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

Optimal. Leaf size=138 \[ \frac{e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )}{\sqrt{b^2-4 a c}}-\frac{e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )}{\sqrt{b^2-4 a c}} \]

[Out]

(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)])/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)]
)/Sqrt[b^2 - 4*a*c]

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Rubi [A]  time = 0.194263, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2268, 2178} \[ \frac{e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )}{\sqrt{b^2-4 a c}}-\frac{e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[E^(d + e*x)/(a + b*x + c*x^2),x]

[Out]

(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)])/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)]
)/Sqrt[b^2 - 4*a*c]

Rule 2268

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]

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

Mathematica [A]  time = 0.15274, size = 127, normalized size = 0.92 \[ \frac{e^{\frac{e \left (\sqrt{b^2-4 a c}-b\right )}{2 c}+d} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )-e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)] - 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)])/Sqrt[b^2 - 4*a*c]

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Maple [A]  time = 0.011, size = 169, normalized size = 1.2 \begin{align*} -{e \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 ) -{{\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 ) \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)/(c*x^2+b*x+a),x)

[Out]

-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)-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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57157, size = 420, normalized size = 3.04 \begin{align*} \frac{c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}{\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 )} - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}{\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^{2} - 4 \, a c\right )} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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) - 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^2 - 4*a*c)*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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