3.37.32 \(\int \frac {e^{3-\frac {20 e^3 x^3}{-3+2 x}} (180 x^2-80 x^3)}{27-36 x+12 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{3} e^{\frac {20 e^3 x^2}{-2+\frac {3}{x}}} \]

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Rubi [A]  time = 0.40, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {27, 12, 1593, 6706} \begin {gather*} \frac {1}{3} e^{\frac {20 e^3 x^3}{3-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(3 - (20*E^3*x^3)/(-3 + 2*x))*(180*x^2 - 80*x^3))/(27 - 36*x + 12*x^2),x]

[Out]

E^((20*E^3*x^3)/(3 - 2*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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{3-\frac {20 e^3 x^3}{-3+2 x}} \left (180 x^2-80 x^3\right )}{3 (-3+2 x)^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{3-\frac {20 e^3 x^3}{-3+2 x}} \left (180 x^2-80 x^3\right )}{(-3+2 x)^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{3-\frac {20 e^3 x^3}{-3+2 x}} (180-80 x) x^2}{(-3+2 x)^2} \, dx\\ &=\frac {1}{3} e^{\frac {20 e^3 x^3}{3-2 x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 21, normalized size = 0.91 \begin {gather*} \frac {1}{3} e^{\frac {20 e^3 x^3}{3-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 - (20*E^3*x^3)/(-3 + 2*x))*(180*x^2 - 80*x^3))/(27 - 36*x + 12*x^2),x]

[Out]

E^((20*E^3*x^3)/(3 - 2*x))/3

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fricas [A]  time = 0.67, size = 26, normalized size = 1.13 \begin {gather*} \frac {1}{3} \, e^{\left (-\frac {20 \, x^{3} e^{3} - 6 \, x + 9}{2 \, x - 3} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-80*x^3+180*x^2)*exp(3)*exp(-20*x^3*exp(3)/(2*x-3))/(12*x^2-36*x+27),x, algorithm="fricas")

[Out]

1/3*e^(-(20*x^3*e^3 - 6*x + 9)/(2*x - 3) - 3)

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giac [A]  time = 0.18, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, e^{\left (-\frac {20 \, x^{3} e^{3}}{2 \, x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-80*x^3+180*x^2)*exp(3)*exp(-20*x^3*exp(3)/(2*x-3))/(12*x^2-36*x+27),x, algorithm="giac")

[Out]

1/3*e^(-20*x^3*e^3/(2*x - 3))

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maple [A]  time = 0.16, size = 18, normalized size = 0.78

method	result	size
gosper	\(\frac {{\mathrm e}^{-\frac {20 x^{3} {\mathrm e}^{3}}{2 x -3}}}{3}\)	\(18\)
risch	\(\frac {{\mathrm e}^{-\frac {20 x^{3} {\mathrm e}^{3}}{2 x -3}}}{3}\)	\(18\)
norman	\(\frac {\frac {2 x \,{\mathrm e}^{-\frac {20 x^{3} {\mathrm e}^{3}}{2 x -3}}}{3}-{\mathrm e}^{-\frac {20 x^{3} {\mathrm e}^{3}}{2 x -3}}}{2 x -3}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-80*x^3+180*x^2)*exp(3)*exp(-20*x^3*exp(3)/(2*x-3))/(12*x^2-36*x+27),x,method=_RETURNVERBOSE)

[Out]

1/3*exp(-20*x^3*exp(3)/(2*x-3))

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maxima [A]  time = 0.88, size = 31, normalized size = 1.35 \begin {gather*} \frac {1}{3} \, e^{\left (-10 \, x^{2} e^{3} - 15 \, x e^{3} - \frac {135 \, e^{3}}{2 \, {\left (2 \, x - 3\right )}} - \frac {45}{2} \, e^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-80*x^3+180*x^2)*exp(3)*exp(-20*x^3*exp(3)/(2*x-3))/(12*x^2-36*x+27),x, algorithm="maxima")

[Out]

1/3*e^(-10*x^2*e^3 - 15*x*e^3 - 135/2*e^3/(2*x - 3) - 45/2*e^3)

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mupad [B]  time = 2.36, size = 17, normalized size = 0.74 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {20\,x^3\,{\mathrm {e}}^3}{2\,x-3}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3)*exp(-(20*x^3*exp(3))/(2*x - 3))*(180*x^2 - 80*x^3))/(12*x^2 - 36*x + 27),x)

[Out]

exp(-(20*x^3*exp(3))/(2*x - 3))/3

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sympy [A]  time = 0.25, size = 17, normalized size = 0.74 \begin {gather*} \frac {e^{- \frac {20 x^{3} e^{3}}{2 x - 3}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-80*x**3+180*x**2)*exp(3)*exp(-20*x**3*exp(3)/(2*x-3))/(12*x**2-36*x+27),x)

[Out]

exp(-20*x**3*exp(3)/(2*x - 3))/3

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