3.472 \(\int \frac{e^{d+e x} x^3}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=232 \[ \frac{\left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c^3}+\frac{\left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c^3}-\frac{b e^{d+e x}}{c^2 e}-\frac{e^{d+e x}}{c e^2}+\frac{x e^{d+e x}}{c e} \]

[Out]

-(E^(d + e*x)/(c*e^2)) - (b*E^(d + e*x))/(c^2*e) + (E^(d + e*x)*x)/(c*e) + ((b^2
- a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)
/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^3) + ((b^
2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*E^(d - ((b + Sqrt[b^2 - 4*a*c])*e
)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^3)

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Rubi [A]  time = 0.88973, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c^3}+\frac{\left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c^3}-\frac{b e^{d+e x}}{c^2 e}-\frac{e^{d+e x}}{c e^2}+\frac{x e^{d+e x}}{c e} \]

Antiderivative was successfully verified.

[In]  Int[(E^(d + e*x)*x^3)/(a + b*x + c*x^2),x]

[Out]

-(E^(d + e*x)/(c*e^2)) - (b*E^(d + e*x))/(c^2*e) + (E^(d + e*x)*x)/(c*e) + ((b^2
- a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)
/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^3) + ((b^
2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*E^(d - ((b + Sqrt[b^2 - 4*a*c])*e
)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^3)

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Rubi in Sympy [A]  time = 70.8033, size = 231, normalized size = 1. \[ - \frac{b e^{d + e x}}{c^{2} e} + \frac{x e^{d + e x}}{c e} - \frac{e^{d + e x}}{c e^{2}} - \frac{\left (b \left (- 3 a c + b^{2}\right ) - \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ) e^{\frac{- b e + 2 c d + e \sqrt{- 4 a c + b^{2}}}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x - \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 c^{3} \sqrt{- 4 a c + b^{2}}} + \frac{\left (b \left (- 3 a c + b^{2}\right ) + \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ) e^{\frac{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 c^{3} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e*x+d)*x**3/(c*x**2+b*x+a),x)

[Out]

-b*exp(d + e*x)/(c**2*e) + x*exp(d + e*x)/(c*e) - exp(d + e*x)/(c*e**2) - (b*(-3
*a*c + b**2) - sqrt(-4*a*c + b**2)*(-a*c + b**2))*exp((-b*e + 2*c*d + e*sqrt(-4*
a*c + b**2))/(2*c))*Ei(e*(b/2 + c*x - sqrt(-4*a*c + b**2)/2)/c)/(2*c**3*sqrt(-4*
a*c + b**2)) + (b*(-3*a*c + b**2) + sqrt(-4*a*c + b**2)*(-a*c + b**2))*exp((2*c*
d - e*(b + sqrt(-4*a*c + b**2)))/(2*c))*Ei(e*(b/2 + c*x + sqrt(-4*a*c + b**2)/2)
/c)/(2*c**3*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.608737, size = 268, normalized size = 1.16 \[ \frac{e^{d-\frac{b e}{c}} \left (-2 c \sqrt{b^2-4 a c} e^{e \left (\frac{b}{c}+x\right )} (b e+c (-e) x+c)+e^2 \left (b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}+3 a b c-b^3\right ) e^{\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )+e^2 \left (b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right ) e^{\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )\right )}{2 c^3 e^2 \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Integrate[(E^(d + e*x)*x^3)/(a + b*x + c*x^2),x]

[Out]

(E^(d - (b*e)/c)*(-2*c*Sqrt[b^2 - 4*a*c]*E^(e*(b/c + x))*(c + b*e - c*e*x) + (-b
^3 + 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*e^2*E^(((b + Sqrt[
b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)]
+ (b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*e^2*E^(((b -
Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(
2*c)]))/(2*c^3*Sqrt[b^2 - 4*a*c]*e^2)

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Maple [B]  time = 0.031, size = 3532, normalized size = 15.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e*x+d)*x^3/(c*x^2+b*x+a),x)

[Out]

1/e^4*(-e^2*exp(e*x+d)*(-(e*x+d)*c+b*e-2*c*d+c)/c^2+1/2/c^3*e^2*(-3*exp(1/2/c*(-
b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*
e^2+b^2*e^2)^(1/2))/c)*a*b*c*e^3+6*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1
/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*c^2*d*e^
2+exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+
2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b^3*e^3-3*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2
+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c
)*b^2*c*d*e^2+3*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*
(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b*c^2*d^2*e-2*exp(1/2/c*(-b*e
+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2
+b^2*e^2)^(1/2))/c)*c^3*d^3+3*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)
*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*b*c*e^3-6*exp
(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d
+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*c^2*d*e^2-exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*
e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b
^3*e^3+3*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)
*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b^2*c*d*e^2-3*exp(-1/2*(b*e-2*c*d+(-
4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^
2)^(1/2))/c)*b*c^2*d^2*e+2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei
(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c^3*d^3+exp(1/2/c*
(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*
c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*a*c*e^2-exp(1/2/c*(-b*e+2*c*
d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*
e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*b^2*e^2+3*exp(1/2/c*(-b*e+2*c*d+(-4*a*
c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/
2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*c*d*e-3*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^
2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(
-4*a*c*e^2+b^2*e^2)^(1/2)*c^2*d^2+exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)
)/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2
+b^2*e^2)^(1/2)*a*c*e^2-exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,
-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^
(1/2)*b^2*e^2+3*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*
(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*
c*d*e-3*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*
c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*c^2*d^2)/(
-4*a*c*e^2+b^2*e^2)^(1/2)+d^3*e^2*(exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1
/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)-exp(-1/2*(
b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*
c*e^2+b^2*e^2)^(1/2))/c))/(-4*a*c*e^2+b^2*e^2)^(1/2)-3/2*d^2*e^2*(-exp(1/2/c*(-b
*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e
^2+b^2*e^2)^(1/2))/c)*b*e+2*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*E
i(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c*d+exp(-1/2*(b*e
-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e
^2+b^2*e^2)^(1/2))/c)*b*e-2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*E
i(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c*d+exp(1/2/c*(-b
*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e
^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)+exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2
+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))
/c)*(-4*a*c*e^2+b^2*e^2)^(1/2))/c/(-4*a*c*e^2+b^2*e^2)^(1/2)-3*d*(e^2/c*exp(e*x+
d)+1/2/c^2*e^2*(2*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-
2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*c*e^2-exp(1/2/c*(-b*e+2*c
*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2
*e^2)^(1/2))/c)*b^2*e^2+2*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(
1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b*c*d*e-2*exp(1/2/c
*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a
*c*e^2+b^2*e^2)^(1/2))/c)*c^2*d^2-2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/
2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*c*e^2+e
xp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c
*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b^2*e^2-2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*
e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b
*c*d*e+2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e*x+d)
*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c^2*d^2+exp(1/2/c*(-b*e+2*c*d+(-4*a*
c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/
2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*b*e-2*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^
2)^(1/2)))*Ei(1,1/2*(-2*(e*x+d)*c-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a
*c*e^2+b^2*e^2)^(1/2)*c*d+exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(
1,-1/2*(2*(e*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2
)^(1/2)*b*e-2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*(e
*x+d)*c+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*c*d)
/(-4*a*c*e^2+b^2*e^2)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (c e x^{3} e^{d} - c x^{2} e^{d} - b x e^{d}\right )} e^{\left (e x\right )}}{c^{2} e^{2} x^{2} + b c e^{2} x + a c e^{2}} - \int -\frac{{\left ({\left (b e e^{d} + 2 \, c e^{d}\right )} a x +{\left (b^{2} e e^{d} - 2 \, a c e e^{d}\right )} x^{2} + a b e^{d}\right )} e^{\left (e x\right )}}{c^{3} e^{2} x^{4} + 2 \, b c^{2} e^{2} x^{3} + 2 \, a b c e^{2} x + a^{2} c e^{2} +{\left (b^{2} c e^{2} + 2 \, a c^{2} e^{2}\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(e*x + d)/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

(c*e*x^3*e^d - c*x^2*e^d - b*x*e^d)*e^(e*x)/(c^2*e^2*x^2 + b*c*e^2*x + a*c*e^2)
- integrate(-((b*e*e^d + 2*c*e^d)*a*x + (b^2*e*e^d - 2*a*c*e*e^d)*x^2 + a*b*e^d)
*e^(e*x)/(c^3*e^2*x^4 + 2*b*c^2*e^2*x^3 + 2*a*b*c*e^2*x + a^2*c*e^2 + (b^2*c*e^2
+ 2*a*c^2*e^2)*x^2), x)

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Fricas [A]  time = 0.266401, size = 413, normalized size = 1.78 \[ -\frac{{\left ({\left (b^{3} - 3 \, a b c\right )} e^{3} -{\left (b^{2} c - a c^{2}\right )} e^{2} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )}{\rm Ei}\left (\frac{2 \, c e x + b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} -{\left ({\left (b^{3} - 3 \, a b c\right )} e^{3} +{\left (b^{2} c - a c^{2}\right )} e^{2} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )}{\rm Ei}\left (\frac{2 \, c e x + b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - 2 \,{\left (c^{3} e x - b c^{2} e - c^{3}\right )} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} e^{\left (e x + d\right )}}{2 \, c^{4} e^{2} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(e*x + d)/(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

-1/2*(((b^3 - 3*a*b*c)*e^3 - (b^2*c - a*c^2)*e^2*sqrt((b^2 - 4*a*c)*e^2/c^2))*Ei
(1/2*(2*c*e*x + b*e - c*sqrt((b^2 - 4*a*c)*e^2/c^2))/c)*e^(1/2*(2*c*d - b*e + c*
sqrt((b^2 - 4*a*c)*e^2/c^2))/c) - ((b^3 - 3*a*b*c)*e^3 + (b^2*c - a*c^2)*e^2*sqr
t((b^2 - 4*a*c)*e^2/c^2))*Ei(1/2*(2*c*e*x + b*e + c*sqrt((b^2 - 4*a*c)*e^2/c^2))
/c)*e^(1/2*(2*c*d - b*e - c*sqrt((b^2 - 4*a*c)*e^2/c^2))/c) - 2*(c^3*e*x - b*c^2
*e - c^3)*sqrt((b^2 - 4*a*c)*e^2/c^2)*e^(e*x + d))/(c^4*e^2*sqrt((b^2 - 4*a*c)*e
^2/c^2))

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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(exp(e*x+d)*x**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(e*x + d)/(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

integrate(x^3*e^(e*x + d)/(c*x^2 + b*x + a), x)