3.472 \(\int \frac{e^{d+e x} x^3}{a+b x+c x^2} \, dx\)
Optimal. Leaf size=232 \[ \frac{\left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )}{2 c^3}+\frac{\left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )}{2 c^3}-\frac{b e^{d+e x}}{c^2 e}-\frac{e^{d+e x}}{c e^2}+\frac{x e^{d+e x}}{c e} \]
[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)
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Rubi [A] time = 0.509539, antiderivative size = 232, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{2270, 2194, 2176, 2178} \[ \frac{\left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) e^{d-\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )}{2 c^3}+\frac{\left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) e^{d-\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )}{2 c^3}-\frac{b e^{d+e x}}{c^2 e}-\frac{e^{d+e x}}{c e^2}+\frac{x e^{d+e x}}{c e} \]
Antiderivative was successfully verified.
[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 2270
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]
Rule 2194
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 2176
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] && !$UseGamma === True
Rule 2178
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !$UseGamma === True
Rubi steps
\begin{align*} \int \frac{e^{d+e x} x^3}{a+b x+c x^2} \, dx &=\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] time = 0.639057, size = 268, normalized size = 1.16 \[ \frac{e^{d-\frac{b e}{c}} \left (e^2 \left (b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}+3 a b c-b^3\right ) e^{\frac{e \left (\sqrt{b^2-4 a c}+b\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x-\sqrt{b^2-4 a c}\right )}{2 c}\right )+e^2 \left (b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right ) e^{\frac{e \left (b-\sqrt{b^2-4 a c}\right )}{2 c}} \text{Ei}\left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{2 c}\right )-2 c \sqrt{b^2-4 a c} e^{e \left (\frac{b}{c}+x\right )} (b e+c (-e) x+c)\right )}{2 c^3 e^2 \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[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)
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Maple [B] time = 0.022, size = 3532, normalized size = 15.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(exp(e*x+d)*x^3/(c*x^2+b*x+a),x)
[Out]
1/e^4*(-e^2*exp(e*x+d)*(-c*(e*x+d)+b*e-2*c*d+c)/c^2+1/2/c^3*e^2*(-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*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*b*c*e^3+6*exp(1/2/c*(-b*e+2*c*d+(-4
*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*c^2*d*e^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*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b^3
*e^3-3*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*Ei(1,1/2*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2
)^(1/2))/c)*b^2*c*d*e^2+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*(-2*c*(e*x+d)-b*e+2*c*d+
(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b*c^2*d^2*e-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*(-2*c
*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c^3*d^3+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*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*b*c*e^3-6*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b
^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*c^2*d*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*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b^3*e^3+3*e
xp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))
/c)*b^2*c*d*e^2-3*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*
e^2+b^2*e^2)^(1/2))/c)*b*c^2*d^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*(2*c*(e*x+d)
+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c^3*d^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*
(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*a*c*e^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*(-2*c*(e*x+d)-b*e+2*c*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^2*e^2+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*(-2*c*(e*x+d)-b*e+2*c*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*c*d*e-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*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*c^2*d^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*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(
-4*a*c*e^2+b^2*e^2)^(1/2)*a*c*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*(2*c*(e*x+d)+b*
e-2*c*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^2*e^2+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*(2*c*(e*x+d)+b*e-2*c*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*
c*d*e-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*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^
2)^(1/2))/c)*(-4*a*c*e^2+b^2*e^2)^(1/2)*c^2*d^2)/(-4*a*c*e^2+b^2*e^2)^(1/2)+d^3*e^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*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(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*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c))/(-4*a*c*e^2+b^
2*e^2)^(1/2)-3/2*d^2*e^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*(-2*c*(e*x+d)-b*e+2*c*d
+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b*e+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*(-2*c*(e*x+d
)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c*d+exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2
*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b*e-2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*E
i(1,-1/2*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c*d+exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^
(1/2)))*Ei(1,1/2*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(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*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*
c*e^2+b^2*e^2)^(1/2))/c/(-4*a*c*e^2+b^2*e^2)^(1/2)-3*d*(e^2/c*exp(e*x+d)+1/2/c^2*e^2*(2*Ei(1,1/2*(-2*c*(e*x+d)
-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*a*c*e^2-Ei(1,1/2*
(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)))*b^2*
e^2+2*Ei(1,1/2*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*exp(1/2/c*(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^
2)^(1/2)))*b*c*d*e-2*Ei(1,1/2*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*exp(1/2/c*(-b*e+2*c*d+(-4
*a*c*e^2+b^2*e^2)^(1/2)))*c^2*d^2-2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(1,-1/2*(2*c*(e*x+d)+
b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*a*c*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*
(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b^2*e^2-2*exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)
)/c)*Ei(1,-1/2*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b*c*d*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*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c^2*d^2+Ei(1,1/2*(-2*c*(e*
x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(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)))*b*e-2*Ei(1,1/2*(-2*c*(e*x+d)-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(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)))*c*d+exp(-1/2*(b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*Ei(
1,-1/2*(2*c*(e*x+d)+b*e-2*c*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*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*(2*c*(e*x+d)+b*e-2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*(-4*a*c*e^2
+b^2*e^2)^(1/2)*c*d)/(-4*a*c*e^2+b^2*e^2)^(1/2)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (c e x^{3} e^{d} - c x^{2} e^{d} - b x e^{d}\right )} e^{\left (e x\right )}}{c^{2} e^{2} x^{2} + b c e^{2} x + a c e^{2}} - \int -\frac{{\left ({\left (b e e^{d} + 2 \, c e^{d}\right )} a x +{\left (b^{2} e e^{d} - 2 \, a c e e^{d}\right )} x^{2} + a b e^{d}\right )} e^{\left (e x\right )}}{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} +{\left (b^{2} c e^{2} + 2 \, a c^{2} e^{2}\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Fricas [A] time = 1.65116, size = 709, normalized size = 3.06 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(exp(e*x+d)*x**3/(c*x**2+b*x+a),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} e^{\left (e x + d\right )}}{c x^{2} + b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)