3.5.71 \(\int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx\) [471]

Optimal. Leaf size=186 \[ \frac {e^{d+e x}}{c e}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^2}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^2} \]

[Out]

exp(e*x+d)/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^
2)/(-4*a*c+b^2)^(1/2))/c^2-1/2*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^2)/(-4*a*c+b^2)^(1/2))/c^2

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Rubi [A]
time = 0.29, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2302, 2225, 2209} \begin {gather*} -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\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^2}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\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^2}+\frac {e^{d+e x}}{c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(d + e*x)/(c*e) - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegr
alEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^2) - ((b + (b^2 - 2*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^2)

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

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Mathematica [A]
time = 0.34, size = 217, normalized size = 1.17 \begin {gather*} -\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \left (-2 c \sqrt {b^2-4 a c} e^{\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}}+\left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right ) e e^{\frac {\sqrt {b^2-4 a c} e}{c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+\left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) e \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{2 c^2 \sqrt {b^2-4 a c} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*(-2*c*Sqrt[b^2 - 4*a*c]*E^((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))
/(2*c)) + (-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*e*E^((Sqrt[b^2 - 4*a*c]*e)/c)*ExpIntegralEi[(e*(b - Sqrt[b^2 -
4*a*c] + 2*c*x))/(2*c)] + (b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*e*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*
x))/(2*c)]))/(c^2*Sqrt[b^2 - 4*a*c]*e)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1729\) vs. \(2(160)=320\).
time = 0.11, size = 1730, normalized size = 9.30

method	result	size
risch	\(\frac {{\mathrm e}^{e x +d}}{c e}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a}{c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2}}{2 c^{2} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a}{c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2}}{2 c^{2} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 c^{2}}+\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 c^{2}}\)	\(561\)
derivativedivides	\(\text {Expression too large to display}\)	\(1730\)
default	\(\text {Expression too large to display}\)	\(1730\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^3*(-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*(-b*e+2*c*d-2*c*(e*x+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*(b*e-2*c*d+2*c*(e*x+d)+(-4*
a*c*e^2+b^2*e^2)^(1/2))/c))/(-4*a*c*e^2+b^2*e^2)^(1/2)+e^2/c*exp(e*x+d)+1/2/c^2*e^2*(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*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*(-b*e+2*c*d-2*c*(e*x+d)+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*b^2*e
^2+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*c*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*(-b*e+2*c*d-2*c*(e*x+d)+(-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*c
*(e*x+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*(
b*e-2*c*d+2*c*(e*x+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*(b*e-2*c*d+2*c*(e*x+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*(b*e-2*c*d+2*c*(e*x+d)+(-4*a*c*e^2+b^2*e^2)^(1/2))/c)*c^2*d^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)*(-4*a*c*e^2+b^
2*e^2)^(1/2)*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*(-b*e+2*c*d-2*c*(e*x+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+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)*(-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*c*(e*x+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)+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*(-b*e+2*c*d-2*c*(e*x+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*(-b*e+2*c*d-2*c*(e*x+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*(b*e-2*c*d+2*c*(e*x+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)*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)*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*(-b*e+2*c*d-2*c*(e*x+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*(b*e-2*c*d+2*c*(e*x+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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 266, normalized size = 1.43 \begin {gather*} \frac {{\left ({\left ({\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - {\left (b^{3} - 4 \, a b c\right )} e\right )} {\rm Ei}\left (-\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - {\left (2 \, c x + b\right )} e}{2 \, c}\right ) e^{\left (\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + 2 \, c d - b e}{2 \, c}\right )} - {\left ({\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + {\left (b^{3} - 4 \, a b c\right )} e\right )} {\rm Ei}\left (\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + {\left (2 \, c x + b\right )} e}{2 \, c}\right ) e^{\left (-\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - 2 \, c d + b e}{2 \, c}\right )} + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{\left (x e + d\right )}\right )} e^{\left (-1\right )}}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(((b^2*c - 2*a*c^2)*sqrt((b^2 - 4*a*c)/c^2)*e - (b^3 - 4*a*b*c)*e)*Ei(-1/2*(c*sqrt((b^2 - 4*a*c)/c^2)*e -
(2*c*x + b)*e)/c)*e^(1/2*(c*sqrt((b^2 - 4*a*c)/c^2)*e + 2*c*d - b*e)/c) - ((b^2*c - 2*a*c^2)*sqrt((b^2 - 4*a*c
)/c^2)*e + (b^3 - 4*a*b*c)*e)*Ei(1/2*(c*sqrt((b^2 - 4*a*c)/c^2)*e + (2*c*x + b)*e)/c)*e^(-1/2*(c*sqrt((b^2 - 4
*a*c)/c^2)*e - 2*c*d + b*e)/c) + 2*(b^2*c - 4*a*c^2)*e^(x*e + d))*e^(-1)/(b^2*c^2 - 4*a*c^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{d} \int \frac {x^{2} e^{e x}}{a + b x + c x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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