3.89.60 \(\int \frac {e^{-x} (-3+3 x-19 x^2-10 x^3)}{9-54 x+21 x^2+180 x^3+100 x^4} \, dx\)

Optimal. Leaf size=22 \[ \frac {e^{-x} x}{-3-9 (-1-x) x+x^2} \]

________________________________________________________________________________________

Rubi [C]  time = 1.18, antiderivative size = 503, normalized size of antiderivative = 22.86, number of steps used = 25, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {6688, 6742, 2177, 2178, 2270} \begin {gather*} \frac {3 \left (9+\sqrt {201}\right ) e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x-\sqrt {201}-9\right )\right )}{1340}-\frac {\left (201-11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x-\sqrt {201}-9\right )\right )}{4020}-\frac {40 e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x-\sqrt {201}-9\right )\right )}{67 \sqrt {201}}-\frac {9}{67} \sqrt {\frac {3}{67}} e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x-\sqrt {201}-9\right )\right )+\frac {2}{67} e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x-\sqrt {201}-9\right )\right )-\frac {\left (201+11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x+\sqrt {201}-9\right )\right )}{4020}+\frac {3 \left (9-\sqrt {201}\right ) e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x+\sqrt {201}-9\right )\right )}{1340}+\frac {40 e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x+\sqrt {201}-9\right )\right )}{67 \sqrt {201}}+\frac {9}{67} \sqrt {\frac {3}{67}} e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x+\sqrt {201}-9\right )\right )+\frac {2}{67} e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-20 x+\sqrt {201}-9\right )\right )+\frac {3 \left (9-\sqrt {201}\right ) e^{-x}}{67 \left (20 x-\sqrt {201}+9\right )}+\frac {40 e^{-x}}{67 \left (20 x-\sqrt {201}+9\right )}+\frac {3 \left (9+\sqrt {201}\right ) e^{-x}}{67 \left (20 x+\sqrt {201}+9\right )}+\frac {40 e^{-x}}{67 \left (20 x+\sqrt {201}+9\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 3*x - 19*x^2 - 10*x^3)/(E^x*(9 - 54*x + 21*x^2 + 180*x^3 + 100*x^4)),x]

[Out]

40/(67*E^x*(9 - Sqrt[201] + 20*x)) + (3*(9 - Sqrt[201]))/(67*E^x*(9 - Sqrt[201] + 20*x)) + 40/(67*E^x*(9 + Sqr
t[201] + 20*x)) + (3*(9 + Sqrt[201]))/(67*E^x*(9 + Sqrt[201] + 20*x)) + (2*E^((9 + Sqrt[201])/20)*ExpIntegralE
i[(-9 - Sqrt[201] - 20*x)/20])/67 - (9*Sqrt[3/67]*E^((9 + Sqrt[201])/20)*ExpIntegralEi[(-9 - Sqrt[201] - 20*x)
/20])/67 - (40*E^((9 + Sqrt[201])/20)*ExpIntegralEi[(-9 - Sqrt[201] - 20*x)/20])/(67*Sqrt[201]) - ((201 - 11*S
qrt[201])*E^((9 + Sqrt[201])/20)*ExpIntegralEi[(-9 - Sqrt[201] - 20*x)/20])/4020 + (3*(9 + Sqrt[201])*E^((9 +
Sqrt[201])/20)*ExpIntegralEi[(-9 - Sqrt[201] - 20*x)/20])/1340 + (2*E^((9 - Sqrt[201])/20)*ExpIntegralEi[(-9 +
Sqrt[201] - 20*x)/20])/67 + (9*Sqrt[3/67]*E^((9 - Sqrt[201])/20)*ExpIntegralEi[(-9 + Sqrt[201] - 20*x)/20])/6
7 + (40*E^((9 - Sqrt[201])/20)*ExpIntegralEi[(-9 + Sqrt[201] - 20*x)/20])/(67*Sqrt[201]) + (3*(9 - Sqrt[201])*
E^((9 - Sqrt[201])/20)*ExpIntegralEi[(-9 + Sqrt[201] - 20*x)/20])/1340 - ((201 + 11*Sqrt[201])*E^((9 - Sqrt[20
1])/20)*ExpIntegralEi[(-9 + Sqrt[201] - 20*x)/20])/4020

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), 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] && LtQ[m, -1] && 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 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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-3+3 x-19 x^2-10 x^3\right )}{\left (3-9 x-10 x^2\right )^2} \, dx\\ &=\int \left (\frac {3 e^{-x} (-2+3 x)}{\left (-3+9 x+10 x^2\right )^2}+\frac {e^{-x} (-1-x)}{-3+9 x+10 x^2}\right ) \, dx\\ &=3 \int \frac {e^{-x} (-2+3 x)}{\left (-3+9 x+10 x^2\right )^2} \, dx+\int \frac {e^{-x} (-1-x)}{-3+9 x+10 x^2} \, dx\\ &=3 \int \left (-\frac {2 e^{-x}}{\left (-3+9 x+10 x^2\right )^2}+\frac {3 e^{-x} x}{\left (-3+9 x+10 x^2\right )^2}\right ) \, dx+\int \left (\frac {\left (-1-\frac {11}{\sqrt {201}}\right ) e^{-x}}{9-\sqrt {201}+20 x}+\frac {\left (-1+\frac {11}{\sqrt {201}}\right ) e^{-x}}{9+\sqrt {201}+20 x}\right ) \, dx\\ &=-\left (6 \int \frac {e^{-x}}{\left (-3+9 x+10 x^2\right )^2} \, dx\right )+9 \int \frac {e^{-x} x}{\left (-3+9 x+10 x^2\right )^2} \, dx+\frac {1}{201} \left (-201+11 \sqrt {201}\right ) \int \frac {e^{-x}}{9+\sqrt {201}+20 x} \, dx-\frac {1}{201} \left (201+11 \sqrt {201}\right ) \int \frac {e^{-x}}{9-\sqrt {201}+20 x} \, dx\\ &=-\frac {\left (201-11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )}{4020}-\frac {\left (201+11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )}{4020}-6 \int \left (\frac {400 e^{-x}}{201 \left (-9+\sqrt {201}-20 x\right )^2}+\frac {400 e^{-x}}{201 \sqrt {201} \left (-9+\sqrt {201}-20 x\right )}+\frac {400 e^{-x}}{201 \left (9+\sqrt {201}+20 x\right )^2}+\frac {400 e^{-x}}{201 \sqrt {201} \left (9+\sqrt {201}+20 x\right )}\right ) \, dx+9 \int \left (\frac {20 \left (-9+\sqrt {201}\right ) e^{-x}}{201 \left (-9+\sqrt {201}-20 x\right )^2}-\frac {20 \sqrt {\frac {3}{67}} e^{-x}}{67 \left (-9+\sqrt {201}-20 x\right )}+\frac {20 \left (-9-\sqrt {201}\right ) e^{-x}}{201 \left (9+\sqrt {201}+20 x\right )^2}-\frac {20 \sqrt {\frac {3}{67}} e^{-x}}{67 \left (9+\sqrt {201}+20 x\right )}\right ) \, dx\\ &=-\frac {\left (201-11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )}{4020}-\frac {\left (201+11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )}{4020}-\frac {800}{67} \int \frac {e^{-x}}{\left (-9+\sqrt {201}-20 x\right )^2} \, dx-\frac {800}{67} \int \frac {e^{-x}}{\left (9+\sqrt {201}+20 x\right )^2} \, dx-\frac {1}{67} \left (180 \sqrt {\frac {3}{67}}\right ) \int \frac {e^{-x}}{-9+\sqrt {201}-20 x} \, dx-\frac {1}{67} \left (180 \sqrt {\frac {3}{67}}\right ) \int \frac {e^{-x}}{9+\sqrt {201}+20 x} \, dx-\frac {800 \int \frac {e^{-x}}{-9+\sqrt {201}-20 x} \, dx}{67 \sqrt {201}}-\frac {800 \int \frac {e^{-x}}{9+\sqrt {201}+20 x} \, dx}{67 \sqrt {201}}-\frac {1}{67} \left (60 \left (9-\sqrt {201}\right )\right ) \int \frac {e^{-x}}{\left (-9+\sqrt {201}-20 x\right )^2} \, dx-\frac {1}{67} \left (60 \left (9+\sqrt {201}\right )\right ) \int \frac {e^{-x}}{\left (9+\sqrt {201}+20 x\right )^2} \, dx\\ &=\frac {40 e^{-x}}{67 \left (9-\sqrt {201}+20 x\right )}+\frac {3 \left (9-\sqrt {201}\right ) e^{-x}}{67 \left (9-\sqrt {201}+20 x\right )}+\frac {40 e^{-x}}{67 \left (9+\sqrt {201}+20 x\right )}+\frac {3 \left (9+\sqrt {201}\right ) e^{-x}}{67 \left (9+\sqrt {201}+20 x\right )}-\frac {9}{67} \sqrt {\frac {3}{67}} e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )-\frac {40 e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )}{67 \sqrt {201}}-\frac {\left (201-11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )}{4020}+\frac {9}{67} \sqrt {\frac {3}{67}} e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )+\frac {40 e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )}{67 \sqrt {201}}-\frac {\left (201+11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )}{4020}-\frac {40}{67} \int \frac {e^{-x}}{-9+\sqrt {201}-20 x} \, dx+\frac {40}{67} \int \frac {e^{-x}}{9+\sqrt {201}+20 x} \, dx-\frac {1}{67} \left (3 \left (9-\sqrt {201}\right )\right ) \int \frac {e^{-x}}{-9+\sqrt {201}-20 x} \, dx+\frac {1}{67} \left (3 \left (9+\sqrt {201}\right )\right ) \int \frac {e^{-x}}{9+\sqrt {201}+20 x} \, dx\\ &=\frac {40 e^{-x}}{67 \left (9-\sqrt {201}+20 x\right )}+\frac {3 \left (9-\sqrt {201}\right ) e^{-x}}{67 \left (9-\sqrt {201}+20 x\right )}+\frac {40 e^{-x}}{67 \left (9+\sqrt {201}+20 x\right )}+\frac {3 \left (9+\sqrt {201}\right ) e^{-x}}{67 \left (9+\sqrt {201}+20 x\right )}+\frac {2}{67} e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )-\frac {9}{67} \sqrt {\frac {3}{67}} e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )-\frac {40 e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )}{67 \sqrt {201}}-\frac {\left (201-11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )}{4020}+\frac {3 \left (9+\sqrt {201}\right ) e^{\frac {1}{20} \left (9+\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9-\sqrt {201}-20 x\right )\right )}{1340}+\frac {2}{67} e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )+\frac {9}{67} \sqrt {\frac {3}{67}} e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )+\frac {40 e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )}{67 \sqrt {201}}+\frac {3 \left (9-\sqrt {201}\right ) e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )}{1340}-\frac {\left (201+11 \sqrt {201}\right ) e^{\frac {1}{20} \left (9-\sqrt {201}\right )} \text {Ei}\left (\frac {1}{20} \left (-9+\sqrt {201}-20 x\right )\right )}{4020}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.27, size = 19, normalized size = 0.86 \begin {gather*} \frac {e^{-x} x}{-3+9 x+10 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 3*x - 19*x^2 - 10*x^3)/(E^x*(9 - 54*x + 21*x^2 + 180*x^3 + 100*x^4)),x]

[Out]

x/(E^x*(-3 + 9*x + 10*x^2))

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 18, normalized size = 0.82 \begin {gather*} \frac {x e^{\left (-x\right )}}{10 \, x^{2} + 9 \, x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-19*x^2+3*x-3)/(100*x^4+180*x^3+21*x^2-54*x+9)/exp(x),x, algorithm="fricas")

[Out]

x*e^(-x)/(10*x^2 + 9*x - 3)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 18, normalized size = 0.82 \begin {gather*} \frac {x e^{\left (-x\right )}}{10 \, x^{2} + 9 \, x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-19*x^2+3*x-3)/(100*x^4+180*x^3+21*x^2-54*x+9)/exp(x),x, algorithm="giac")

[Out]

x*e^(-x)/(10*x^2 + 9*x - 3)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 19, normalized size = 0.86

method	result	size
gosper	\(\frac {x \,{\mathrm e}^{-x}}{10 x^{2}+9 x -3}\)	\(19\)
norman	\(\frac {x \,{\mathrm e}^{-x}}{10 x^{2}+9 x -3}\)	\(19\)
risch	\(\frac {x \,{\mathrm e}^{-x}}{10 x^{2}+9 x -3}\)	\(19\)
default	\(\frac {{\mathrm e}^{-x} \left (20 x +9\right )}{670 x^{2}+603 x -201}+\frac {3 \,{\mathrm e}^{-x} \left (3 x -2\right )}{67 \left (10 x^{2}+9 x -3\right )}+\frac {19 \,{\mathrm e}^{-x} \left (47 x -9\right )}{670 \left (10 x^{2}+9 x -3\right )}-\frac {3 \,{\mathrm e}^{-x} \left (171 x -47\right )}{670 \left (10 x^{2}+9 x -3\right )}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*x^3-19*x^2+3*x-3)/(100*x^4+180*x^3+21*x^2-54*x+9)/exp(x),x,method=_RETURNVERBOSE)

[Out]

x/exp(x)/(10*x^2+9*x-3)

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 18, normalized size = 0.82 \begin {gather*} \frac {x e^{\left (-x\right )}}{10 \, x^{2} + 9 \, x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-19*x^2+3*x-3)/(100*x^4+180*x^3+21*x^2-54*x+9)/exp(x),x, algorithm="maxima")

[Out]

x*e^(-x)/(10*x^2 + 9*x - 3)

________________________________________________________________________________________

mupad [B]  time = 0.21, size = 18, normalized size = 0.82 \begin {gather*} \frac {x\,{\mathrm {e}}^{-x}}{10\,x^2+9\,x-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(19*x^2 - 3*x + 10*x^3 + 3))/(21*x^2 - 54*x + 180*x^3 + 100*x^4 + 9),x)

[Out]

(x*exp(-x))/(9*x + 10*x^2 - 3)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 14, normalized size = 0.64 \begin {gather*} \frac {x e^{- x}}{10 x^{2} + 9 x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x**3-19*x**2+3*x-3)/(100*x**4+180*x**3+21*x**2-54*x+9)/exp(x),x)

[Out]

x*exp(-x)/(10*x**2 + 9*x - 3)

________________________________________________________________________________________