3.47.69 \(\int (2+2 e^{2 x}+2 x+256 x^3+e^x (8+2 x)) \, dx\)

Optimal. Leaf size=20 \[ 4+\left (3+e^x+x\right )^2-4 \left (10+x-16 x^4\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.50, number of steps used = 4, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2194, 2176} \begin {gather*} 64 x^4+x^2+2 x-2 e^x+e^{2 x}+2 e^x (x+4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + 2*E^(2*x) + 2*x + 256*x^3 + E^x*(8 + 2*x),x]

[Out]

-2*E^x + E^(2*x) + 2*x + x^2 + 64*x^4 + 2*E^x*(4 + 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 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} &=2 x+x^2+64 x^4+2 \int e^{2 x} \, dx+\int e^x (8+2 x) \, dx\\ &=e^{2 x}+2 x+x^2+64 x^4+2 e^x (4+x)-2 \int e^x \, dx\\ &=-2 e^x+e^{2 x}+2 x+x^2+64 x^4+2 e^x (4+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.60 \begin {gather*} 2 \left (\frac {e^{2 x}}{2}+x+\frac {x^2}{2}+32 x^4+e^x (3+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + 2*E^(2*x) + 2*x + 256*x^3 + E^x*(8 + 2*x),x]

[Out]

2*(E^(2*x)/2 + x + x^2/2 + 32*x^4 + E^x*(3 + x))

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fricas [A]  time = 0.83, size = 23, normalized size = 1.15 \begin {gather*} 64 \, x^{4} + x^{2} + 2 \, {\left (x + 3\right )} e^{x} + 2 \, x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)^2+(2*x+8)*exp(x)+256*x^3+2*x+2,x, algorithm="fricas")

[Out]

64*x^4 + x^2 + 2*(x + 3)*e^x + 2*x + e^(2*x)

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giac [A]  time = 0.17, size = 23, normalized size = 1.15 \begin {gather*} 64 \, x^{4} + x^{2} + 2 \, {\left (x + 3\right )} e^{x} + 2 \, x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)^2+(2*x+8)*exp(x)+256*x^3+2*x+2,x, algorithm="giac")

[Out]

64*x^4 + x^2 + 2*(x + 3)*e^x + 2*x + e^(2*x)

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maple [A]  time = 0.03, size = 25, normalized size = 1.25

method	result	size
risch	\({\mathrm e}^{2 x}+\left (2 x +6\right ) {\mathrm e}^{x}+64 x^{4}+x^{2}+2 x\)	\(25\)
default	\(2 x +2 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}+x^{2}+64 x^{4}+{\mathrm e}^{2 x}\)	\(26\)
norman	\(2 x +2 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}+x^{2}+64 x^{4}+{\mathrm e}^{2 x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(x)^2+(2*x+8)*exp(x)+256*x^3+2*x+2,x,method=_RETURNVERBOSE)

[Out]

exp(2*x)+(2*x+6)*exp(x)+64*x^4+x^2+2*x

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maxima [A]  time = 0.36, size = 23, normalized size = 1.15 \begin {gather*} 64 \, x^{4} + x^{2} + 2 \, {\left (x + 3\right )} e^{x} + 2 \, x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)^2+(2*x+8)*exp(x)+256*x^3+2*x+2,x, algorithm="maxima")

[Out]

64*x^4 + x^2 + 2*(x + 3)*e^x + 2*x + e^(2*x)

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mupad [B]  time = 3.13, size = 25, normalized size = 1.25 \begin {gather*} 2\,x+{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^x+x^2+64\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 2*exp(2*x) + exp(x)*(2*x + 8) + 256*x^3 + 2,x)

[Out]

2*x + exp(2*x) + 6*exp(x) + 2*x*exp(x) + x^2 + 64*x^4

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sympy [A]  time = 0.10, size = 24, normalized size = 1.20 \begin {gather*} 64 x^{4} + x^{2} + 2 x + \left (2 x + 6\right ) e^{x} + e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)**2+(2*x+8)*exp(x)+256*x**3+2*x+2,x)

[Out]

64*x**4 + x**2 + 2*x + (2*x + 6)*exp(x) + exp(2*x)

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