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

Optimal. Leaf size=158 \[ \frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c}+\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c} \]

[Out]

((1 - b/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegral
Ei[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c) + ((1 + b/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)

_______________________________________________________________________________________

Rubi [A]  time = 0.371302, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c}+\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

((1 - b/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegral
Ei[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c) + ((1 + b/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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.5924, size = 160, normalized size = 1.01 \[ - \frac{\left (b - \sqrt{- 4 a c + b^{2}}\right ) e^{\frac{- b e + 2 c d + e \sqrt{- 4 a c + b^{2}}}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x - \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 c \sqrt{- 4 a c + b^{2}}} + \frac{\left (b + \sqrt{- 4 a c + b^{2}}\right ) e^{\frac{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 c \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b - sqrt(-4*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*sqrt(-4*a*c + b**2)) + (b + sqrt(-
4*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*sqrt(-4*a*c + b**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.208282, size = 153, normalized size = 0.97 \[ \frac{e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \left (\left (\sqrt{b^2-4 a c}-b\right ) e^{\frac{e \sqrt{b^2-4 a c}}{c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )+\left (\sqrt{b^2-4 a c}+b\right ) \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )\right )}{2 c \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.019, size = 685, normalized size = 4.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e^2*(-1/2*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)))*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*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)*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*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)+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)-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))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.238608, size = 285, normalized size = 1.8 \[ -\frac{{\left (b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )}{\rm Ei}\left (\frac{2 \, c e x + b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} -{\left (b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )}{\rm Ei}\left (\frac{2 \, c e x + b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, c^{2} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*((b*e - c*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*e + c*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))/(c^2*sq
rt((b^2 - 4*a*c)*e^2/c^2))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{d} \int \frac{x e^{e x}}{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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