\(\int (6 e^{\frac {2}{3} (-4+3 e^5+3 x^3)} x^2-12 e^{2+\frac {1}{3} (-4+3 e^5+3 x^3)} x^2) \, dx\) [8563]

Optimal result

Integrand size = 49, antiderivative size = 27 \[ \int \left (6 e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2-12 e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2\right ) \, dx=\left (-2 e^2+e^{e^5+\frac {1}{3} \left (-1+3 \left (-1+x^3\right )\right )}\right )^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {2257, 2240} \[ \int \left (6 e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2-12 e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2\right ) \, dx=e^{2 x^3-\frac {2}{3} \left (4-3 e^5\right )}-4 e^{x^3+\frac {1}{3} \left (2+3 e^5\right )} \]

[In]

Int[6*E^((2*(-4 + 3*E^5 + 3*x^3))/3)*x^2 - 12*E^(2 + (-4 + 3*E^5 + 3*x^3)/3)*x^2,x]

[Out]

-4*E^((2 + 3*E^5)/3 + x^3) + E^((-2*(4 - 3*E^5))/3 + 2*x^3)

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2257

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && BinomialQ[v, x] &&  !(LinearMatchQ[u, x] && BinomialMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = 6 \int e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2 \, dx-12 \int e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2 \, dx \\ & = 6 \int e^{-\frac {2}{3} \left (4-3 e^5\right )+2 x^3} x^2 \, dx-12 \int e^{\frac {1}{3} \left (2+3 e^5\right )+x^3} x^2 \, dx \\ & = -4 e^{\frac {1}{3} \left (2+3 e^5\right )+x^3}+e^{-\frac {2}{3} \left (4-3 e^5\right )+2 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \left (6 e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2-12 e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2\right ) \, dx=\frac {\left (2 e^{10/3}-e^{e^5+x^3}\right )^2}{e^{8/3}} \]

[In]

Integrate[6*E^((2*(-4 + 3*E^5 + 3*x^3))/3)*x^2 - 12*E^(2 + (-4 + 3*E^5 + 3*x^3)/3)*x^2,x]

[Out]

(2*E^(10/3) - E^(E^5 + x^3))^2/E^(8/3)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

method	result	size
default	\({\mathrm e}^{2 \,{\mathrm e}^{5}+2 x^{3}-\frac {8}{3}}-4 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{5}+x^{3}-\frac {4}{3}}\)	\(24\)
norman	\({\mathrm e}^{2 \,{\mathrm e}^{5}+2 x^{3}-\frac {8}{3}}-4 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{5}+x^{3}-\frac {4}{3}}\)	\(24\)
risch	\({\mathrm e}^{2 \,{\mathrm e}^{5}+2 x^{3}-\frac {8}{3}}-4 \,{\mathrm e}^{\frac {2}{3}+{\mathrm e}^{5}+x^{3}}\)	\(24\)
parallelrisch	\({\mathrm e}^{2 \,{\mathrm e}^{5}+2 x^{3}-\frac {8}{3}}-4 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{5}+x^{3}-\frac {4}{3}}\)	\(24\)
parts	\({\mathrm e}^{2 \,{\mathrm e}^{5}+2 x^{3}-\frac {8}{3}}-4 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{5}+x^{3}-\frac {4}{3}}\)	\(24\)

[In]

int(6*x^2*exp(exp(5)+x^3-4/3)^2-12*x^2*exp(2)*exp(exp(5)+x^3-4/3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \left (6 e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2-12 e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2\right ) \, dx={\left (e^{\left (2 \, x^{3} + 2 \, e^{5} + \frac {4}{3}\right )} - 4 \, e^{\left (x^{3} + e^{5} + \frac {14}{3}\right )}\right )} e^{\left (-4\right )} \]

[In]

integrate(6*x^2*exp(exp(5)+x^3-4/3)^2-12*x^2*exp(2)*exp(exp(5)+x^3-4/3),x, algorithm="fricas")

[Out]

(e^(2*x^3 + 2*e^5 + 4/3) - 4*e^(x^3 + e^5 + 14/3))*e^(-4)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \left (6 e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2-12 e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2\right ) \, dx=- 4 e^{2} e^{x^{3} - \frac {4}{3} + e^{5}} + e^{2 x^{3} - \frac {8}{3} + 2 e^{5}} \]

[In]

integrate(6*x**2*exp(exp(5)+x**3-4/3)**2-12*x**2*exp(2)*exp(exp(5)+x**3-4/3),x)

[Out]

-4*exp(2)*exp(x**3 - 4/3 + exp(5)) + exp(2*x**3 - 8/3 + 2*exp(5))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \left (6 e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2-12 e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2\right ) \, dx=e^{\left (2 \, x^{3} + 2 \, e^{5} - \frac {8}{3}\right )} - 4 \, e^{\left (x^{3} + e^{5} + \frac {2}{3}\right )} \]

[In]

integrate(6*x^2*exp(exp(5)+x^3-4/3)^2-12*x^2*exp(2)*exp(exp(5)+x^3-4/3),x, algorithm="maxima")

[Out]

e^(2*x^3 + 2*e^5 - 8/3) - 4*e^(x^3 + e^5 + 2/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \left (6 e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2-12 e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2\right ) \, dx=e^{\left (2 \, x^{3} + 2 \, e^{5} - \frac {8}{3}\right )} - 4 \, e^{\left (x^{3} + e^{5} + \frac {2}{3}\right )} \]

[In]

integrate(6*x^2*exp(exp(5)+x^3-4/3)^2-12*x^2*exp(2)*exp(exp(5)+x^3-4/3),x, algorithm="giac")

[Out]

e^(2*x^3 + 2*e^5 - 8/3) - 4*e^(x^3 + e^5 + 2/3)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \left (6 e^{\frac {2}{3} \left (-4+3 e^5+3 x^3\right )} x^2-12 e^{2+\frac {1}{3} \left (-4+3 e^5+3 x^3\right )} x^2\right ) \, dx=-{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-\frac {8}{3}}\,{\mathrm {e}}^{{\mathrm {e}}^5}\,\left (4\,{\mathrm {e}}^{10/3}-{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{{\mathrm {e}}^5}\right ) \]

[In]

int(6*x^2*exp(2*exp(5) + 2*x^3 - 8/3) - 12*x^2*exp(exp(5) + x^3 - 4/3)*exp(2),x)

[Out]

-exp(x^3)*exp(-8/3)*exp(exp(5))*(4*exp(10/3) - exp(x^3)*exp(exp(5)))