3.82.27 \(\int (-1+e^{196+28 x+x^2} (-28-2 x)+16 x+4 x^3+e^{2+x} (-16 x-8 x^2-8 x^3-2 x^4)+e^{4+2 x} (4 x^3+2 x^4)) \, dx\)

Optimal. Leaf size=29 \[ -e^{(14+x)^2}-x+\left (4+x \left (x-e^{2+x} x\right )\right )^2 \]

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Rubi [A]  time = 0.29, antiderivative size = 52, normalized size of antiderivative = 1.79, number of steps used = 31, number of rules used = 6, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2227, 2209, 2196, 2176, 2194, 1593} \begin {gather*} -2 e^{x+2} x^4+e^{2 x+4} x^4+x^4-8 e^{x+2} x^2+8 x^2-x-e^{(x+14)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + E^(196 + 28*x + x^2)*(-28 - 2*x) + 16*x + 4*x^3 + E^(2 + x)*(-16*x - 8*x^2 - 8*x^3 - 2*x^4) + E^(4 +
2*x)*(4*x^3 + 2*x^4),x]

[Out]

-E^(14 + x)^2 - x + 8*x^2 - 8*E^(2 + x)*x^2 + x^4 - 2*E^(2 + x)*x^4 + E^(4 + 2*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

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2227

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x+8 x^2+x^4+\int e^{196+28 x+x^2} (-28-2 x) \, dx+\int e^{2+x} \left (-16 x-8 x^2-8 x^3-2 x^4\right ) \, dx+\int e^{4+2 x} \left (4 x^3+2 x^4\right ) \, dx\\ &=-x+8 x^2+x^4+\int e^{(14+x)^2} (-28-2 x) \, dx+\int e^{4+2 x} x^3 (4+2 x) \, dx+\int \left (-16 e^{2+x} x-8 e^{2+x} x^2-8 e^{2+x} x^3-2 e^{2+x} x^4\right ) \, dx\\ &=-e^{(14+x)^2}-x+8 x^2+x^4-2 \int e^{2+x} x^4 \, dx-8 \int e^{2+x} x^2 \, dx-8 \int e^{2+x} x^3 \, dx-16 \int e^{2+x} x \, dx+\int \left (4 e^{4+2 x} x^3+2 e^{4+2 x} x^4\right ) \, dx\\ &=-e^{(14+x)^2}-x-16 e^{2+x} x+8 x^2-8 e^{2+x} x^2-8 e^{2+x} x^3+x^4-2 e^{2+x} x^4+2 \int e^{4+2 x} x^4 \, dx+4 \int e^{4+2 x} x^3 \, dx+8 \int e^{2+x} x^3 \, dx+16 \int e^{2+x} \, dx+16 \int e^{2+x} x \, dx+24 \int e^{2+x} x^2 \, dx\\ &=16 e^{2+x}-e^{(14+x)^2}-x+8 x^2+16 e^{2+x} x^2+2 e^{4+2 x} x^3+x^4-2 e^{2+x} x^4+e^{4+2 x} x^4-4 \int e^{4+2 x} x^3 \, dx-6 \int e^{4+2 x} x^2 \, dx-16 \int e^{2+x} \, dx-24 \int e^{2+x} x^2 \, dx-48 \int e^{2+x} x \, dx\\ &=-e^{(14+x)^2}-x-48 e^{2+x} x+8 x^2-8 e^{2+x} x^2-3 e^{4+2 x} x^2+x^4-2 e^{2+x} x^4+e^{4+2 x} x^4+6 \int e^{4+2 x} x \, dx+6 \int e^{4+2 x} x^2 \, dx+48 \int e^{2+x} \, dx+48 \int e^{2+x} x \, dx\\ &=48 e^{2+x}-e^{(14+x)^2}-x+3 e^{4+2 x} x+8 x^2-8 e^{2+x} x^2+x^4-2 e^{2+x} x^4+e^{4+2 x} x^4-3 \int e^{4+2 x} \, dx-6 \int e^{4+2 x} x \, dx-48 \int e^{2+x} \, dx\\ &=-e^{(14+x)^2}-\frac {3}{2} e^{4+2 x}-x+8 x^2-8 e^{2+x} x^2+x^4-2 e^{2+x} x^4+e^{4+2 x} x^4+3 \int e^{4+2 x} \, dx\\ &=-e^{(14+x)^2}-x+8 x^2-8 e^{2+x} x^2+x^4-2 e^{2+x} x^4+e^{4+2 x} x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 47, normalized size = 1.62 \begin {gather*} -e^{(14+x)^2}-x+8 x^2+x^4+e^{4+2 x} x^4-2 e^{2+x} x \left (4 x+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + E^(196 + 28*x + x^2)*(-28 - 2*x) + 16*x + 4*x^3 + E^(2 + x)*(-16*x - 8*x^2 - 8*x^3 - 2*x^4) + E
^(4 + 2*x)*(4*x^3 + 2*x^4),x]

[Out]

-E^(14 + x)^2 - x + 8*x^2 + x^4 + E^(4 + 2*x)*x^4 - 2*E^(2 + x)*x*(4*x + x^3)

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fricas [A]  time = 0.70, size = 48, normalized size = 1.66 \begin {gather*} x^{4} e^{\left (2 \, x + 4\right )} + x^{4} + 8 \, x^{2} - 2 \, {\left (x^{4} + 4 \, x^{2}\right )} e^{\left (x + 2\right )} - x - e^{\left (x^{2} + 28 \, x + 196\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-28)*exp(x^2+28*x+196)+(2*x^4+4*x^3)*exp(2+x)^2+(-2*x^4-8*x^3-8*x^2-16*x)*exp(2+x)+4*x^3+16*x-1
,x, algorithm="fricas")

[Out]

x^4*e^(2*x + 4) + x^4 + 8*x^2 - 2*(x^4 + 4*x^2)*e^(x + 2) - x - e^(x^2 + 28*x + 196)

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giac [A]  time = 0.15, size = 48, normalized size = 1.66 \begin {gather*} x^{4} e^{\left (2 \, x + 4\right )} + x^{4} + 8 \, x^{2} - 2 \, {\left (x^{4} + 4 \, x^{2}\right )} e^{\left (x + 2\right )} - x - e^{\left (x^{2} + 28 \, x + 196\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-28)*exp(x^2+28*x+196)+(2*x^4+4*x^3)*exp(2+x)^2+(-2*x^4-8*x^3-8*x^2-16*x)*exp(2+x)+4*x^3+16*x-1
,x, algorithm="giac")

[Out]

x^4*e^(2*x + 4) + x^4 + 8*x^2 - 2*(x^4 + 4*x^2)*e^(x + 2) - x - e^(x^2 + 28*x + 196)

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maple [A]  time = 0.05, size = 47, normalized size = 1.62

method	result	size
risch	\(-{\mathrm e}^{\left (x +14\right )^{2}}+{\mathrm e}^{2 x +4} x^{4}+\left (-2 x^{4}-8 x^{2}\right ) {\mathrm e}^{2+x}+x^{4}+8 x^{2}-x\)	\(47\)
norman	\(x^{4}+{\mathrm e}^{2 x +4} x^{4}-x +8 x^{2}-8 x^{2} {\mathrm e}^{2+x}-2 \,{\mathrm e}^{2+x} x^{4}-{\mathrm e}^{x^{2}+28 x +196}\)	\(52\)
default	\(-x -{\mathrm e}^{x^{2}+28 x +196}+{\mathrm e}^{2 x +4} \left (2+x \right )^{4}-8 \,{\mathrm e}^{2 x +4} \left (2+x \right )^{3}+24 \,{\mathrm e}^{2 x +4} \left (2+x \right )^{2}-32 \left (2+x \right ) {\mathrm e}^{2 x +4}+16 \,{\mathrm e}^{2 x +4}-56 \,{\mathrm e}^{2+x} \left (2+x \right )^{2}+16 \,{\mathrm e}^{2+x} \left (2+x \right )^{3}+96 \,{\mathrm e}^{2+x} \left (2+x \right )-64 \,{\mathrm e}^{2+x}-2 \,{\mathrm e}^{2+x} \left (2+x \right )^{4}+8 x^{2}+x^{4}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x-28)*exp(x^2+28*x+196)+(2*x^4+4*x^3)*exp(2+x)^2+(-2*x^4-8*x^3-8*x^2-16*x)*exp(2+x)+4*x^3+16*x-1,x,met
hod=_RETURNVERBOSE)

[Out]

-exp((x+14)^2)+exp(2*x+4)*x^4+(-2*x^4-8*x^2)*exp(2+x)+x^4+8*x^2-x

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maxima [A]  time = 0.37, size = 51, normalized size = 1.76 \begin {gather*} x^{4} e^{\left (2 \, x + 4\right )} + x^{4} + 8 \, x^{2} - 2 \, {\left (x^{4} e^{2} + 4 \, x^{2} e^{2}\right )} e^{x} - x - e^{\left (x^{2} + 28 \, x + 196\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-28)*exp(x^2+28*x+196)+(2*x^4+4*x^3)*exp(2+x)^2+(-2*x^4-8*x^3-8*x^2-16*x)*exp(2+x)+4*x^3+16*x-1
,x, algorithm="maxima")

[Out]

x^4*e^(2*x + 4) + x^4 + 8*x^2 - 2*(x^4*e^2 + 4*x^2*e^2)*e^x - x - e^(x^2 + 28*x + 196)

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mupad [B]  time = 0.21, size = 51, normalized size = 1.76 \begin {gather*} x^4\,{\mathrm {e}}^{2\,x+4}-{\mathrm {e}}^{x^2+28\,x+196}-8\,x^2\,{\mathrm {e}}^{x+2}-2\,x^4\,{\mathrm {e}}^{x+2}-x+8\,x^2+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x - exp(28*x + x^2 + 196)*(2*x + 28) - exp(x + 2)*(16*x + 8*x^2 + 8*x^3 + 2*x^4) + exp(2*x + 4)*(4*x^3
+ 2*x^4) + 4*x^3 - 1,x)

[Out]

x^4*exp(2*x + 4) - exp(28*x + x^2 + 196) - 8*x^2*exp(x + 2) - 2*x^4*exp(x + 2) - x + 8*x^2 + x^4

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sympy [B]  time = 0.20, size = 46, normalized size = 1.59 \begin {gather*} x^{4} e^{2 x + 4} + x^{4} + 8 x^{2} - x + \left (- 2 x^{4} - 8 x^{2}\right ) e^{x + 2} - e^{x^{2} + 28 x + 196} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-28)*exp(x**2+28*x+196)+(2*x**4+4*x**3)*exp(2+x)**2+(-2*x**4-8*x**3-8*x**2-16*x)*exp(2+x)+4*x**
3+16*x-1,x)

[Out]

x**4*exp(2*x + 4) + x**4 + 8*x**2 - x + (-2*x**4 - 8*x**2)*exp(x + 2) - exp(x**2 + 28*x + 196)

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