3.12.92 \(\int (-54-24 e-72 e^5-24 e^{10}+16 e^{4 x}+e^{3 x} (-72-48 e^5)+e^{2 x} (84+16 e+144 e^5+48 e^{10}-48 x)+72 x+e^x (18-24 e-72 e^{10}-16 e^{15}+72 x+e^5 (-60-16 e+48 x))) \, dx\)

Optimal. Leaf size=27 \[ 4 \left (-e-\left (\frac {3}{2}+e^5-e^x\right )^2+3 x\right )^2 \]

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Rubi [B]  time = 0.09, antiderivative size = 119, normalized size of antiderivative = 4.41, number of steps used = 8, number of rules used = 3, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2194, 2176, 2187} \begin {gather*} 36 x^2-6 \left (9+4 e+12 e^5+4 e^{10}\right ) x+12 e^{2 x}+4 e^{4 x}+2 e^{2 x} \left (-12 x+12 e^{10}+36 e^5+4 e+21\right )+2 \left (3+2 e^5\right ) e^x \left (12 x-4 e^{10}-12 e^5-4 e+3\right )-24 \left (3+2 e^5\right ) e^x-8 \left (3+2 e^5\right ) e^{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-54 - 24*E - 72*E^5 - 24*E^10 + 16*E^(4*x) + E^(3*x)*(-72 - 48*E^5) + E^(2*x)*(84 + 16*E + 144*E^5 + 48*E^
10 - 48*x) + 72*x + E^x*(18 - 24*E - 72*E^10 - 16*E^15 + 72*x + E^5*(-60 - 16*E + 48*x)),x]

[Out]

12*E^(2*x) + 4*E^(4*x) - 24*E^x*(3 + 2*E^5) - 8*E^(3*x)*(3 + 2*E^5) + 2*E^(2*x)*(21 + 4*E + 36*E^5 + 12*E^10 -
12*x) - 6*(9 + 4*E + 12*E^5 + 4*E^10)*x + 36*x^2 + 2*E^x*(3 + 2*E^5)*(3 - 4*E - 12*E^5 - 4*E^10 + 12*x)

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 2187

Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*(u_)^(m_.), x_Symbol] :> Int[NormalizePowerOfLinear[u, x]^
m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ
[u, x] &&  !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) && IntegerQ[m]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-6 \left (9+4 e+12 e^5+4 e^{10}\right ) x+36 x^2+16 \int e^{4 x} \, dx-\left (24 \left (3+2 e^5\right )\right ) \int e^{3 x} \, dx+\int e^{2 x} \left (84+16 e+144 e^5+48 e^{10}-48 x\right ) \, dx+\int e^x \left (18-24 e-72 e^{10}-16 e^{15}+72 x+e^5 (-60-16 e+48 x)\right ) \, dx\\ &=4 e^{4 x}-8 e^{3 x} \left (3+2 e^5\right )+2 e^{2 x} \left (21+4 e+36 e^5+12 e^{10}-12 x\right )-6 \left (9+4 e+12 e^5+4 e^{10}\right ) x+36 x^2+24 \int e^{2 x} \, dx+\int e^x \left (2 \left (3+2 e^5\right ) \left (3-4 e-12 e^5-4 e^{10}\right )+24 \left (3+2 e^5\right ) x\right ) \, dx\\ &=12 e^{2 x}+4 e^{4 x}-8 e^{3 x} \left (3+2 e^5\right )+2 e^{2 x} \left (21+4 e+36 e^5+12 e^{10}-12 x\right )-6 \left (9+4 e+12 e^5+4 e^{10}\right ) x+36 x^2+2 e^x \left (3+2 e^5\right ) \left (3-4 e-12 e^5-4 e^{10}+12 x\right )-\left (24 \left (3+2 e^5\right )\right ) \int e^x \, dx\\ &=12 e^{2 x}+4 e^{4 x}-24 e^x \left (3+2 e^5\right )-8 e^{3 x} \left (3+2 e^5\right )+2 e^{2 x} \left (21+4 e+36 e^5+12 e^{10}-12 x\right )-6 \left (9+4 e+12 e^5+4 e^{10}\right ) x+36 x^2+2 e^x \left (3+2 e^5\right ) \left (3-4 e-12 e^5-4 e^{10}+12 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 43, normalized size = 1.59 \begin {gather*} \frac {1}{4} \left (9+4 e+12 e^5+4 e^{10}-12 e^x+4 e^{2 x}-8 e^{5+x}-12 x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-54 - 24*E - 72*E^5 - 24*E^10 + 16*E^(4*x) + E^(3*x)*(-72 - 48*E^5) + E^(2*x)*(84 + 16*E + 144*E^5 +
48*E^10 - 48*x) + 72*x + E^x*(18 - 24*E - 72*E^10 - 16*E^15 + 72*x + E^5*(-60 - 16*E + 48*x)),x]

[Out]

(9 + 4*E + 12*E^5 + 4*E^10 - 12*E^x + 4*E^(2*x) - 8*E^(5 + x) - 12*x)^2/4

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fricas [B]  time = 0.77, size = 99, normalized size = 3.67 \begin {gather*} 36 \, x^{2} - 24 \, x e^{10} - 72 \, x e^{5} - 24 \, x e - 8 \, {\left (2 \, e^{5} + 3\right )} e^{\left (3 \, x\right )} - 2 \, {\left (12 \, x - 12 \, e^{10} - 36 \, e^{5} - 4 \, e - 27\right )} e^{\left (2 \, x\right )} + 2 \, {\left (6 \, {\left (4 \, x - 9\right )} e^{5} + 36 \, x - 8 \, e^{15} - 36 \, e^{10} - 8 \, e^{6} - 12 \, e - 27\right )} e^{x} - 54 \, x + 4 \, e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*exp(x)^4+(-48*exp(5)-72)*exp(x)^3+(48*exp(5)^2+144*exp(5)+16*exp(1)-48*x+84)*exp(x)^2+(-16*exp(5)
^3-72*exp(5)^2+(-16*exp(1)+48*x-60)*exp(5)-24*exp(1)+72*x+18)*exp(x)-24*exp(5)^2-72*exp(5)-24*exp(1)+72*x-54,x
, algorithm="fricas")

[Out]

36*x^2 - 24*x*e^10 - 72*x*e^5 - 24*x*e - 8*(2*e^5 + 3)*e^(3*x) - 2*(12*x - 12*e^10 - 36*e^5 - 4*e - 27)*e^(2*x
) + 2*(6*(4*x - 9)*e^5 + 36*x - 8*e^15 - 36*e^10 - 8*e^6 - 12*e - 27)*e^x - 54*x + 4*e^(4*x)

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giac [B]  time = 0.33, size = 121, normalized size = 4.48 \begin {gather*} 36 \, x^{2} - 24 \, x e^{10} - 72 \, x e^{5} - 24 \, x e - 8 \, {\left (2 \, e^{5} + 3\right )} e^{\left (3 \, x\right )} - 6 \, {\left (4 \, x - 9\right )} e^{\left (2 \, x\right )} + 12 \, {\left (4 \, x - 9\right )} e^{\left (x + 5\right )} + 18 \, {\left (4 \, x - 3\right )} e^{x} - 54 \, x + 4 \, e^{\left (4 \, x\right )} + 24 \, e^{\left (2 \, x + 10\right )} + 72 \, e^{\left (2 \, x + 5\right )} + 8 \, e^{\left (2 \, x + 1\right )} - 16 \, e^{\left (x + 15\right )} - 72 \, e^{\left (x + 10\right )} - 16 \, e^{\left (x + 6\right )} - 24 \, e^{\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*exp(x)^4+(-48*exp(5)-72)*exp(x)^3+(48*exp(5)^2+144*exp(5)+16*exp(1)-48*x+84)*exp(x)^2+(-16*exp(5)
^3-72*exp(5)^2+(-16*exp(1)+48*x-60)*exp(5)-24*exp(1)+72*x+18)*exp(x)-24*exp(5)^2-72*exp(5)-24*exp(1)+72*x-54,x
, algorithm="giac")

[Out]

36*x^2 - 24*x*e^10 - 72*x*e^5 - 24*x*e - 8*(2*e^5 + 3)*e^(3*x) - 6*(4*x - 9)*e^(2*x) + 12*(4*x - 9)*e^(x + 5)
+ 18*(4*x - 3)*e^x - 54*x + 4*e^(4*x) + 24*e^(2*x + 10) + 72*e^(2*x + 5) + 8*e^(2*x + 1) - 16*e^(x + 15) - 72*
e^(x + 10) - 16*e^(x + 6) - 24*e^(x + 1)

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maple [B]  time = 0.04, size = 100, normalized size = 3.70

method	result	size
risch	\(4 \,{\mathrm e}^{4 x}-16 \,{\mathrm e}^{3 x} {\mathrm e}^{5}-24 \,{\mathrm e}^{3 x}+\left (24 \,{\mathrm e}^{10}+72 \,{\mathrm e}^{5}+8 \,{\mathrm e}-24 x +54\right ) {\mathrm e}^{2 x}+\left (-16 \,{\mathrm e}^{15}-72 \,{\mathrm e}^{10}-16 \,{\mathrm e}^{6}+48 x \,{\mathrm e}^{5}-108 \,{\mathrm e}^{5}-24 \,{\mathrm e}+72 x -54\right ) {\mathrm e}^{x}-24 x \,{\mathrm e}^{10}-72 x \,{\mathrm e}^{5}-24 x \,{\mathrm e}+36 x^{2}-54 x\)	\(100\)
norman	\(\left (-16 \,{\mathrm e}^{5}-24\right ) {\mathrm e}^{3 x}+\left (54+24 \,{\mathrm e}^{10}+72 \,{\mathrm e}^{5}+8 \,{\mathrm e}\right ) {\mathrm e}^{2 x}+\left (-24 \,{\mathrm e}^{10}-72 \,{\mathrm e}^{5}-24 \,{\mathrm e}-54\right ) x +\left (-54-108 \,{\mathrm e}^{5}-16 \,{\mathrm e}^{15}-72 \,{\mathrm e}^{10}-16 \,{\mathrm e} \,{\mathrm e}^{5}-24 \,{\mathrm e}\right ) {\mathrm e}^{x}+\left (48 \,{\mathrm e}^{5}+72\right ) x \,{\mathrm e}^{x}+36 x^{2}+4 \,{\mathrm e}^{4 x}-24 x \,{\mathrm e}^{2 x}\)	\(111\)
default	\(-54 x +\frac {\left (-48 \,{\mathrm e}^{5}-72\right ) {\mathrm e}^{3 x}}{3}+54 \,{\mathrm e}^{2 x}-24 x \,{\mathrm e}^{2 x}+8 \,{\mathrm e} \,{\mathrm e}^{2 x}+72 \,{\mathrm e}^{5} {\mathrm e}^{2 x}+24 \,{\mathrm e}^{10} {\mathrm e}^{2 x}+72 \,{\mathrm e}^{x} x -54 \,{\mathrm e}^{x}-24 \,{\mathrm e} \,{\mathrm e}^{x}-60 \,{\mathrm e}^{5} {\mathrm e}^{x}-72 \,{\mathrm e}^{10} {\mathrm e}^{x}-16 \,{\mathrm e}^{15} {\mathrm e}^{x}+48 \,{\mathrm e}^{5} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )-16 \,{\mathrm e}^{5} {\mathrm e}^{x} {\mathrm e}+36 x^{2}-24 x \,{\mathrm e}^{10}+4 \,{\mathrm e}^{4 x}-24 x \,{\mathrm e}-72 x \,{\mathrm e}^{5}\)	\(142\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*exp(x)^4+(-48*exp(5)-72)*exp(x)^3+(48*exp(5)^2+144*exp(5)+16*exp(1)-48*x+84)*exp(x)^2+(-16*exp(5)^3-72*
exp(5)^2+(-16*exp(1)+48*x-60)*exp(5)-24*exp(1)+72*x+18)*exp(x)-24*exp(5)^2-72*exp(5)-24*exp(1)+72*x-54,x,metho
d=_RETURNVERBOSE)

[Out]

4*exp(4*x)-16*exp(3*x)*exp(5)-24*exp(3*x)+(24*exp(10)+72*exp(5)+8*exp(1)-24*x+54)*exp(2*x)+(-16*exp(15)-72*exp
(10)-16*exp(6)+48*x*exp(5)-108*exp(5)-24*exp(1)+72*x-54)*exp(x)-24*x*exp(10)-72*x*exp(5)-24*x*exp(1)+36*x^2-54
*x

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maxima [B]  time = 0.41, size = 100, normalized size = 3.70 \begin {gather*} 36 \, x^{2} - 24 \, x e^{10} - 72 \, x e^{5} - 24 \, x e - 8 \, {\left (2 \, e^{5} + 3\right )} e^{\left (3 \, x\right )} - 2 \, {\left (12 \, x - 12 \, e^{10} - 36 \, e^{5} - 4 \, e - 27\right )} e^{\left (2 \, x\right )} + 2 \, {\left (12 \, x {\left (2 \, e^{5} + 3\right )} - 8 \, e^{15} - 36 \, e^{10} - 8 \, e^{6} - 54 \, e^{5} - 12 \, e - 27\right )} e^{x} - 54 \, x + 4 \, e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*exp(x)^4+(-48*exp(5)-72)*exp(x)^3+(48*exp(5)^2+144*exp(5)+16*exp(1)-48*x+84)*exp(x)^2+(-16*exp(5)
^3-72*exp(5)^2+(-16*exp(1)+48*x-60)*exp(5)-24*exp(1)+72*x+18)*exp(x)-24*exp(5)^2-72*exp(5)-24*exp(1)+72*x-54,x
, algorithm="maxima")

[Out]

36*x^2 - 24*x*e^10 - 72*x*e^5 - 24*x*e - 8*(2*e^5 + 3)*e^(3*x) - 2*(12*x - 12*e^10 - 36*e^5 - 4*e - 27)*e^(2*x
) + 2*(12*x*(2*e^5 + 3) - 8*e^15 - 36*e^10 - 8*e^6 - 54*e^5 - 12*e - 27)*e^x - 54*x + 4*e^(4*x)

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mupad [B]  time = 0.13, size = 103, normalized size = 3.81 \begin {gather*} 4\,{\mathrm {e}}^{4\,x}-24\,x\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{2\,x}\,\left (8\,\mathrm {e}+72\,{\mathrm {e}}^5+24\,{\mathrm {e}}^{10}+54\right )-{\mathrm {e}}^x\,\left (24\,\mathrm {e}+108\,{\mathrm {e}}^5+16\,{\mathrm {e}}^6+72\,{\mathrm {e}}^{10}+16\,{\mathrm {e}}^{15}+54\right )-x\,\left (24\,\mathrm {e}+72\,{\mathrm {e}}^5+24\,{\mathrm {e}}^{10}+54\right )-{\mathrm {e}}^{3\,x}\,\left (16\,{\mathrm {e}}^5+24\right )+36\,x^2+x\,{\mathrm {e}}^x\,\left (48\,{\mathrm {e}}^5+72\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(72*x + 16*exp(4*x) - 24*exp(1) - 72*exp(5) - 24*exp(10) + exp(2*x)*(16*exp(1) - 48*x + 144*exp(5) + 48*exp
(10) + 84) - exp(x)*(24*exp(1) - 72*x + 72*exp(10) + 16*exp(15) + exp(5)*(16*exp(1) - 48*x + 60) - 18) - exp(3
*x)*(48*exp(5) + 72) - 54,x)

[Out]

4*exp(4*x) - 24*x*exp(2*x) + exp(2*x)*(8*exp(1) + 72*exp(5) + 24*exp(10) + 54) - exp(x)*(24*exp(1) + 108*exp(5
) + 16*exp(6) + 72*exp(10) + 16*exp(15) + 54) - x*(24*exp(1) + 72*exp(5) + 24*exp(10) + 54) - exp(3*x)*(16*exp
(5) + 24) + 36*x^2 + x*exp(x)*(48*exp(5) + 72)

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sympy [B]  time = 0.21, size = 110, normalized size = 4.07 \begin {gather*} 36 x^{2} + x \left (- 24 e^{10} - 72 e^{5} - 24 e - 54\right ) + \left (- 24 x + 8 e + 54 + 72 e^{5} + 24 e^{10}\right ) e^{2 x} + \left (72 x + 48 x e^{5} - 16 e^{15} - 72 e^{10} - 108 e^{5} - 16 e^{6} - 24 e - 54\right ) e^{x} + 4 e^{4 x} + \left (- 16 e^{5} - 24\right ) e^{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*exp(x)**4+(-48*exp(5)-72)*exp(x)**3+(48*exp(5)**2+144*exp(5)+16*exp(1)-48*x+84)*exp(x)**2+(-16*ex
p(5)**3-72*exp(5)**2+(-16*exp(1)+48*x-60)*exp(5)-24*exp(1)+72*x+18)*exp(x)-24*exp(5)**2-72*exp(5)-24*exp(1)+72
*x-54,x)

[Out]

36*x**2 + x*(-24*exp(10) - 72*exp(5) - 24*E - 54) + (-24*x + 8*E + 54 + 72*exp(5) + 24*exp(10))*exp(2*x) + (72
*x + 48*x*exp(5) - 16*exp(15) - 72*exp(10) - 108*exp(5) - 16*exp(6) - 24*E - 54)*exp(x) + 4*exp(4*x) + (-16*ex
p(5) - 24)*exp(3*x)

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