Integrand size = 23, antiderivative size = 212 \[ \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=-\frac {e^{d+e x}}{a x}-\frac {b e^d \operatorname {ExpIntegralEi}(e x)}{a^2}+\frac {e e^d \operatorname {ExpIntegralEi}(e x)}{a}+\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 a^2} \]
-exp(e*x+d)/a/x-b*exp(d)*Ei(e*x)/a^2+e*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)*(b+(2*a*c -b^2)/(-4*a*c+b^2)^(1/2))/a^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)*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2)) /a^2
Time = 0.88 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.51 \[ \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \left (-2 a \sqrt {b^2-4 a c} e^{\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}}-2 \sqrt {b^2-4 a c} (b-a e) e^{\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} x \operatorname {ExpIntegralEi}(e x)+\left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) e^{\frac {\sqrt {b^2-4 a c} e}{c}} x \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )-b^2 x \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+2 a c x \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+b \sqrt {b^2-4 a c} x \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{2 a^2 \sqrt {b^2-4 a c} x} \]
(E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*(-2*a*Sqrt[b^2 - 4*a*c]*E^((e*( b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)) - 2*Sqrt[b^2 - 4*a*c]*(b - a*e)*E^( ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*x*ExpIntegralEi[e*x] + (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 - Sq rt[b^2 - 4*a*c] + 2*c*x))/(2*c)] - b^2*x*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)] + 2*a*c*x*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)] + b*Sqrt[b^2 - 4*a*c]*x*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)]))/(2*a^2*Sqrt[b^2 - 4*a*c]*x)
Time = 0.73 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2700, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2700 |
| \(\displaystyle \int \left (\frac {e^{d+e x} \left (-a c+b^2+b c x\right )}{a^2 \left (a+b x+c x^2\right )}-\frac {b e^{d+e x}}{a^2 x}+\frac {e^{d+e x}}{a x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a^2}-\frac {b e^d \operatorname {ExpIntegralEi}(e x)}{a^2}+\frac {e^d e \operatorname {ExpIntegralEi}(e x)}{a}-\frac {e^{d+e x}}{a x}\) |
-(E^(d + e*x)/(a*x)) - (b*E^d*ExpIntegralEi[e*x])/a^2 + (e*E^d*ExpIntegral Ei[e*x])/a + ((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) + ((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)
3.5.67.3.1 Defintions of rubi rules used
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] && Polynomia lQ[u, x] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(560\) vs. \(2(184)=368\).
Time = 0.63 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.65
| method | result | size |
| derivativedivides | \(e \left (-\frac {{\mathrm e}^{e x +d}}{a e x}-\frac {\left (a e -b \right ) {\mathrm e}^{d} \operatorname {Ei}_{1}\left (-e x \right )}{a^{2} e}-\frac {-2 \,{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a c e +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2} e +2 \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a c e -{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2} e +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, b +{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, b}{2 a^{2} e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )\) | \(561\) |
| default | \(e \left (-\frac {{\mathrm e}^{e x +d}}{a e x}-\frac {\left (a e -b \right ) {\mathrm e}^{d} \operatorname {Ei}_{1}\left (-e x \right )}{a^{2} e}-\frac {-2 \,{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a c e +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2} e +2 \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a c e -{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2} e +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, b +{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, b}{2 a^{2} e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\right )\) | \(561\) |
| risch | \(-\frac {{\mathrm e}^{e x +d}}{a x}-\frac {e \,{\mathrm e}^{d} \operatorname {Ei}_{1}\left (-e x \right )}{a}+\frac {{\mathrm e}^{d} \operatorname {Ei}_{1}\left (-e x \right ) b}{a^{2}}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c}{a \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2}}{2 a^{2} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c}{a \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2}}{2 a^{2} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 a^{2}}-\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 a^{2}}\) | \(589\) |
e*(-exp(e*x+d)/a/e/x-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*(-b*e+2*c*d-2*c*(e*x+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*(-b*e+2*c*d-2*c*(e*x+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*(b *e-2*c*d+2*c*(e*x+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*(b*e-2*c*d+2*c*(e*x+d)+(-4*a* c*e^2+b^2*e^2)^(1/2))/c)*b^2*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*(-b*e+2*c*d-2*c*(e*x+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*(b*e-2*c*d+2*c*(e*x+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))
Time = 0.35 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.48 \[ \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} e^{2} - {\left (b^{3} - 4 \, a b c\right )} e\right )} x {\rm Ei}\left (e x\right ) e^{d} - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e e^{\left (e x + d\right )} + {\left ({\left (b^{3} - 4 \, a b c\right )} e x + {\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} x\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left ({\left (b^{3} - 4 \, a b c\right )} e x - {\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} x\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e x} \]
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)
Timed out. \[ \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\int { \frac {e^{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} x^{2}} \,d x } \]
\[ \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\int { \frac {e^{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{d+e\,x}}{x^2\,\left (c\,x^2+b\,x+a\right )} \,d x \]