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]

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

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Rubi [A]  time = 0.19, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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

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]

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*}

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Mathematica [A]  time = 0.15, 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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fricas [A]  time = 0.43, size = 192, normalized size = 1.39 \[ \frac {c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{{\left (b^{2} - 4 \, a c\right )} e} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (e x + d\right )}}{c x^{2} + b x + a}\,{d x} \]

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)

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maple [A]  time = 0.02, size = 169, normalized size = 1.22 \[ -\frac {\left (\Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}-\Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}\right ) e}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}} \]

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*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,1/2*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (e x + d\right )}}{c x^{2} + b x + a}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d + e*x)/(a + b*x + c*x^2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{d} \int \frac {e^{e x}}{a + b x + c x^{2}}\, dx \]

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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