Integrand size = 21, antiderivative size = 158 \[ \int \frac {e^{d+e x} x}{a+b x+c x^2} \, dx=\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c}+\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c} \]
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))/c+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))/c
Time = 0.33 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.97 \[ \int \frac {e^{d+e x} x}{a+b x+c x^2} \, dx=\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \left (\left (-b+\sqrt {b^2-4 a c}\right ) e^{\frac {\sqrt {b^2-4 a c} e}{c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+\left (b+\sqrt {b^2-4 a c}\right ) \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{2 c \sqrt {b^2-4 a c}} \]
(E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*((-b + Sqrt[b^2 - 4*a*c])*E^((S qrt[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)]))/(2*c*Sqrt[b^2 - 4*a*c])
Time = 0.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {x e^{d+e x}}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 2700 |
| \(\displaystyle \int \left (\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{-\sqrt {b^2-4 a c}+b+2 c x}+\frac {\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) e^{d+e x}}{\sqrt {b^2-4 a c}+b+2 c x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c}+\frac {\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c}\) |
((1 - b/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIn tegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c) + ((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*c)
3.5.70.3.1 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. \(349\) vs. \(2(132)=264\).
Time = 0.43 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.22
| method | result | size |
| risch | \(\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e}{2 c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e}{2 c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{2 c}-\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{2 c}\) | \(350\) |
| derivativedivides | \(\frac {-\frac {e^{2} \left (-{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e +2 \,{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c d +{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e -2 \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c d +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}{2 c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {d \,e^{2} \left ({\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )-{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}}{e^{2}}\) | \(685\) |
| default | \(\frac {-\frac {e^{2} \left (-{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e +2 \,{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c d +{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e -2 \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c d +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}{2 c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {d \,e^{2} \left ({\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )-{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}}{e^{2}}\) | \(685\) |
1/2/c/(-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/c/(-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/c*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/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*c*(e*x+d)+(-4 *a*c*e^2+b^2*e^2)^(1/2))/c)
Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.42 \[ \int \frac {e^{d+e x} x}{a+b x+c x^2} \, dx=-\frac {{\left (b c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} - {\left (b^{2} - 4 \, a c\right )} e\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - {\left (b c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} + {\left (b^{2} - 4 \, a c\right )} e\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e} \]
-1/2*((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))/((b^2*c - 4*a*c^2)*e)
\[ \int \frac {e^{d+e x} x}{a+b x+c x^2} \, dx=e^{d} \int \frac {x e^{e x}}{a + b x + c x^{2}}\, dx \]
\[ \int \frac {e^{d+e x} x}{a+b x+c x^2} \, dx=\int { \frac {x e^{\left (e x + d\right )}}{c x^{2} + b x + a} \,d x } \]
x*e^(e*x + d)/(c*e*x^2 + b*e*x + a*e) + integrate((c*x^2*e^d - a*e^d)*e^(e *x)/(c^2*e*x^4 + 2*b*c*e*x^3 + 2*a*b*e*x + a^2*e + (b^2*e + 2*a*c*e)*x^2), x)
\[ \int \frac {e^{d+e x} x}{a+b x+c x^2} \, dx=\int { \frac {x e^{\left (e x + d\right )}}{c x^{2} + b x + a} \,d x } \]
Timed out. \[ \int \frac {e^{d+e x} x}{a+b x+c x^2} \, dx=\int \frac {x\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \]