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]
-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
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Rubi [A] time = 0.51, antiderivative size = 232, normalized size of antiderivative = 1.00,
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 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
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 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]
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*}
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Mathematica [A] time = 0.65, size = 268, normalized size = 1.16 \[ \frac {e^{d-\frac {b e}{c}} \left (-2 c \sqrt {b^2-4 a c} e^{e \left (\frac {b}{c}+x\right )} (b e+c (-e) x+c)+e^2 \left (b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}+3 a b c-b^3\right ) e^{\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )+e^2 \left (b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}-3 a b c+b^3\right ) e^{\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )\right )}{2 c^3 e^2 \sqrt {b^2-4 a c}} \]
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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fricas [A] time = 0.43, size = 330, normalized size = 1.42 \[ \frac {{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{2} + {\left (b^{3} c - 3 \, a b c^{2}\right )} e \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - 2 \, {\left (b^{2} c^{2} - 4 \, a c^{3} - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x + {\left (b^{3} c - 4 \, a b c^{2}\right )} e\right )} e^{\left (e x + d\right )}}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e^{2}} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} e^{\left (e x + d\right )}}{c x^{2} + b x + a}\,{d x} \]
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)
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maple [B] time = 0.05, size = 3532, normalized size = 15.22 \[ \text {output too large to display} \]
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)*(-(e*x+d)*c+b*e-2*c*d+c)/c^2+1/2/c^3*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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b^3
*e^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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+
(-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*(-b*e
+2*c*d-2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2
*(e*x+d)*c+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c^3*d^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*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e
*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(-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*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-4*a*c*e^2+b^2*e^2)^(1/2))/c))/(-4*a*c*e^2+b^
2*e^2)^(1/2)-3/2*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*(-b*e+2*c*d-2*(e*x+d)*c
+(-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)*Ei(1,1/2*(-b*e+2*c*d-
2*(e*x+d)*c+(-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*(b
*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(1/c*e^2*exp(e*x+d)+1/2/c^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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-4
*a*c*e^2+b^2*e^2)^(1/2))/c)*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*(b*e-2*c*d+
2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*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*(-b*e+2*c*d-2*(e*x+d)*c+(-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*(-b*e+2*c*d-2*(e*x+d)*c+(-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+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*(b*e-2*c*d+2*(e*x+d)*c+(-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*(b*e-2*c*d+2*(e*x+d)*c+(-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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (c e x^{3} e^{d} - c x^{2} e^{d} - b x e^{d}\right )} e^{\left (e x\right )}}{c^{2} e^{2} x^{2} + b c e^{2} x + a c e^{2}} - \int -\frac {{\left ({\left (b e e^{d} + 2 \, c e^{d}\right )} a x + {\left (b^{2} e e^{d} - 2 \, a c e e^{d}\right )} x^{2} + a b e^{d}\right )} e^{\left (e x\right )}}{c^{3} e^{2} x^{4} + 2 \, b c^{2} e^{2} x^{3} + 2 \, a b c e^{2} x + a^{2} c e^{2} + {\left (b^{2} c e^{2} + 2 \, a c^{2} e^{2}\right )} x^{2}}\,{d x} \]
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{d} \int \frac {x^{3} e^{e x}}{a + b x + c x^{2}}\, dx \]
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]
exp(d)*Integral(x**3*exp(e*x)/(a + b*x + c*x**2), x)
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