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

Optimal. Leaf size=186 \[ -\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c^2}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c^2}+\frac{e^{d+e x}}{c e} \]

[Out]

E^(d + e*x)/(c*e) - ((b - (b^2 - 2*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^2) - ((b + (b^2 - 2*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^2)

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Rubi [A]  time = 0.736465, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c^2}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 c^2}+\frac{e^{d+e x}}{c e} \]

Antiderivative was successfully verified.

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

[Out]

E^(d + e*x)/(c*e) - ((b - (b^2 - 2*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^2) - ((b + (b^2 - 2*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^2)

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Rubi in Sympy [A]  time = 64.5475, size = 190, normalized size = 1.02 \[ \frac{e^{d + e x}}{c e} + \frac{\left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) e^{\frac{- b e + 2 c d + e \sqrt{- 4 a c + b^{2}}}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x - \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 c^{2} \sqrt{- 4 a c + b^{2}}} - \frac{\left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) e^{\frac{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 c^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.555614, size = 217, normalized size = 1.17 \[ -\frac{e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \left (e \left (b \sqrt{b^2-4 a c}+2 a c-b^2\right ) e^{\frac{e \sqrt{b^2-4 a c}}{c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )+e \left (b \sqrt{b^2-4 a c}-2 a c+b^2\right ) \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )-2 c \sqrt{b^2-4 a c} e^{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}}\right )}{2 c^2 e \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.024, size = 1730, normalized size = 9.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e^3*(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+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*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)-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)-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)+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*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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255121, size = 358, normalized size = 1.92 \[ \frac{2 \, c^{2} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} e^{\left (e x + d\right )} -{\left (b c e \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} -{\left (b^{2} - 2 \, a c\right )} e^{2}\right )}{\rm Ei}\left (\frac{2 \, c e x + b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} -{\left (b c e \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} +{\left (b^{2} - 2 \, a c\right )} e^{2}\right )}{\rm Ei}\left (\frac{2 \, c e x + b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, c^{3} e \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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