\(\int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx\) [472]

Optimal result

Integrand size = 23, antiderivative size = 232 \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=-\frac {e^{d+e x}}{c e^2}-\frac {b e^{d+e x}}{c^2 e}+\frac {e^{d+e x} x}{c e}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^3}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^3} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2302, 2225, 2207, 2209} \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\frac {\left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^3}+\frac {\left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^3}-\frac {b e^{d+e x}}{c^2 e}-\frac {e^{d+e x}}{c e^2}+\frac {x e^{d+e x}}{c e} \]

[In]

Int[(E^(d + e*x)*x^3)/(a + b*x + c*x^2),x]

[Out]

-(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))/Sqr
t[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))*E
xpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^3)

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

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 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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 {b e^{d+e x}}{c^2}+\frac {e^{d+e x} x}{c}+\frac {e^{d+e x} \left (a b+\left (b^2-a c\right ) x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {e^{d+e x} \left (a b+\left (b^2-a c\right ) x\right )}{a+b x+c x^2} \, dx}{c^2}-\frac {b \int e^{d+e x} \, dx}{c^2}+\frac {\int e^{d+e x} x \, dx}{c} \\ & = -\frac {b e^{d+e x}}{c^2 e}+\frac {e^{d+e x} x}{c e}+\frac {\int \left (\frac {\left (b^2-a c+\frac {b \left (-b^2+3 a c\right )}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (b^2-a c-\frac {b \left (-b^2+3 a c\right )}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{c^2}-\frac {\int e^{d+e x} \, dx}{c e} \\ & = -\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 ) \int \frac {e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{c^2}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{c^2} \\ & = -\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}} \text {Ei}\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}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.16 \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\frac {e^{d-\frac {b e}{c}} \left (-2 c \sqrt {b^2-4 a c} e^{e \left (\frac {b}{c}+x\right )} (c+b e-c e x)+\left (-b^3+3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) e^2 e^{\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+\left (b^3-3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) e^2 e^{\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{2 c^3 \sqrt {b^2-4 a c} e^2} \]

[In]

Integrate[(E^(d + e*x)*x^3)/(a + b*x + c*x^2),x]

[Out]

(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)

Maple [B] (verified)

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	\(\text {Expression too large to display}\)	\(3532\)

[In]

int(exp(e*x+d)*x^3/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

-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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.42 \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\frac {{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2} + {\left (b^{3} c - 3 \, a b c^{2}\right )} e \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - 2 \, {\left (b^{2} c^{2} - 4 \, a c^{3} - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x + {\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} e^{\left (e x + d\right )}}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e^{2}} \]

[In]

integrate(exp(e*x+d)*x^3/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

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)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\text {Timed out} \]

[In]

integrate(exp(e*x+d)*x**3/(c*x**2+b*x+a),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate(exp(e*x+d)*x^3/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

(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)

Giac [F]

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

[In]

integrate(exp(e*x+d)*x^3/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

integrate(x^3*e^(e*x + d)/(c*x^2 + b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{d+e x} x^3}{a+b x+c x^2} \, dx=\int \frac {x^3\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \]

[In]

int((x^3*exp(d + e*x))/(a + b*x + c*x^2),x)

[Out]

int((x^3*exp(d + e*x))/(a + b*x + c*x^2), x)