3.60.2 \(\int e^{-6-4 x^3} (-60 x^2+3 e^{6+4 x^3} x^2) \, dx\)

Optimal. Leaf size=15 \[ 5 e^{-6-4 x^3}+x^3 \]

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Rubi [A]  time = 0.08, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6688, 12, 14, 2209} \begin {gather*} x^3+5 e^{-4 x^3-6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-6 - 4*x^3)*(-60*x^2 + 3*E^(6 + 4*x^3)*x^2),x]

[Out]

5*E^(-6 - 4*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

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 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} &=\int 3 \left (1-20 e^{-6-4 x^3}\right ) x^2 \, dx\\ &=3 \int \left (1-20 e^{-6-4 x^3}\right ) x^2 \, dx\\ &=3 \int \left (x^2-20 e^{-6-4 x^3} x^2\right ) \, dx\\ &=x^3-60 \int e^{-6-4 x^3} x^2 \, dx\\ &=5 e^{-6-4 x^3}+x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 15, normalized size = 1.00 \begin {gather*} 5 e^{-6-4 x^3}+x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-6 - 4*x^3)*(-60*x^2 + 3*E^(6 + 4*x^3)*x^2),x]

[Out]

5*E^(-6 - 4*x^3) + x^3

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fricas [A]  time = 0.69, size = 23, normalized size = 1.53 \begin {gather*} {\left (x^{3} e^{\left (4 \, x^{3} + 6\right )} + 5\right )} e^{\left (-4 \, x^{3} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*exp(4*x^3+6)-60*x^2)/exp(4*x^3+6),x, algorithm="fricas")

[Out]

(x^3*e^(4*x^3 + 6) + 5)*e^(-4*x^3 - 6)

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giac [B]  time = 0.13, size = 30, normalized size = 2.00 \begin {gather*} \frac {1}{4} \, {\left (4 \, x^{3} e^{6} - {\left (e^{\left (4 \, x^{3} + 6\right )} - 20\right )} e^{\left (-4 \, x^{3}\right )}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*exp(4*x^3+6)-60*x^2)/exp(4*x^3+6),x, algorithm="giac")

[Out]

1/4*(4*x^3*e^6 - (e^(4*x^3 + 6) - 20)*e^(-4*x^3))*e^(-6)

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maple [A]  time = 0.06, size = 15, normalized size = 1.00

method	result	size
risch	\(x^{3}+5 \,{\mathrm e}^{-4 x^{3}-6}\)	\(15\)
default	\(x^{3}+5 \,{\mathrm e}^{-4 x^{3}} {\mathrm e}^{-6}\)	\(17\)
norman	\(\left (5+x^{3} {\mathrm e}^{4 x^{3}+6}\right ) {\mathrm e}^{-4 x^{3}-6}\)	\(26\)
meijerg	\(-\frac {{\mathrm e}^{-4 x^{3}+4 x^{3} {\mathrm e}^{-6}} \left (1-{\mathrm e}^{4 x^{3} \left (1-{\mathrm e}^{-6}\right )}\right )}{4 \left (1-{\mathrm e}^{-6}\right )}-5 \,{\mathrm e}^{-4 x^{3}+4 x^{3} {\mathrm e}^{-6}} \left (1-{\mathrm e}^{-4 x^{3} {\mathrm e}^{-6}}\right )\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*exp(4*x^3+6)-60*x^2)/exp(4*x^3+6),x,method=_RETURNVERBOSE)

[Out]

x^3+5*exp(-4*x^3-6)

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maxima [A]  time = 0.36, size = 14, normalized size = 0.93 \begin {gather*} x^{3} + 5 \, e^{\left (-4 \, x^{3} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*exp(4*x^3+6)-60*x^2)/exp(4*x^3+6),x, algorithm="maxima")

[Out]

x^3 + 5*e^(-4*x^3 - 6)

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mupad [B]  time = 4.77, size = 14, normalized size = 0.93 \begin {gather*} 5\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{-4\,x^3}+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(- 4*x^3 - 6)*(3*x^2*exp(4*x^3 + 6) - 60*x^2),x)

[Out]

5*exp(-6)*exp(-4*x^3) + x^3

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sympy [A]  time = 0.10, size = 14, normalized size = 0.93 \begin {gather*} x^{3} + 5 e^{- 4 x^{3} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2*exp(4*x**3+6)-60*x**2)/exp(4*x**3+6),x)

[Out]

x**3 + 5*exp(-4*x**3 - 6)

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