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

Optimal. Leaf size=169 \[ -\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 a}-\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 a}+\frac{e^d \text{ExpIntegralEi}(e x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.705651, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 a}-\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )}{2 a}+\frac{e^d \text{ExpIntegralEi}(e x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 58.1935, size = 170, normalized size = 1.01 \[ \frac{\left (b - \sqrt{- 4 a c + b^{2}}\right ) e^{\frac{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 a \sqrt{- 4 a c + b^{2}}} - \frac{\left (b + \sqrt{- 4 a c + b^{2}}\right ) e^{\frac{- b e + 2 c d + e \sqrt{- 4 a c + b^{2}}}{2 c}} \operatorname{Ei}{\left (\frac{e \left (\frac{b}{2} + c x - \frac{\sqrt{- 4 a c + b^{2}}}{2}\right )}{c} \right )}}{2 a \sqrt{- 4 a c + b^{2}}} + \frac{e^{d} \operatorname{Ei}{\left (e x \right )}}{a} \]

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

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Mathematica [A]  time = 0.642598, size = 163, normalized size = 0.96 \[ \frac{e^d \left (\frac{e^{-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \left (\left (b-\sqrt{b^2-4 a c}\right ) \text{ExpIntegralEi}\left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )-\left (\sqrt{b^2-4 a c}+b\right ) e^{\frac{e \sqrt{b^2-4 a c}}{c}} \text{ExpIntegralEi}\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}\right )\right )}{\sqrt{b^2-4 a c}}+2 \text{ExpIntegralEi}(e x)\right )}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.027, size = 369, normalized size = 2.2 \[ -{\frac{{{\rm e}^{d}}{\it Ei} \left ( 1,-ex \right ) }{a}}+{\frac{1}{2\,a} \left ({{\rm e}^{{\frac{1}{2\,c} \left ( -be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) }}}{\it Ei} \left ( 1,{\frac{1}{2\,c} \left ( -2\, \left ( ex+d \right ) c-be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) } \right ) be-{{\rm e}^{-{\frac{1}{2\,c} \left ( be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{2\,c} \left ( 2\, \left ( ex+d \right ) c+be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) } \right ) be+{{\rm e}^{{\frac{1}{2\,c} \left ( -be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) }}}{\it Ei} \left ( 1,{\frac{1}{2\,c} \left ( -2\, \left ( ex+d \right ) c-be+2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) } \right ) \sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}+{{\rm e}^{-{\frac{1}{2\,c} \left ( be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{2\,c} \left ( 2\, \left ( ex+d \right ) c+be-2\,cd+\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ) } \right ) \sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}} \right ){\frac{1}{\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}}}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.26084, size = 324, normalized size = 1.92 \[ \frac{2 \, c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}{\rm Ei}\left (e x\right ) e^{d} -{\left (b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )}{\rm Ei}\left (\frac{2 \, c e x + b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} +{\left (b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )}{\rm Ei}\left (\frac{2 \, c e x + b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, a c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{d} \int \frac{e^{e x}}{a x + b x^{2} + c x^{3}}\, dx \]

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(exp(e*x)/(a*x + b*x**2 + c*x**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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