3.19.27 \(\int \frac {1}{9} e^{\frac {1}{3} (-12+15 x^2-80 e^{2 x/3} x^2-160 e^{x/3} x^3-80 x^4)} (9+90 x^2-960 x^4+e^{2 x/3} (-480 x^2-160 x^3)+e^{x/3} (-1440 x^3-160 x^4)) \, dx\)

Optimal. Leaf size=28 \[ e^{-4+\frac {5}{3} x^2 \left (3-16 \left (e^{x/3}+x\right )^2\right )} x \]

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Rubi [B]  time = 0.91, antiderivative size = 147, normalized size of antiderivative = 5.25, number of steps used = 2, number of rules used = 2, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {12, 2288} \begin {gather*} \frac {\left (-96 x^4+9 x^2-16 e^{x/3} \left (x^4+9 x^3\right )-16 e^{2 x/3} \left (x^3+3 x^2\right )\right ) \exp \left (\frac {1}{3} \left (-80 x^4-160 e^{x/3} x^3-80 e^{2 x/3} x^2+15 x^2-12\right )\right )}{-16 e^{x/3} x^3-96 x^3-144 e^{x/3} x^2-16 e^{2 x/3} x^2-48 e^{2 x/3} x+9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-12 + 15*x^2 - 80*E^((2*x)/3)*x^2 - 160*E^(x/3)*x^3 - 80*x^4)/3)*(9 + 90*x^2 - 960*x^4 + E^((2*x)/3)*
(-480*x^2 - 160*x^3) + E^(x/3)*(-1440*x^3 - 160*x^4)))/9,x]

[Out]

(E^((-12 + 15*x^2 - 80*E^((2*x)/3)*x^2 - 160*E^(x/3)*x^3 - 80*x^4)/3)*(9*x^2 - 96*x^4 - 16*E^((2*x)/3)*(3*x^2
+ x^3) - 16*E^(x/3)*(9*x^3 + x^4)))/(9*x - 48*E^((2*x)/3)*x - 144*E^(x/3)*x^2 - 16*E^((2*x)/3)*x^2 - 96*x^3 -
16*E^(x/3)*x^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \exp \left (\frac {1}{3} \left (-12+15 x^2-80 e^{2 x/3} x^2-160 e^{x/3} x^3-80 x^4\right )\right ) \left (9+90 x^2-960 x^4+e^{2 x/3} \left (-480 x^2-160 x^3\right )+e^{x/3} \left (-1440 x^3-160 x^4\right )\right ) \, dx\\ &=\frac {\exp \left (\frac {1}{3} \left (-12+15 x^2-80 e^{2 x/3} x^2-160 e^{x/3} x^3-80 x^4\right )\right ) \left (9 x^2-96 x^4-16 e^{2 x/3} \left (3 x^2+x^3\right )-16 e^{x/3} \left (9 x^3+x^4\right )\right )}{9 x-48 e^{2 x/3} x-144 e^{x/3} x^2-16 e^{2 x/3} x^2-96 x^3-16 e^{x/3} x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 42, normalized size = 1.50 \begin {gather*} e^{\frac {1}{3} \left (-12+\left (15-80 e^{2 x/3}\right ) x^2-160 e^{x/3} x^3-80 x^4\right )} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-12 + 15*x^2 - 80*E^((2*x)/3)*x^2 - 160*E^(x/3)*x^3 - 80*x^4)/3)*(9 + 90*x^2 - 960*x^4 + E^((2*
x)/3)*(-480*x^2 - 160*x^3) + E^(x/3)*(-1440*x^3 - 160*x^4)))/9,x]

[Out]

E^((-12 + (15 - 80*E^((2*x)/3))*x^2 - 160*E^(x/3)*x^3 - 80*x^4)/3)*x

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fricas [A]  time = 0.86, size = 33, normalized size = 1.18 \begin {gather*} x e^{\left (-\frac {80}{3} \, x^{4} - \frac {160}{3} \, x^{3} e^{\left (\frac {1}{3} \, x\right )} - \frac {80}{3} \, x^{2} e^{\left (\frac {2}{3} \, x\right )} + 5 \, x^{2} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-160*x^3-480*x^2)*exp(1/3*x)^2+(-160*x^4-1440*x^3)*exp(1/3*x)-960*x^4+90*x^2+9)*exp(-80/3*x^2*
exp(1/3*x)^2-160/3*x^3*exp(1/3*x)-80/3*x^4+5*x^2-4),x, algorithm="fricas")

[Out]

x*e^(-80/3*x^4 - 160/3*x^3*e^(1/3*x) - 80/3*x^2*e^(2/3*x) + 5*x^2 - 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {1}{9} \, {\left (960 \, x^{4} - 90 \, x^{2} + 160 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{\left (\frac {2}{3} \, x\right )} + 160 \, {\left (x^{4} + 9 \, x^{3}\right )} e^{\left (\frac {1}{3} \, x\right )} - 9\right )} e^{\left (-\frac {80}{3} \, x^{4} - \frac {160}{3} \, x^{3} e^{\left (\frac {1}{3} \, x\right )} - \frac {80}{3} \, x^{2} e^{\left (\frac {2}{3} \, x\right )} + 5 \, x^{2} - 4\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-160*x^3-480*x^2)*exp(1/3*x)^2+(-160*x^4-1440*x^3)*exp(1/3*x)-960*x^4+90*x^2+9)*exp(-80/3*x^2*
exp(1/3*x)^2-160/3*x^3*exp(1/3*x)-80/3*x^4+5*x^2-4),x, algorithm="giac")

[Out]

integrate(-1/9*(960*x^4 - 90*x^2 + 160*(x^3 + 3*x^2)*e^(2/3*x) + 160*(x^4 + 9*x^3)*e^(1/3*x) - 9)*e^(-80/3*x^4
- 160/3*x^3*e^(1/3*x) - 80/3*x^2*e^(2/3*x) + 5*x^2 - 4), x)

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maple [A]  time = 0.10, size = 34, normalized size = 1.21

method	result	size
risch	\(x \,{\mathrm e}^{-\frac {80 x^{2} {\mathrm e}^{\frac {2 x}{3}}}{3}-\frac {160 x^{3} {\mathrm e}^{\frac {x}{3}}}{3}-\frac {80 x^{4}}{3}+5 x^{2}-4}\)	\(34\)
norman	\(x \,{\mathrm e}^{-\frac {80 x^{2} {\mathrm e}^{\frac {2 x}{3}}}{3}-\frac {160 x^{3} {\mathrm e}^{\frac {x}{3}}}{3}-\frac {80 x^{4}}{3}+5 x^{2}-4}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((-160*x^3-480*x^2)*exp(1/3*x)^2+(-160*x^4-1440*x^3)*exp(1/3*x)-960*x^4+90*x^2+9)*exp(-80/3*x^2*exp(1/
3*x)^2-160/3*x^3*exp(1/3*x)-80/3*x^4+5*x^2-4),x,method=_RETURNVERBOSE)

[Out]

x*exp(-80/3*x^2*exp(2/3*x)-160/3*x^3*exp(1/3*x)-80/3*x^4+5*x^2-4)

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maxima [A]  time = 0.70, size = 33, normalized size = 1.18 \begin {gather*} x e^{\left (-\frac {80}{3} \, x^{4} - \frac {160}{3} \, x^{3} e^{\left (\frac {1}{3} \, x\right )} - \frac {80}{3} \, x^{2} e^{\left (\frac {2}{3} \, x\right )} + 5 \, x^{2} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-160*x^3-480*x^2)*exp(1/3*x)^2+(-160*x^4-1440*x^3)*exp(1/3*x)-960*x^4+90*x^2+9)*exp(-80/3*x^2*
exp(1/3*x)^2-160/3*x^3*exp(1/3*x)-80/3*x^4+5*x^2-4),x, algorithm="maxima")

[Out]

x*e^(-80/3*x^4 - 160/3*x^3*e^(1/3*x) - 80/3*x^2*e^(2/3*x) + 5*x^2 - 4)

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mupad [B]  time = 1.28, size = 36, normalized size = 1.29 \begin {gather*} x\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{5\,x^2}\,{\mathrm {e}}^{-\frac {80\,x^4}{3}}\,{\mathrm {e}}^{-\frac {80\,x^2\,{\mathrm {e}}^{\frac {2\,x}{3}}}{3}}\,{\mathrm {e}}^{-\frac {160\,x^3\,{\mathrm {e}}^{x/3}}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5*x^2 - (160*x^3*exp(x/3))/3 - (80*x^2*exp((2*x)/3))/3 - (80*x^4)/3 - 4)*(exp((2*x)/3)*(480*x^2 + 16
0*x^3) + exp(x/3)*(1440*x^3 + 160*x^4) - 90*x^2 + 960*x^4 - 9))/9,x)

[Out]

x*exp(-4)*exp(5*x^2)*exp(-(80*x^4)/3)*exp(-(80*x^2*exp((2*x)/3))/3)*exp(-(160*x^3*exp(x/3))/3)

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sympy [A]  time = 0.34, size = 41, normalized size = 1.46 \begin {gather*} x e^{- \frac {80 x^{4}}{3} - \frac {160 x^{3} e^{\frac {x}{3}}}{3} - \frac {80 x^{2} e^{\frac {2 x}{3}}}{3} + 5 x^{2} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((-160*x**3-480*x**2)*exp(1/3*x)**2+(-160*x**4-1440*x**3)*exp(1/3*x)-960*x**4+90*x**2+9)*exp(-80
/3*x**2*exp(1/3*x)**2-160/3*x**3*exp(1/3*x)-80/3*x**4+5*x**2-4),x)

[Out]

x*exp(-80*x**4/3 - 160*x**3*exp(x/3)/3 - 80*x**2*exp(2*x/3)/3 + 5*x**2 - 4)

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