f**2*i + 2*sqrt(4*a*c - b**2)*atan((2*e**(h + i*x)*c + b)/sqrt(4*a*c - 
b**2))*b*d*f**2*i + 8*e**h*int((e**(i*x)*x**2)/(e**(2*h + 2*i*x)*c + e**(h 
 + i*x)*b + a),x)*a**2*c*e*g**2 - 2*e**h*int((e**(i*x)*x**2)/(e**(2*h + 2* 
i*x)*c + e**(h + i*x)*b + a),x)*a*b**2*e*g**2 + 16*e**h*int((e**(i*x)*x)/( 
e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a**2*c*e*f*g - 4*e**h*int((e** 
(i*x)*x)/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a*b**2*e*f*g + 8*int 
(x**2/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a**2*c*d*g**2 - 2*int(x 
**2/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a*b**2*d*g**2 + 16*int(x/ 
(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a**2*c*d*f*g - 4*int(x/(e**(2 
*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a*b**2*d*f*g + 4*log(e**(2*h + 2*i* 
x)*c + e**(h + i*x)*b + a)*a*c*d*f**2*i - log(e**(2*h + 2*i*x)*c + e**(h + 
 i*x)*b + a)*b**2*d*f**2*i + 8*a*c*d*f**2*x - 2*b**2*d*f**2*x)/(2*a*(4*a*c 
 - b**2))