\(\int \frac {e^{d+e x}}{x (a+b x+c x^2)} \, dx\) [468]

Optimal result

Integrand size = 23, antiderivative size = 169 \[ \int \frac {e^{d+e x}}{x \left (a+b x+c x^2\right )} \, dx=\frac {e^d \operatorname {ExpIntegralEi}(e x)}{a}-\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2302, 2209} \[ \int \frac {e^{d+e x}}{x \left (a+b x+c x^2\right )} \, dx=-\frac {\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a}+\frac {e^d \operatorname {ExpIntegralEi}(e x)}{a} \]

[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 2209

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

Rule 2302

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.37 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96 \[ \int \frac {e^{d+e x}}{x \left (a+b x+c x^2\right )} \, dx=\frac {e^d \left (2 \operatorname {ExpIntegralEi}(e x)+\frac {e^{-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \left (-\left (\left (b+\sqrt {b^2-4 a c}\right ) e^{\frac {\sqrt {b^2-4 a c} e}{c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )+\left (b-\sqrt {b^2-4 a c}\right ) \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{\sqrt {b^2-4 a c}}\right )}{2 a} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(362\) vs. \(2(142)=284\).

Time = 0.44 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.15

method	result	size
risch	\(-\frac {{\mathrm e}^{d} \operatorname {Ei}_{1}\left (-e x \right )}{a}+\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e}{2 a \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e}{2 a \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{2 a}+\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{2 a}\)	\(363\)
derivativedivides	\(-\frac {{\mathrm e}^{d} \operatorname {Ei}_{1}\left (-e x \right )}{a}+\frac {{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e -{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 a \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\)	\(369\)
default	\(-\frac {{\mathrm e}^{d} \operatorname {Ei}_{1}\left (-e x \right )}{a}+\frac {{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e -{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 a \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\)	\(369\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.43 \[ \int \frac {e^{d+e x}}{x \left (a+b x+c x^2\right )} \, dx=\frac {2 \, {\left (b^{2} - 4 \, a c\right )} e {\rm Ei}\left (e x\right ) e^{d} - {\left (b c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} + {\left (b^{2} - 4 \, a c\right )} e\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left (b c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} - {\left (b^{2} - 4 \, a c\right )} e\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e} \]

[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]

\[ \int \frac {e^{d+e x}}{x \left (a+b x+c x^2\right )} \, dx=e^{d} \int \frac {e^{e x}}{a x + b x^{2} + c x^{3}}\, dx \]

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{d+e x}}{x \left (a+b x+c x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{d+e\,x}}{x\,\left (c\,x^2+b\,x+a\right )} \,d x \]

[In]

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

[Out]

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