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

Optimal. Leaf size=17 \[ -\left (6+e-3 e^{2 x} x^2\right )^2 \]

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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 {gather*} -9 e^{4 x} x^4+36 e^{2 x} x^2+6 e^{2 x+1} x^2 \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int e^{4 x} \left (-36 x^3-36 x^4\right ) \, dx+\int e^{2 x} \left (72 x+72 x^2+e \left (12 x+12 x^2\right )\right ) \, dx\\ &=\int e^{4 x} (-36-36 x) x^3 \, dx+\int \left (72 e^{2 x} x+72 e^{2 x} x^2+12 e^{1+2 x} x (1+x)\right ) \, dx\\ &=12 \int e^{1+2 x} x (1+x) \, dx+72 \int e^{2 x} x \, dx+72 \int e^{2 x} x^2 \, dx+\int \left (-36 e^{4 x} x^3-36 e^{4 x} x^4\right ) \, dx\\ &=36 e^{2 x} x+36 e^{2 x} x^2+12 \int \left (e^{1+2 x} x+e^{1+2 x} x^2\right ) \, dx-36 \int e^{2 x} \, dx-36 \int e^{4 x} x^3 \, dx-36 \int e^{4 x} x^4 \, dx-72 \int e^{2 x} x \, dx\\ &=-18 e^{2 x}+36 e^{2 x} x^2-9 e^{4 x} x^3-9 e^{4 x} x^4+12 \int e^{1+2 x} x \, dx+12 \int e^{1+2 x} x^2 \, dx+27 \int e^{4 x} x^2 \, dx+36 \int e^{2 x} \, dx+36 \int e^{4 x} x^3 \, dx\\ &=6 e^{1+2 x} x+36 e^{2 x} x^2+\frac {27}{4} e^{4 x} x^2+6 e^{1+2 x} x^2-9 e^{4 x} x^4-6 \int e^{1+2 x} \, dx-12 \int e^{1+2 x} x \, dx-\frac {27}{2} \int e^{4 x} x \, dx-27 \int e^{4 x} x^2 \, dx\\ &=-3 e^{1+2 x}-\frac {27}{8} e^{4 x} x+36 e^{2 x} x^2+6 e^{1+2 x} x^2-9 e^{4 x} x^4+\frac {27}{8} \int e^{4 x} \, dx+6 \int e^{1+2 x} \, dx+\frac {27}{2} \int e^{4 x} x \, dx\\ &=\frac {27 e^{4 x}}{32}+36 e^{2 x} x^2+6 e^{1+2 x} x^2-9 e^{4 x} x^4-\frac {27}{8} \int e^{4 x} \, dx\\ &=36 e^{2 x} x^2+6 e^{1+2 x} x^2-9 e^{4 x} x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 19, normalized size = 1.12 \begin {gather*} -\left (-6-e+3 e^{2 x} x^2\right )^2 \end {gather*}

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 {gather*} -9 \, x^{4} e^{\left (4 \, x\right )} + 6 \, {\left (x^{2} e + 6 \, x^{2}\right )} e^{\left (2 \, x\right )} \end {gather*}

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 {gather*} -9 \, x^{4} e^{\left (4 \, x\right )} + 36 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} e^{\left (2 \, x + 1\right )} \end {gather*}

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

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 {gather*} -9 \, x^{4} e^{\left (4 \, x\right )} + 3 \, {\left (2 \, x^{2} e - 2 \, x e + e\right )} e^{\left (2 \, x\right )} + 18 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + 3 \, {\left (2 \, x e - e\right )} e^{\left (2 \, x\right )} + 18 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end {gather*}

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 {gather*} 3\,x^2\,{\mathrm {e}}^{2\,x}\,\left (2\,\mathrm {e}-3\,x^2\,{\mathrm {e}}^{2\,x}+12\right ) \end {gather*}

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 {gather*} - 9 x^{4} e^{4 x} + \left (6 e x^{2} + 36 x^{2}\right ) e^{2 x} \end {gather*}

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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