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

Optimal. Leaf size=599 \[ \frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3}{3 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3}{3 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^2 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^2 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x) \text {Li}_2\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^3}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 \text {Li}_3\left (-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3} \]

[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))

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Rubi [A]
time = 0.69, antiderivative size = 599, normalized size of antiderivative = 1.00, 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 = {2297, 2215, 2221, 2611, 2320, 6724} \begin {gather*} -\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 )} \end {gather*}

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 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*} \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 ) \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+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 ) \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+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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1419\) vs. \(2(599)=1198\).
time = 1.56, size = 1419, normalized size = 2.37 \begin {gather*} -\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 \tan ^{-1}\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 \tan ^{-1}\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) \text {Li}_2\left (\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) \text {Li}_2\left (-\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 \text {Li}_3\left (\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 \text {Li}_3\left (\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 \text {Li}_3\left (\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\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} \end {gather*}

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]

-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)

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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (d +e \,{\mathrm e}^{i x +h}\right ) \left (g x +f \right )^{2}}{a +b \,{\mathrm e}^{i x +h}+c \,{\mathrm e}^{2 i x +2 h}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> 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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1834 vs. \(2 (545) = 1090\).
time = 0.47, size = 1834, normalized size = 3.06 \begin {gather*} \text {Too large to display} \end {gather*}

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*(I*b^2 - 4*I*a*c)*d*g^2*h^3 + 2*(b^2 - 4*a*c)*d*g^2*x^3 + 6*(b^2 - 4*a*c)*d*f*g*x^2 + 6*(b^2 - 4*a*c)*d
*f^2*x + 6*(-I*b^2 + 4*I*a*c)*d*f^2 + 6*(b^2 - 4*a*c)*d*f*g + 2*(I*b^2 - 4*I*a*c)*d*g^2 + 6*((b^2 - 4*a*c)*d*f
*g + (I*b^2 - 4*I*a*c)*d*g^2)*h^2 + 6*((-I*b^2 + 4*I*a*c)*d*f^2 + 2*(b^2 - 4*a*c)*d*f*g + (I*b^2 - 4*I*a*c)*d*
g^2)*h + 6*((b^2 - 4*a*c)*d*g^2*x + (b^2 - 4*a*c)*d*f*g - (2*(a^2*g^2*x + a^2*f*g)*e^2 - (a*b*d*g^2*x + a*b*d*
f*g)*e)*sqrt((b^2 - 4*a*c)/a^2)*e^(-1))*dilog(-1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(h + I*x + 1) + 2*a*e + b*e^(h
+ I*x + 1))*e^(-1)/a + 1) + 6*((b^2 - 4*a*c)*d*g^2*x + (b^2 - 4*a*c)*d*f*g + (2*(a^2*g^2*x + a^2*f*g)*e^2 - (
a*b*d*g^2*x + a*b*d*f*g)*e)*sqrt((b^2 - 4*a*c)/a^2)*e^(-1))*dilog(1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(h + I*x +
1) - 2*a*e - b*e^(h + I*x + 1))*e^(-1)/a + 1) + 3*((I*b^2 - 4*I*a*c)*d*g^2*h^2 + (I*b^2 - 4*I*a*c)*d*g^2*x^2 +
2*(I*b^2 - 4*I*a*c)*d*f*g*x + 2*(b^2 - 4*a*c)*d*f*g + (I*b^2 - 4*I*a*c)*d*g^2 + (2*(-I*a^2*g^2*h^2 - I*a^2*g^
2*x^2 - 2*I*a^2*f*g*x - 2*a^2*f*g - I*a^2*g^2 - 2*(a^2*f*g + I*a^2*g^2)*h)*e^2 + (I*a*b*d*g^2*h^2 + I*a*b*d*g^
2*x^2 + 2*I*a*b*d*f*g*x + 2*a*b*d*f*g + I*a*b*d*g^2 + 2*(a*b*d*f*g + I*a*b*d*g^2)*h)*e)*sqrt((b^2 - 4*a*c)/a^2
)*e^(-1) + 2*((b^2 - 4*a*c)*d*f*g + (I*b^2 - 4*I*a*c)*d*g^2)*h)*log(1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(h + I*x
+ 1) + 2*a*e + b*e^(h + I*x + 1))*e^(-1)/a) + 3*((I*b^2 - 4*I*a*c)*d*g^2*h^2 + (I*b^2 - 4*I*a*c)*d*g^2*x^2 + 2
*(I*b^2 - 4*I*a*c)*d*f*g*x + 2*(b^2 - 4*a*c)*d*f*g + (I*b^2 - 4*I*a*c)*d*g^2 + (2*(I*a^2*g^2*h^2 + I*a^2*g^2*x
^2 + 2*I*a^2*f*g*x + 2*a^2*f*g + I*a^2*g^2 + 2*(a^2*f*g + I*a^2*g^2)*h)*e^2 + (-I*a*b*d*g^2*h^2 - I*a*b*d*g^2*
x^2 - 2*I*a*b*d*f*g*x - 2*a*b*d*f*g - I*a*b*d*g^2 - 2*(a*b*d*f*g + I*a*b*d*g^2)*h)*e)*sqrt((b^2 - 4*a*c)/a^2)*
e^(-1) + 2*((b^2 - 4*a*c)*d*f*g + (I*b^2 - 4*I*a*c)*d*g^2)*h)*log(-1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(h + I*x +
1) - 2*a*e - b*e^(h + I*x + 1))*e^(-1)/a) + 3*((-I*b^2 + 4*I*a*c)*d*g^2*h^2 + (I*b^2 - 4*I*a*c)*d*f^2 - 2*(b^
2 - 4*a*c)*d*f*g + (-I*b^2 + 4*I*a*c)*d*g^2 + (2*(-I*a^2*g^2*h^2 + I*a^2*f^2 - 2*a^2*f*g - I*a^2*g^2 - 2*(a^2*
f*g + I*a^2*g^2)*h)*e^2 + (I*a*b*d*g^2*h^2 - I*a*b*d*f^2 + 2*a*b*d*f*g + I*a*b*d*g^2 + 2*(a*b*d*f*g + I*a*b*d*
g^2)*h)*e)*sqrt((b^2 - 4*a*c)/a^2)*e^(-1) - 2*((b^2 - 4*a*c)*d*f*g - (-I*b^2 + 4*I*a*c)*d*g^2)*h)*log(1/2*(a*s
qrt((b^2 - 4*a*c)/a^2)*e + b*e + 2*c*e^(h + I*x + 1))/c) + 3*((-I*b^2 + 4*I*a*c)*d*g^2*h^2 + (I*b^2 - 4*I*a*c)
*d*f^2 - 2*(b^2 - 4*a*c)*d*f*g + (-I*b^2 + 4*I*a*c)*d*g^2 + (2*(I*a^2*g^2*h^2 - I*a^2*f^2 + 2*a^2*f*g + I*a^2*
g^2 + 2*(a^2*f*g + I*a^2*g^2)*h)*e^2 + (-I*a*b*d*g^2*h^2 + I*a*b*d*f^2 - 2*a*b*d*f*g - I*a*b*d*g^2 - 2*(a*b*d*
f*g + I*a*b*d*g^2)*h)*e)*sqrt((b^2 - 4*a*c)/a^2)*e^(-1) - 2*((b^2 - 4*a*c)*d*f*g - (-I*b^2 + 4*I*a*c)*d*g^2)*h
)*log(-1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e - b*e - 2*c*e^(h + I*x + 1))/c) + 6*((I*b^2 - 4*I*a*c)*d*g^2 + (I*a*b*
d*g^2*e - 2*I*a^2*g^2*e^2)*sqrt((b^2 - 4*a*c)/a^2)*e^(-1))*polylog(3, -1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(h + I
*x + 1) + b*e^(h + I*x + 1))*e^(-1)/a) + 6*((I*b^2 - 4*I*a*c)*d*g^2 + (-I*a*b*d*g^2*e + 2*I*a^2*g^2*e^2)*sqrt(
(b^2 - 4*a*c)/a^2)*e^(-1))*polylog(3, 1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(h + I*x + 1) - b*e^(h + I*x + 1))*e^(-
1)/a))/(a*b^2 - 4*a^2*c)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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*(d + e^(h + I*x + 1))/(c*e^(2*h + 2*I*x) + b*e^(h + I*x) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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