3.46.10 \(\int (e^{2 x} (-6 x-6 x^2)+e^x (-18 x-6 x^2+x^3)) \, dx\)

Optimal. Leaf size=16 \[ e^x x^2 \left (-3 \left (3+e^x\right )+x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 26, normalized size of antiderivative = 1.62, number of steps used = 21, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1593, 2196, 2176, 2194, 1594} \begin {gather*} e^x x^3-9 e^x x^2-3 e^{2 x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*x)*(-6*x - 6*x^2) + E^x*(-18*x - 6*x^2 + x^3),x]

[Out]

-9*E^x*x^2 - 3*E^(2*x)*x^2 + E^x*x^3

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 1594

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

Rule 2176

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] &&  !$UseGamma === True

Rule 2194

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]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{2 x} \left (-6 x-6 x^2\right ) \, dx+\int e^x \left (-18 x-6 x^2+x^3\right ) \, dx\\ &=\int e^{2 x} (-6-6 x) x \, dx+\int e^x x \left (-18-6 x+x^2\right ) \, dx\\ &=\int \left (-6 e^{2 x} x-6 e^{2 x} x^2\right ) \, dx+\int \left (-18 e^x x-6 e^x x^2+e^x x^3\right ) \, dx\\ &=-\left (6 \int e^{2 x} x \, dx\right )-6 \int e^x x^2 \, dx-6 \int e^{2 x} x^2 \, dx-18 \int e^x x \, dx+\int e^x x^3 \, dx\\ &=-18 e^x x-3 e^{2 x} x-6 e^x x^2-3 e^{2 x} x^2+e^x x^3+3 \int e^{2 x} \, dx-3 \int e^x x^2 \, dx+6 \int e^{2 x} x \, dx+12 \int e^x x \, dx+18 \int e^x \, dx\\ &=18 e^x+\frac {3 e^{2 x}}{2}-6 e^x x-9 e^x x^2-3 e^{2 x} x^2+e^x x^3-3 \int e^{2 x} \, dx+6 \int e^x x \, dx-12 \int e^x \, dx\\ &=6 e^x-9 e^x x^2-3 e^{2 x} x^2+e^x x^3-6 \int e^x \, dx\\ &=-9 e^x x^2-3 e^{2 x} x^2+e^x x^3\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 18, normalized size = 1.12 \begin {gather*} -e^x \left (9+3 e^x-x\right ) x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)*(-6*x - 6*x^2) + E^x*(-18*x - 6*x^2 + x^3),x]

[Out]

-(E^x*(9 + 3*E^x - x)*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 22, normalized size = 1.38 \begin {gather*} -3 \, x^{2} e^{\left (2 \, x\right )} + {\left (x^{3} - 9 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2-6*x)*exp(x)^2+(x^3-6*x^2-18*x)*exp(x),x, algorithm="fricas")

[Out]

-3*x^2*e^(2*x) + (x^3 - 9*x^2)*e^x

________________________________________________________________________________________

giac [A]  time = 0.17, size = 22, normalized size = 1.38 \begin {gather*} -3 \, x^{2} e^{\left (2 \, x\right )} + {\left (x^{3} - 9 \, x^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2-6*x)*exp(x)^2+(x^3-6*x^2-18*x)*exp(x),x, algorithm="giac")

[Out]

-3*x^2*e^(2*x) + (x^3 - 9*x^2)*e^x

________________________________________________________________________________________

maple [A]  time = 0.03, size = 23, normalized size = 1.44

method	result	size
risch	\(-3 \,{\mathrm e}^{2 x} x^{2}+\left (x^{3}-9 x^{2}\right ) {\mathrm e}^{x}\)	\(23\)
default	\(-3 \,{\mathrm e}^{2 x} x^{2}+{\mathrm e}^{x} x^{3}-9 \,{\mathrm e}^{x} x^{2}\)	\(24\)
norman	\(-3 \,{\mathrm e}^{2 x} x^{2}+{\mathrm e}^{x} x^{3}-9 \,{\mathrm e}^{x} x^{2}\)	\(24\)
meijerg	\(-\frac {\left (12 x^{2}-12 x +6\right ) {\mathrm e}^{2 x}}{4}+\frac {3 \left (-4 x +2\right ) {\mathrm e}^{2 x}}{4}-\frac {\left (-4 x^{3}+12 x^{2}-24 x +24\right ) {\mathrm e}^{x}}{4}-2 \left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}+9 \left (-2 x +2\right ) {\mathrm e}^{x}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^2-6*x)*exp(x)^2+(x^3-6*x^2-18*x)*exp(x),x,method=_RETURNVERBOSE)

[Out]

-3*exp(2*x)*x^2+(x^3-9*x^2)*exp(x)

________________________________________________________________________________________

maxima [B]  time = 0.37, size = 45, normalized size = 2.81 \begin {gather*} -3 \, x^{2} e^{\left (2 \, x\right )} + {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} - 6 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 18 \, {\left (x - 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2-6*x)*exp(x)^2+(x^3-6*x^2-18*x)*exp(x),x, algorithm="maxima")

[Out]

-3*x^2*e^(2*x) + (x^3 - 3*x^2 + 6*x - 6)*e^x - 6*(x^2 - 2*x + 2)*e^x - 18*(x - 1)*e^x

________________________________________________________________________________________

mupad [B]  time = 3.03, size = 16, normalized size = 1.00 \begin {gather*} -x^2\,{\mathrm {e}}^x\,\left (3\,{\mathrm {e}}^x-x+9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- exp(2*x)*(6*x + 6*x^2) - exp(x)*(18*x + 6*x^2 - x^3),x)

[Out]

-x^2*exp(x)*(3*exp(x) - x + 9)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 20, normalized size = 1.25 \begin {gather*} - 3 x^{2} e^{2 x} + \left (x^{3} - 9 x^{2}\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**2-6*x)*exp(x)**2+(x**3-6*x**2-18*x)*exp(x),x)

[Out]

-3*x**2*exp(2*x) + (x**3 - 9*x**2)*exp(x)

________________________________________________________________________________________