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

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

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

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Rubi [A]  time = 0.620086, 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, Rules used = {2270, 2177, 2178} \[ \frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )}{2 a^2}+\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )}{2 a^2}-\frac{b e^d \text{Ei}(e x)}{a^2}+\frac{e^d e \text{Ei}(e x)}{a}-\frac{e^{d+e x}}{a x} \]

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

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 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

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

Mathematica [A]  time = 1.13093, size = 232, normalized size = 1.09 \[ \frac{e^d \left (\frac{e^{-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \left (x \left (b \sqrt{b^2-4 a c}-2 a c+b^2\right ) e^{\frac{e \sqrt{b^2-4 a c}}{c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )+x \left (b \sqrt{b^2-4 a c}+2 a c-b^2\right ) \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )-2 a \sqrt{b^2-4 a c} e^{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c}}\right )}{x \sqrt{b^2-4 a c}}-2 (b-a e) \text{Ei}(e x)\right )}{2 a^2} \]

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 = 0.023, size = 561, normalized size = 2.7 \begin{align*} e \left ( -{\frac{{{\rm e}^{ex+d}}}{aex}}-{\frac{ \left ( ae-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\,c \left ( ex+d \right ) -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\,c \left ( ex+d \right ) -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\,c \left ( ex+d \right ) +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\,c \left ( ex+d \right ) +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\,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}}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\,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}}b \right ){\frac{1}{\sqrt{-4\,ac{e}^{2}+{b}^{2}{e}^{2}}}}} \right ) \end{align*}

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*c*(e*x+d)-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*c*(e*x+d)-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*c*(e*x+d)+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)*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^2*e+e
xp(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)*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*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)*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. \begin{align*} \int \frac{e^{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} x^{2}}\,{d x} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58163, size = 682, normalized size = 3.22 \begin{align*} \frac{2 \,{\left ({\left (a b^{2} - 4 \, a^{2} c\right )} e^{2} -{\left (b^{3} - 4 \, a b c\right )} e\right )} x{\rm Ei}\left (e x\right ) e^{d} - 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} e e^{\left (e x + d\right )} +{\left ({\left (b^{3} - 4 \, a b c\right )} e x +{\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} 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 ({\left (b^{3} - 4 \, a b c\right )} e x -{\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} 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 (a^{2} b^{2} - 4 \, a^{3} c\right )} e x} \end{align*}

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, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 [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^{2}}\,{d x} \end{align*}

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, algorithm="giac")

[Out]

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