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

Optimal. Leaf size=770 \[ \frac{6 g^2 (f+g x) \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^3 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{6 g^2 (f+g x) \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^3 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{3 g (f+g x)^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^2 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 g (f+g x)^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{6 g^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (4,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^4 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{6 g^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (4,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^4 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(f+g x)^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )}{i \left (b-\sqrt{b^2-4 a c}\right )}-\frac{(f+g x)^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )}{i \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^4 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )}{4 g \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^4 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right )}{4 g \left (b-\sqrt{b^2-4 a c}\right )} \]

[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*Log[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)

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Rubi [A]  time = 2.4576, antiderivative size = 770, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159 \[ \frac{6 g^2 (f+g x) \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^3 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{6 g^2 (f+g x) \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (3,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^3 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{3 g (f+g x)^2 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^2 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{3 g (f+g x)^2 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{6 g^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \text{PolyLog}\left (4,-\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}\right )}{i^4 \left (b-\sqrt{b^2-4 a c}\right )}-\frac{6 g^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (4,-\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}\right )}{i^4 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(f+g x)^3 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\frac{2 c e^{h+i x}}{b-\sqrt{b^2-4 a c}}+1\right )}{i \left (b-\sqrt{b^2-4 a c}\right )}-\frac{(f+g x)^3 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c e^{h+i x}}{\sqrt{b^2-4 a c}+b}+1\right )}{i \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^4 \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right )}{4 g \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f+g x)^4 \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right )}{4 g \left (b-\sqrt{b^2-4 a c}\right )} \]

Antiderivative was successfully verified.

[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*Log[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)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

Timed out

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Mathematica [B]  time = 6.47288, size = 2441, normalized size = 3.17 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[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]

-(-4*Sqrt[-(b^2 - 4*a*c)^2]*d*f^3*i^4*x - 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*Sqrt[b^2 - 4*a*c]*e*f^3*i^3*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*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*Sqr
t[-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 + 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*Lo
g[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])] + 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*Po
lyLog[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*PolyLo
g[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 + Sqr
t[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*Pol
yLog[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*S
qrt[-(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 + Sq
rt[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*Pol
yLog[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*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])])/(4*a
*Sqrt[-(b^2 - 4*a*c)^2]*i^4)

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Maple [F]  time = 0.17, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d+e{{\rm e}^{ix+h}} \right ) \left ( gx+f \right ) ^{3}}{a+b{{\rm e}^{ix+h}}+c{{\rm e}^{2\,ix+2\,h}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.380238, size = 2531, normalized size = 3.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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(-(2*c*e^(i*x + h) + a*sqrt((b^2 - 4*a*c)/a^2) +
b)/(a*sqrt((b^2 - 4*a*c)/a^2) + b) + 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((2*c*e^(i*x + h) - a*sqrt((b^2 - 4*a*c)/a^2) + b)/(a*sqrt
((b^2 - 4*a*c)/a^2) - b) + 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)*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)/(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)*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)/(a*sqrt((b^2 - 4*a*c)/a^2) - b)) -
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))*polylo
g(4, -2*c*e^(i*x + h)/(a*sqrt((b^2 - 4*a*c)/a^2) + b)) - 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, 2*c*e^(i*x + h)/(a*
sqrt((b^2 - 4*a*c)/a^2) - b)) + 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, -2*c*e^(i*x + h)/(a*sqrt((b^2 - 4*a*c)/a^2) + b)) + 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, 2*c*e^(i*x + h)/(a*s
qrt((b^2 - 4*a*c)/a^2) - b)))/((a*b^2 - 4*a^2*c)*i^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x + f\right )}^{3}{\left (e e^{\left (i x + h\right )} + d\right )}}{c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)