3.20.63 \(\int \frac {e^{2 x} (-6-12 x)+6 x^2}{e} \, dx\) [1963]

Optimal. Leaf size=20 \[ \frac {2 x^2 \left (-\frac {3 e^{2 x}}{x}+x\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.60, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2207, 2225} \begin {gather*} \frac {2 x^3}{e}+3 e^{2 x-1}-3 e^{2 x-1} (2 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*E^(-1 + 2*x) + (2*x^3)/E - 3*E^(-1 + 2*x)*(1 + 2*x)

Rule 12

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

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 18, normalized size = 0.90 \begin {gather*} \frac {-6 e^{2 x} x+2 x^3}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*E^(2*x)*x + 2*x^3)/E

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Maple [A]
time = 0.06, size = 19, normalized size = 0.95

method	result	size
risch	\(2 \,{\mathrm e}^{-1} x^{3}-6 x \,{\mathrm e}^{2 x -1}\)	\(18\)
default	\({\mathrm e}^{-1} \left (-6 x \,{\mathrm e}^{2 x}+2 x^{3}\right )\)	\(19\)
norman	\(2 \,{\mathrm e}^{-1} x^{3}-6 x \,{\mathrm e}^{-1} {\mathrm e}^{2 x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x-6)*exp(x)^2+6*x^2)/exp(1),x,method=_RETURNVERBOSE)

[Out]

1/exp(1)*(-6*x*exp(x)^2+2*x^3)

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Maxima [A]
time = 0.26, size = 26, normalized size = 1.30 \begin {gather*} {\left (2 \, x^{3} - 3 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 3 \, e^{\left (2 \, x\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x-6)*exp(x)^2+6*x^2)/exp(1),x, algorithm="maxima")

[Out]

(2*x^3 - 3*(2*x - 1)*e^(2*x) - 3*e^(2*x))*e^(-1)

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Fricas [A]
time = 0.35, size = 15, normalized size = 0.75 \begin {gather*} 2 \, {\left (x^{3} - 3 \, x e^{\left (2 \, x\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x-6)*exp(x)^2+6*x^2)/exp(1),x, algorithm="fricas")

[Out]

2*(x^3 - 3*x*e^(2*x))*e^(-1)

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Sympy [A]
time = 0.04, size = 19, normalized size = 0.95 \begin {gather*} \frac {2 x^{3}}{e} - \frac {6 x e^{2 x}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x-6)*exp(x)**2+6*x**2)/exp(1),x)

[Out]

2*x**3*exp(-1) - 6*x*exp(-1)*exp(2*x)

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Giac [A]
time = 0.38, size = 15, normalized size = 0.75 \begin {gather*} 2 \, {\left (x^{3} - 3 \, x e^{\left (2 \, x\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x-6)*exp(x)^2+6*x^2)/exp(1),x, algorithm="giac")

[Out]

2*(x^3 - 3*x*e^(2*x))*e^(-1)

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Mupad [B]
time = 0.05, size = 17, normalized size = 0.85 \begin {gather*} -2\,x\,{\mathrm {e}}^{-1}\,\left (3\,{\mathrm {e}}^{2\,x}-x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-1)*(exp(2*x)*(12*x + 6) - 6*x^2),x)

[Out]

-2*x*exp(-1)*(3*exp(2*x) - x^2)

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