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

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

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

________________________________________________________________________________________

Rubi [A]  time = 0.411215, 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, Rules used = {2270, 2178} \[ -\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )}{2 a}-\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )}{2 a}+\frac{e^d \text{Ei}(e x)}{a} \]

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

Rule 2270

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{e^{d+e x}}{x \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{e^{d+e x}}{a x}+\frac{e^{d+e x} (-b-c x)}{a \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{e^{d+e x}}{x} \, dx}{a}+\frac{\int \frac{e^{d+e x} (-b-c x)}{a+b x+c x^2} \, dx}{a}\\ &=\frac{e^d \text{Ei}(e x)}{a}+\frac{\int \left (\frac{\left (-c-\frac{b c}{\sqrt{b^2-4 a c}}\right ) e^{d+e x}}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (-c+\frac{b c}{\sqrt{b^2-4 a c}}\right ) e^{d+e x}}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{a}\\ &=\frac{e^d \text{Ei}(e x)}{a}-\frac{\left (c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{e^{d+e x}}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{a}-\frac{\left (c \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{e^{d+e x}}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{a}\\ &=\frac{e^d \text{Ei}(e x)}{a}-\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c}} \text{Ei}\left (\frac{e \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 a}-\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c}} \text{Ei}\left (\frac{e \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.554525, size = 163, normalized size = 0.96 \[ \frac{e^d \left (\frac{e^{-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \left (\left (b-\sqrt{b^2-4 a c}\right ) \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )-\left (\sqrt{b^2-4 a c}+b\right ) e^{\frac{e \sqrt{b^2-4 a c}}{c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )\right )}{\sqrt{b^2-4 a c}}+2 \text{Ei}(e x)\right )}{2 a} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.018, size = 369, normalized size = 2.2 \begin{align*} -{\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\,c \left ( ex+d \right ) -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\,c \left ( ex+d \right ) +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\,c \left ( ex+d \right ) -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\,c \left ( ex+d \right ) +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}}}}} \end{align*}

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)))*Ei(1,1/2*(-2*c*(e*x+d)-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*c*(e*x+d)
+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*
c*(e*x+d)-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*c*(e*x+d)+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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.61726, size = 533, normalized size = 3.15 \begin{align*} \frac{2 \,{\left (b^{2} - 4 \, a c\right )} e{\rm Ei}\left (e x\right ) e^{d} -{\left (b c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} +{\left (b^{2} - 4 \, a c\right )} e\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}}} -{\left (b^{2} - 4 \, a c\right )} e\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 (a b^{2} - 4 \, a^{2} c\right )} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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