3.45.18 \(\int \frac {e^{-4+2 x} (4-x^2)^2 (4000-4300 x+200 x^2+350 x^3-50 x^4+\frac {e^{4-2 x} (324-81 x^2)}{(4-x^2)^2})}{-324+81 x^2} \, dx\)

Optimal. Leaf size=32 \[ 5-x-\frac {25}{81} e^{-4+2 x} (5-x)^2 \left (4-x^2\right )^2 \]

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Rubi [B]  time = 0.85, antiderivative size = 97, normalized size of antiderivative = 3.03, number of steps used = 33, number of rules used = 5, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {21, 6688, 2196, 2194, 2176} \begin {gather*} -\frac {25}{81} e^{2 x-4} x^6+\frac {250}{81} e^{2 x-4} x^5-\frac {425}{81} e^{2 x-4} x^4-\frac {2000}{81} e^{2 x-4} x^3+\frac {4600}{81} e^{2 x-4} x^2+\frac {4000}{81} e^{2 x-4} x-x-\frac {10000}{81} e^{2 x-4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-4 + 2*x)*(4 - x^2)^2*(4000 - 4300*x + 200*x^2 + 350*x^3 - 50*x^4 + (E^(4 - 2*x)*(324 - 81*x^2))/(4 -
x^2)^2))/(-324 + 81*x^2),x]

[Out]

(-10000*E^(-4 + 2*x))/81 - x + (4000*E^(-4 + 2*x)*x)/81 + (4600*E^(-4 + 2*x)*x^2)/81 - (2000*E^(-4 + 2*x)*x^3)
/81 - (425*E^(-4 + 2*x)*x^4)/81 + (250*E^(-4 + 2*x)*x^5)/81 - (25*E^(-4 + 2*x)*x^6)/81

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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 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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\frac {1}{81} \int e^{-4+2 x} \left (4-x^2\right ) \left (4000-4300 x+200 x^2+350 x^3-50 x^4+\frac {e^{4-2 x} \left (324-81 x^2\right )}{\left (4-x^2\right )^2}\right ) \, dx\right )\\ &=-\left (\frac {1}{81} \int \left (81+50 e^{-4+2 x} \left (320-344 x-64 x^2+114 x^3-8 x^4-7 x^5+x^6\right )\right ) \, dx\right )\\ &=-x-\frac {50}{81} \int e^{-4+2 x} \left (320-344 x-64 x^2+114 x^3-8 x^4-7 x^5+x^6\right ) \, dx\\ &=-x-\frac {50}{81} \int \left (320 e^{-4+2 x}-344 e^{-4+2 x} x-64 e^{-4+2 x} x^2+114 e^{-4+2 x} x^3-8 e^{-4+2 x} x^4-7 e^{-4+2 x} x^5+e^{-4+2 x} x^6\right ) \, dx\\ &=-x-\frac {50}{81} \int e^{-4+2 x} x^6 \, dx+\frac {350}{81} \int e^{-4+2 x} x^5 \, dx+\frac {400}{81} \int e^{-4+2 x} x^4 \, dx+\frac {3200}{81} \int e^{-4+2 x} x^2 \, dx-\frac {1900}{27} \int e^{-4+2 x} x^3 \, dx-\frac {16000}{81} \int e^{-4+2 x} \, dx+\frac {17200}{81} \int e^{-4+2 x} x \, dx\\ &=-\frac {8000}{81} e^{-4+2 x}-x+\frac {8600}{81} e^{-4+2 x} x+\frac {1600}{81} e^{-4+2 x} x^2-\frac {950}{27} e^{-4+2 x} x^3+\frac {200}{81} e^{-4+2 x} x^4+\frac {175}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6+\frac {50}{27} \int e^{-4+2 x} x^5 \, dx-\frac {800}{81} \int e^{-4+2 x} x^3 \, dx-\frac {875}{81} \int e^{-4+2 x} x^4 \, dx-\frac {3200}{81} \int e^{-4+2 x} x \, dx+\frac {950}{9} \int e^{-4+2 x} x^2 \, dx-\frac {8600}{81} \int e^{-4+2 x} \, dx\\ &=-\frac {4100}{27} e^{-4+2 x}-x+\frac {7000}{81} e^{-4+2 x} x+\frac {5875}{81} e^{-4+2 x} x^2-\frac {3250}{81} e^{-4+2 x} x^3-\frac {475}{162} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6-\frac {125}{27} \int e^{-4+2 x} x^4 \, dx+\frac {400}{27} \int e^{-4+2 x} x^2 \, dx+\frac {1600}{81} \int e^{-4+2 x} \, dx+\frac {1750}{81} \int e^{-4+2 x} x^3 \, dx-\frac {950}{9} \int e^{-4+2 x} x \, dx\\ &=-\frac {11500}{81} e^{-4+2 x}-x+\frac {2725}{81} e^{-4+2 x} x+\frac {6475}{81} e^{-4+2 x} x^2-\frac {2375}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6+\frac {250}{27} \int e^{-4+2 x} x^3 \, dx-\frac {400}{27} \int e^{-4+2 x} x \, dx-\frac {875}{27} \int e^{-4+2 x} x^2 \, dx+\frac {475}{9} \int e^{-4+2 x} \, dx\\ &=-\frac {18725}{162} e^{-4+2 x}-x+\frac {2125}{81} e^{-4+2 x} x+\frac {10325}{162} e^{-4+2 x} x^2-\frac {2000}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6+\frac {200}{27} \int e^{-4+2 x} \, dx-\frac {125}{9} \int e^{-4+2 x} x^2 \, dx+\frac {875}{27} \int e^{-4+2 x} x \, dx\\ &=-\frac {18125}{162} e^{-4+2 x}-x+\frac {6875}{162} e^{-4+2 x} x+\frac {4600}{81} e^{-4+2 x} x^2-\frac {2000}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6+\frac {125}{9} \int e^{-4+2 x} x \, dx-\frac {875}{54} \int e^{-4+2 x} \, dx\\ &=-\frac {38875}{324} e^{-4+2 x}-x+\frac {4000}{81} e^{-4+2 x} x+\frac {4600}{81} e^{-4+2 x} x^2-\frac {2000}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6-\frac {125}{18} \int e^{-4+2 x} \, dx\\ &=-\frac {10000}{81} e^{-4+2 x}-x+\frac {4000}{81} e^{-4+2 x} x+\frac {4600}{81} e^{-4+2 x} x^2-\frac {2000}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 36, normalized size = 1.12 \begin {gather*} -\frac {81 e^4 x+25 e^{2 x} \left (20-4 x-5 x^2+x^3\right )^2}{81 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 + 2*x)*(4 - x^2)^2*(4000 - 4300*x + 200*x^2 + 350*x^3 - 50*x^4 + (E^(4 - 2*x)*(324 - 81*x^2))
/(4 - x^2)^2))/(-324 + 81*x^2),x]

[Out]

-1/81*(81*E^4*x + 25*E^(2*x)*(20 - 4*x - 5*x^2 + x^3)^2)/E^4

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fricas [A]  time = 0.66, size = 30, normalized size = 0.94 \begin {gather*} -\frac {25}{81} \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x + 2 \, \log \left (-x^{2} + 4\right ) - 4\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-81*x^2+324)*exp(-log(-x^2+4)+2-x)^2-50*x^4+350*x^3+200*x^2-4300*x+4000)/(81*x^2-324)/exp(-log(-x^
2+4)+2-x)^2,x, algorithm="fricas")

[Out]

-25/81*(x^2 - 10*x + 25)*e^(2*x + 2*log(-x^2 + 4) - 4) - x

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giac [B]  time = 0.16, size = 68, normalized size = 2.12 \begin {gather*} -\frac {1}{81} \, {\left (25 \, x^{6} e^{\left (2 \, x\right )} - 250 \, x^{5} e^{\left (2 \, x\right )} + 425 \, x^{4} e^{\left (2 \, x\right )} + 2000 \, x^{3} e^{\left (2 \, x\right )} - 4600 \, x^{2} e^{\left (2 \, x\right )} + 81 \, x e^{4} - 4000 \, x e^{\left (2 \, x\right )} + 10000 \, e^{\left (2 \, x\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-81*x^2+324)*exp(-log(-x^2+4)+2-x)^2-50*x^4+350*x^3+200*x^2-4300*x+4000)/(81*x^2-324)/exp(-log(-x^
2+4)+2-x)^2,x, algorithm="giac")

[Out]

-1/81*(25*x^6*e^(2*x) - 250*x^5*e^(2*x) + 425*x^4*e^(2*x) + 2000*x^3*e^(2*x) - 4600*x^2*e^(2*x) + 81*x*e^4 - 4
000*x*e^(2*x) + 10000*e^(2*x))*e^(-4)

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maple [A]  time = 0.22, size = 42, normalized size = 1.31

method	result	size
risch	\(-x +\left (-\frac {10000}{81}+\frac {4000}{81} x +\frac {4600}{81} x^{2}-\frac {2000}{81} x^{3}-\frac {425}{81} x^{4}+\frac {250}{81} x^{5}-\frac {25}{81} x^{6}\right ) {\mathrm e}^{2 x -4}\)	\(42\)
default	\(-x -\frac {100 \,{\mathrm e}^{2 x -4} \left (2 x -4\right )^{2}}{9}+\frac {25 \left (2 x -4\right )^{3} {\mathrm e}^{2 x -4}}{27}+\frac {575 \left (2 x -4\right )^{4} {\mathrm e}^{2 x -4}}{1296}-\frac {25 \left (2 x -4\right )^{5} {\mathrm e}^{2 x -4}}{1296}-\frac {25 \,{\mathrm e}^{2 x -4} \left (2 x -4\right )^{6}}{5184}\)	\(80\)
meijerg	\(-\frac {250 \,2^{2+4 \,{\mathrm e}^{-4}} {\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} \hypergeom \left (\left [\frac {1}{2}, 1-2 \,{\mathrm e}^{-4}\right ], \left [\frac {3}{2}\right ], \frac {x^{2}}{4}\right ) x \,{\mathrm e}^{-4}}{81}+\frac {2 \,2^{-3-4 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{-4}} {\mathrm e}^{4} \left (-x^{2}+4\right )^{2 \,{\mathrm e}^{4}+2-2 \,{\mathrm e}^{-4}} \hypergeom \left (\left [\frac {3}{2}, -2 \,{\mathrm e}^{-4}+2 \,{\mathrm e}^{4}+1\right ], \left [\frac {5}{2}\right ], \frac {x^{2}}{4}\right ) x^{3} {\mathrm e}^{-4}}{3 \left (x^{2}-4\right )^{2}}-\frac {2 \,2^{-1-4 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{-4}} {\mathrm e}^{4} \left (-x^{2}+4\right )^{2 \,{\mathrm e}^{4}+2-2 \,{\mathrm e}^{-4}} \hypergeom \left (\left [\frac {1}{2}, -2 \,{\mathrm e}^{-4}+2 \,{\mathrm e}^{4}+1\right ], \left [\frac {3}{2}\right ], \frac {x^{2}}{4}\right ) x \,{\mathrm e}^{-4}}{\left (x^{2}-4\right )^{2}}+\frac {10 \,2^{-2+4 \,{\mathrm e}^{-4}} {\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} \hypergeom \left (\left [\frac {5}{2}, 1-2 \,{\mathrm e}^{-4}\right ], \left [\frac {7}{2}\right ], \frac {x^{2}}{4}\right ) x^{5} {\mathrm e}^{-4}}{81}-\frac {350 \,2^{-4+4 \,{\mathrm e}^{-4}} {\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} \hypergeom \left (\left [2, 1-2 \,{\mathrm e}^{-4}\right ], \relax [3], \frac {x^{2}}{4}\right ) x^{4} {\mathrm e}^{-4}}{81}+\frac {2150 \,2^{-2+4 \,{\mathrm e}^{-4}} {\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} \hypergeom \left (\left [1, 1-2 \,{\mathrm e}^{-4}\right ], \relax [2], \frac {x^{2}}{4}\right ) x^{2} {\mathrm e}^{-4}}{81}-\frac {50 \,{\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} 2^{4 \,{\mathrm e}^{-4}} \hypergeom \left (\left [\frac {3}{2}, 1-2 \,{\mathrm e}^{-4}\right ], \left [\frac {5}{2}\right ], \frac {x^{2}}{4}\right ) x^{3} {\mathrm e}^{-4}}{243}\)	\(373\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-81*x^2+324)*exp(-ln(-x^2+4)+2-x)^2-50*x^4+350*x^3+200*x^2-4300*x+4000)/(81*x^2-324)/exp(-ln(-x^2+4)+2-x
)^2,x,method=_RETURNVERBOSE)

[Out]

-x+(-10000/81+4000/81*x+4600/81*x^2-2000/81*x^3-425/81*x^4+250/81*x^5-25/81*x^6)*exp(2*x-4)

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maxima [B]  time = 0.43, size = 130, normalized size = 4.06 \begin {gather*} -\frac {25}{81} \, {\left (x^{6} - 10 \, x^{5} + 17 \, x^{4} + 80 \, x^{3} - 184 \, x^{2} - 160 \, x + 400\right )} e^{\left (2 \, x - 4\right )} - x + \frac {9 \, x^{3} - 28 \, x}{2 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} - \frac {3 \, {\left (5 \, x^{3} - 12 \, x\right )}}{2 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} + \frac {3 \, x^{3} - 20 \, x}{2 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} + \frac {3 \, {\left (x^{3} + 4 \, x\right )}}{2 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-81*x^2+324)*exp(-log(-x^2+4)+2-x)^2-50*x^4+350*x^3+200*x^2-4300*x+4000)/(81*x^2-324)/exp(-log(-x^
2+4)+2-x)^2,x, algorithm="maxima")

[Out]

-25/81*(x^6 - 10*x^5 + 17*x^4 + 80*x^3 - 184*x^2 - 160*x + 400)*e^(2*x - 4) - x + 1/2*(9*x^3 - 28*x)/(x^4 - 8*
x^2 + 16) - 3/2*(5*x^3 - 12*x)/(x^4 - 8*x^2 + 16) + 1/2*(3*x^3 - 20*x)/(x^4 - 8*x^2 + 16) + 3/2*(x^3 + 4*x)/(x
^4 - 8*x^2 + 16)

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mupad [B]  time = 0.26, size = 76, normalized size = 2.38 \begin {gather*} \frac {4000\,x\,{\mathrm {e}}^{2\,x-4}}{81}-\frac {10000\,{\mathrm {e}}^{2\,x-4}}{81}-x+\frac {4600\,x^2\,{\mathrm {e}}^{2\,x-4}}{81}-\frac {2000\,x^3\,{\mathrm {e}}^{2\,x-4}}{81}-\frac {425\,x^4\,{\mathrm {e}}^{2\,x-4}}{81}+\frac {250\,x^5\,{\mathrm {e}}^{2\,x-4}}{81}-\frac {25\,x^6\,{\mathrm {e}}^{2\,x-4}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x + 2*log(4 - x^2) - 4)*(4300*x - 200*x^2 - 350*x^3 + 50*x^4 + exp(4 - 2*log(4 - x^2) - 2*x)*(81*x
^2 - 324) - 4000))/(81*x^2 - 324),x)

[Out]

(4000*x*exp(2*x - 4))/81 - (10000*exp(2*x - 4))/81 - x + (4600*x^2*exp(2*x - 4))/81 - (2000*x^3*exp(2*x - 4))/
81 - (425*x^4*exp(2*x - 4))/81 + (250*x^5*exp(2*x - 4))/81 - (25*x^6*exp(2*x - 4))/81

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sympy [A]  time = 0.19, size = 39, normalized size = 1.22 \begin {gather*} - x + \frac {\left (- 25 x^{6} + 250 x^{5} - 425 x^{4} - 2000 x^{3} + 4600 x^{2} + 4000 x - 10000\right ) e^{2 x - 4}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-81*x**2+324)*exp(-ln(-x**2+4)+2-x)**2-50*x**4+350*x**3+200*x**2-4300*x+4000)/(81*x**2-324)/exp(-l
n(-x**2+4)+2-x)**2,x)

[Out]

-x + (-25*x**6 + 250*x**5 - 425*x**4 - 2000*x**3 + 4600*x**2 + 4000*x - 10000)*exp(2*x - 4)/81

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