3.41.53 $\int (e^{4x}(-36x^3-36x^4)+e^{2x}(72x+72x^2+e(12x+12x^2)))dx$

Optimal. Leaf size=17 $$-{(6+e-3e^{2x}{x}^2)}^2$$

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Rubi [A]  time = 0.22, antiderivative size = 33, normalized size of antiderivative = 1.94, number of steps used = 27, number of rules used = 4, integrand size = 44, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.091, Rules used = {1593, 2196, 2176, 2194} $$\begin{array}{c}-9e^{4x}{x}^4+36e^{2x}{x}^2+6e^{2x+1}{x}^2\end{array}$$

Antiderivative was successfully verified.

[In]

Int[E^(4*x)*(-36*x^3 - 36*x^4) + E^(2*x)*(72*x + 72*x^2 + E*(12*x + 12*x^2)),x]

[Out]

36*E^(2*x)*x^2 + 6*E^(1 + 2*x)*x^2 - 9*E^(4*x)*x^4

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 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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int e^{4x}(-36x^3-36x^4)dx+\int e^{2x}(72x+72x^2+e(12x+12x^2))dx\\ & =\int e^{4x}(-36-36x)x^{3}dx+\int (72e^{2x}x+72e^{2x}{x}^2+12e^{1+2x}x(1+x))dx\\ & =12\int e^{1+2x}x(1+x)dx+72\int e^{2x}xdx+72\int e^{2x}{x}^{2}dx+\int (-36e^{4x}{x}^3-36e^{4x}{x}^4)dx\\ & =36e^{2x}x+36e^{2x}{x}^2+12\int (e^{1+2x}x+e^{1+2x}{x}^2)dx-36\int e^{2x}dx-36\int e^{4x}{x}^{3}dx-36\int e^{4x}{x}^{4}dx-72\int e^{2x}xdx\\ & =-18e^{2x}+36e^{2x}{x}^2-9e^{4x}{x}^3-9e^{4x}{x}^4+12\int e^{1+2x}xdx+12\int e^{1+2x}{x}^{2}dx+27\int e^{4x}{x}^{2}dx+36\int e^{2x}dx+36\int e^{4x}{x}^{3}dx\\ & =6e^{1+2x}x+36e^{2x}{x}^2+\frac{27}{4}{e}^{4x}{x}^2+6e^{1+2x}{x}^2-9e^{4x}{x}^4-6\int e^{1+2x}dx-12\int e^{1+2x}xdx-\frac{27}{2}\int e^{4x}xdx-27\int e^{4x}{x}^{2}dx\\ & =-3e^{1+2x}-\frac{27}{8}{e}^{4x}x+36e^{2x}{x}^2+6e^{1+2x}{x}^2-9e^{4x}{x}^4+\frac{27}{8}\int e^{4x}dx+6\int e^{1+2x}dx+\frac{27}{2}\int e^{4x}xdx\\ & =\frac{27e^{4x}}{32}+36e^{2x}{x}^2+6e^{1+2x}{x}^2-9e^{4x}{x}^4-\frac{27}{8}\int e^{4x}dx\\ & =36e^{2x}{x}^2+6e^{1+2x}{x}^2-9e^{4x}{x}^4\end{array}\end{array}$$

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Mathematica [A]  time = 0.03, size = 19, normalized size = 1.12 $$\begin{array}{c}-{(-6-e+3e^{2x}{x}^2)}^2\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[E^(4*x)*(-36*x^3 - 36*x^4) + E^(2*x)*(72*x + 72*x^2 + E*(12*x + 12*x^2)),x]

[Out]

-(-6 - E + 3*E^(2*x)*x^2)^2

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fricas [A]  time = 0.65, size = 28, normalized size = 1.65 $$\begin{array}{c}-9x^4{e}^{\left(4x\right)}+6(x^{2}e+6x^2)e^{\left(2x\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*x^4-36*x^3)*exp(x)^4+((12*x^2+12*x)*exp(1)+72*x^2+72*x)*exp(x)^2,x, algorithm="fricas")

[Out]

-9*x^4*e^(4*x) + 6*(x^2*e + 6*x^2)*e^(2*x)

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giac [A]  time = 0.16, size = 30, normalized size = 1.76 $$\begin{array}{c}-9x^4{e}^{\left(4x\right)}+36x^2{e}^{\left(2x\right)}+6x^2{e}^{(2x+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*x^4-36*x^3)*exp(x)^4+((12*x^2+12*x)*exp(1)+72*x^2+72*x)*exp(x)^2,x, algorithm="giac")

[Out]

-9*x^4*e^(4*x) + 36*x^2*e^(2*x) + 6*x^2*e^(2*x + 1)

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maple [A]  time = 0.05, size = 24, normalized size = 1.41

method	result	size
risch	$-9x^4{\mathrm{e}}^{4x}+6(\mathrm{e}+6)x^2{\mathrm{e}}^{2x}$	$24$
norman	$(6\mathrm{e}+36)x^2{\mathrm{e}}^{2x}-9x^4{\mathrm{e}}^{4x}$	$25$
default	$36{\mathrm{e}}^{2x}{x}^2+12\mathrm{e}(\frac{{\mathrm{e}}^{2x}{x}^2}{2}-\frac{x{\mathrm{e}}^{2x}}{2}+\frac{{\mathrm{e}}^{2x}}{4})+12\mathrm{e}(\frac{x{\mathrm{e}}^{2x}}{2}-\frac{{\mathrm{e}}^{2x}}{4})-9x^4{\mathrm{e}}^{4x}$	$65$
meijerg	$-\frac{9(1280x^4-1280x^3+960x^2-480x+120){\mathrm{e}}^{4x}}{1280}+\frac{9(-256x^3+192x^2-96x+24){\mathrm{e}}^{4x}}{256}-\frac{3(\mathrm{e}+6)(2-\frac{(12x^2-12x+6){\mathrm{e}}^{2x}}{3})}{2}+3(\mathrm{e}+6)(1-\frac{(-4x+2){\mathrm{e}}^{2x}}{2})$	$92$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-36*x^4-36*x^3)*exp(x)^4+((12*x^2+12*x)*exp(1)+72*x^2+72*x)*exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

-9*x^4*exp(4*x)+6*(exp(1)+6)*x^2*exp(2*x)

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maxima [B]  time = 0.37, size = 74, normalized size = 4.35 $$\begin{array}{c}-9x^4{e}^{\left(4x\right)}+3(2x^{2}e-2xe+e)e^{\left(2x\right)}+18(2x^2-2x+1)e^{\left(2x\right)}+3(2xe-e)e^{\left(2x\right)}+18(2x-1)e^{\left(2x\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*x^4-36*x^3)*exp(x)^4+((12*x^2+12*x)*exp(1)+72*x^2+72*x)*exp(x)^2,x, algorithm="maxima")

[Out]

-9*x^4*e^(4*x) + 3*(2*x^2*e - 2*x*e + e)*e^(2*x) + 18*(2*x^2 - 2*x + 1)*e^(2*x) + 3*(2*x*e - e)*e^(2*x) + 18*(
2*x - 1)*e^(2*x)

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mupad [B]  time = 0.07, size = 24, normalized size = 1.41 $$\begin{array}{c}3x^2{\mathrm{e}}^{2x}(2\mathrm{e}-3x^2{\mathrm{e}}^{2x}+12)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*(72*x + exp(1)*(12*x + 12*x^2) + 72*x^2) - exp(4*x)*(36*x^3 + 36*x^4),x)

[Out]

3*x^2*exp(2*x)*(2*exp(1) - 3*x^2*exp(2*x) + 12)

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sympy [A]  time = 0.14, size = 27, normalized size = 1.59 $$\begin{array}{c}-9x^4{e}^{4x}+(6e{x}^2+36x^2)e^{2x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-36*x**4-36*x**3)*exp(x)**4+((12*x**2+12*x)*exp(1)+72*x**2+72*x)*exp(x)**2,x)

[Out]

-9*x**4*exp(4*x) + (6*E*x**2 + 36*x**2)*exp(2*x)

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