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

Optimal. Leaf size=212 \[ \frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 a^2}+\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 a^2}-\frac{b e^d \text{ExpIntegralEi}(e x)}{a^2}+\frac{e^d e \text{ExpIntegralEi}(e x)}{a}-\frac{e^{d+e x}}{a x} \]

[Out]

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

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Rubi [A]  time = 1.03839, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 a^2}+\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 a^2}-\frac{b e^d \text{ExpIntegralEi}(e x)}{a^2}+\frac{e^d e \text{ExpIntegralEi}(e x)}{a}-\frac{e^{d+e x}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 74.701, size = 216, normalized size = 1.02 \[ \frac{e e^{d} \operatorname{Ei}{\left (e x \right )}}{a} - \frac{e^{d + e x}}{a x} - \frac{b e^{d} \operatorname{Ei}{\left (e x \right )}}{a^{2}} - \frac{\left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) e^{\frac{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 a^{2} \sqrt{- 4 a c + b^{2}}} + \frac{\left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) e^{\frac{- b e + 2 c d + e \sqrt{- 4 a c + b^{2}}}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x - \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 a^{2} \sqrt{- 4 a c + b^{2}}} \]

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]

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

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Mathematica [A]  time = 1.36289, size = 232, normalized size = 1.09 \[ \frac{e^d \left (\frac{e^{-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \left (x \left (b \sqrt{b^2-4 a c}-2 a c+b^2\right ) e^{\frac{e \sqrt{b^2-4 a c}}{c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )+x \left (b \sqrt{b^2-4 a c}+2 a c-b^2\right ) \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )-2 a \sqrt{b^2-4 a c} e^{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}}\right )}{x \sqrt{b^2-4 a c}}-2 (b-a e) \text{ExpIntegralEi}(e x)\right )}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.039, size = 561, normalized size = 2.7 \[ e \left ( -{\frac{{{\rm e}^{ex+d}}}{aex}}-{\frac{ \left ( ea-b \right ){{\rm e}^{d}}{\it Ei} \left ( 1,-ex \right ) }{{a}^{2}e}}-{\frac{1}{2\,{a}^{2}e} \left ( -2\,{{\rm e}^{1/2\,{\frac{-be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}}{c}}}}{\it Ei} \left ( 1,1/2\,{\frac{-2\, \left ( ex+d \right ) c-be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}}{c}} \right ) ace+{{\rm e}^{{\frac{1}{2\,c} \left ( -be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) }}}{\it Ei} \left ( 1,{\frac{1}{2\,c} \left ( -2\, \left ( ex+d \right ) c-be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) } \right ){b}^{2}e+2\,{{\rm e}^{-1/2\,{\frac{be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}}{c}}}}{\it Ei} \left ( 1,-1/2\,{\frac{2\, \left ( ex+d \right ) c+be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}}{c}} \right ) ace-{{\rm e}^{-{\frac{1}{2\,c} \left ( be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{2\,c} \left ( 2\, \left ( ex+d \right ) c+be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) } \right ){b}^{2}e+{{\rm e}^{{\frac{1}{2\,c} \left ( -be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) }}}{\it Ei} \left ( 1,{\frac{1}{2\,c} \left ( -2\, \left ( ex+d \right ) c-be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) } \right ) \sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}b+{{\rm e}^{-{\frac{1}{2\,c} \left ( be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{2\,c} \left ( 2\, \left ( ex+d \right ) c+be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) } \right ) \sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}b \right ){\frac{1}{\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}}}} \right ) \]

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]

e*(-exp(e*x+d)/a/x/e-1/a^2/e*(a*e-b)*exp(d)*Ei(1,-e*x)-1/2*(-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+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*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-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)*b^2*e+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+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)/a^2/e/(-4*a*c*e^2+b^2*e^2)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273715, size = 382, normalized size = 1.8 \[ \frac{{\left (b c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} x +{\left (b^{2} - 2 \, a c\right )} e x\right )}{\rm Ei}\left (\frac{2 \, c e x + b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} +{\left (b c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} x -{\left (b^{2} - 2 \, a c\right )} e x\right )}{\rm Ei}\left (\frac{2 \, c e x + b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + 2 \,{\left ({\left (a c e - b c\right )} x{\rm Ei}\left (e x\right ) e^{d} - a c e^{\left (e x + d\right )}\right )} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, a^{2} c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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**2/(c*x**2+b*x+a),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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