Integrand size = 23, antiderivative size = 232 \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=-\frac {e^{d+e x}}{c e^2}-\frac {b e^{d+e x}}{c^2 e}+\frac {e^{d+e x} x}{c e}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^3}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^3} \]
-exp(e*x+d)/c/e^2-b*exp(e*x+d)/c^2/e+exp(e*x+d)*x/c/e+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* (-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/c^3+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*(-3*a*c+b^2)/(-4* a*c+b^2)^(1/2))/c^3
Time = 0.46 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.16 \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\frac {e^{d-\frac {b e}{c}} \left (-2 c \sqrt {b^2-4 a c} e^{e \left (\frac {b}{c}+x\right )} (c+b e-c e x)+\left (-b^3+3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) e^2 e^{\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+\left (b^3-3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) e^2 e^{\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{2 c^3 \sqrt {b^2-4 a c} e^2} \]
(E^(d - (b*e)/c)*(-2*c*Sqrt[b^2 - 4*a*c]*E^(e*(b/c + x))*(c + b*e - c*e*x) + (-b^3 + 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*e^2*E^ (((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c ] + 2*c*x))/(2*c)] + (b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*e^2*E^(((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*Sqrt[b^2 - 4*a*c]*e^2)
Time = 0.65 (sec) , antiderivative size = 232, 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.
| \(\displaystyle \int \frac {x^3 e^{d+e x}}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 2700 |
| \(\displaystyle \int \left (\frac {e^{d+e x} \left (x \left (b^2-a c\right )+a b\right )}{c^2 \left (a+b x+c x^2\right )}-\frac {b e^{d+e x}}{c^2}+\frac {x e^{d+e x}}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^3}+\frac {\left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^3}-\frac {b e^{d+e x}}{c^2 e}-\frac {e^{d+e x}}{c e^2}+\frac {x e^{d+e x}}{c e}\) |
-(E^(d + e*x)/(c*e^2)) - (b*E^(d + e*x))/(c^2*e) + (E^(d + e*x)*x)/(c*e) + ((b^2 - a*c - (b*(b^2 - 3*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*c^3) + ((b^2 - a*c + (b*(b^2 - 3*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*c^3)
3.5.72.3.1 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. \(754\) vs. \(2(203)=406\).
Time = 0.56 (sec) , antiderivative size = 755, normalized size of antiderivative = 3.25
| method | result | size |
| risch | \(-\frac {3 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 ) a b}{2 c^{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 ) b^{3}}{2 c^{3} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {3 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 ) a b}{2 c^{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 ) b^{3}}{2 c^{3} \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 ) a}{2 c^{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}}{2 c^{3}}+\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 ) a}{2 c^{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}}{2 c^{3}}+\frac {{\mathrm e}^{e x +d} x}{c e}-\frac {b \,{\mathrm e}^{e x +d}}{c^{2} e}-\frac {{\mathrm e}^{e x +d}}{c \,e^{2}}\) | \(755\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3532\) |
| default | \(\text {Expression too large to display}\) | \(3532\) |
-3/2*e/c^2/(-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)*a*b+1/2*e/c^3/(-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^3+3/2*e/c^2/(-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)*a*b-1/2*e/c^3/(-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^3+1/2/c^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-1/2/c^3*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+1/2/c^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-1/2/c^3*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+exp(e*x+d)*x/c/e-b*exp(e*x+d)/c^2/e-exp(e*x+d)/c/e^2
Time = 0.28 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.42 \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\frac {{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2} + {\left (b^{3} c - 3 \, a b c^{2}\right )} e \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - 2 \, {\left (b^{2} c^{2} - 4 \, a c^{3} - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x + {\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} e^{\left (e x + d\right )}}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e^{2}} \]
1/2*(((b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^2 - (b^3*c - 3*a*b*c^2)*e*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^4 - 5*a*b ^2*c + 4*a^2*c^2)*e^2 + (b^3*c - 3*a*b*c^2)*e*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) - 2*(b^2*c^2 - 4*a*c^3 - (b^2*c^2 - 4*a*c^3)*e*x + (b^3*c - 4*a*b*c^2)*e)*e^(e*x + d))/((b^2*c^3 - 4*a*c^4) *e^2)
Timed out. \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\text {Timed out} \]
\[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\int { \frac {x^{3} e^{\left (e x + d\right )}}{c x^{2} + b x + a} \,d x } \]
(c*e*x^3*e^d - c*x^2*e^d - b*x*e^d)*e^(e*x)/(c^2*e^2*x^2 + b*c*e^2*x + a*c *e^2) - integrate(-((b*e*e^d + 2*c*e^d)*a*x + (b^2*e*e^d - 2*a*c*e*e^d)*x^ 2 + a*b*e^d)*e^(e*x)/(c^3*e^2*x^4 + 2*b*c^2*e^2*x^3 + 2*a*b*c*e^2*x + a^2* c*e^2 + (b^2*c*e^2 + 2*a*c^2*e^2)*x^2), x)
\[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\int { \frac {x^{3} e^{\left (e x + d\right )}}{c x^{2} + b x + a} \,d x } \]
Timed out. \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\int \frac {x^3\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \]