3.62.12 \(\int (8+12 x+3 x^2+2 e^{2 x} (1+2 x)) \, dx\)

Optimal. Leaf size=18 \[ x+x \left (-2+2 e^{2 x}+(3+x)^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.67, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2176, 2194} \begin {gather*} x^3+6 x^2+8 x-e^{2 x}+e^{2 x} (2 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8 + 12*x + 3*x^2 + 2*E^(2*x)*(1 + 2*x),x]

[Out]

-E^(2*x) + 8*x + 6*x^2 + x^3 + E^(2*x)*(1 + 2*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} &=8 x+6 x^2+x^3+2 \int e^{2 x} (1+2 x) \, dx\\ &=8 x+6 x^2+x^3+e^{2 x} (1+2 x)-2 \int e^{2 x} \, dx\\ &=-e^{2 x}+8 x+6 x^2+x^3+e^{2 x} (1+2 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} 8 x+2 e^{2 x} x+6 x^2+x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8 + 12*x + 3*x^2 + 2*E^(2*x)*(1 + 2*x),x]

[Out]

8*x + 2*E^(2*x)*x + 6*x^2 + x^3

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fricas [A]  time = 0.94, size = 21, normalized size = 1.17 \begin {gather*} x^{3} + 6 \, x^{2} + x e^{\left (2 \, x + \log \relax (2)\right )} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+1)*exp(log(2)+2*x)+3*x^2+12*x+8,x, algorithm="fricas")

[Out]

x^3 + 6*x^2 + x*e^(2*x + log(2)) + 8*x

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giac [A]  time = 0.13, size = 21, normalized size = 1.17 \begin {gather*} x^{3} + 6 \, x^{2} + x e^{\left (2 \, x + \log \relax (2)\right )} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+1)*exp(log(2)+2*x)+3*x^2+12*x+8,x, algorithm="giac")

[Out]

x^3 + 6*x^2 + x*e^(2*x + log(2)) + 8*x

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maple [A]  time = 0.07, size = 20, normalized size = 1.11

method	result	size
risch	\(x^{3}+2 x \,{\mathrm e}^{2 x}+8 x +6 x^{2}\)	\(20\)
norman	\(x^{3}+{\mathrm e}^{\ln \relax (2)+2 x} x +8 x +6 x^{2}\)	\(22\)
default	\(8 x +\frac {{\mathrm e}^{\ln \relax (2)+2 x} \left (\ln \relax (2)+2 x \right )}{2}-\frac {{\mathrm e}^{\ln \relax (2)+2 x} \ln \relax (2)}{2}+6 x^{2}+x^{3}\)	\(39\)
derivativedivides	\(4 \ln \relax (2)+8 x +\frac {3 \left (\ln \relax (2)+2 x \right )^{2}}{2}+x^{3}+\frac {{\mathrm e}^{\ln \relax (2)+2 x} \left (\ln \relax (2)+2 x \right )}{2}-\frac {{\mathrm e}^{\ln \relax (2)+2 x} \ln \relax (2)}{2}-3 \ln \relax (2) \left (\ln \relax (2)+2 x \right )\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+1)*exp(ln(2)+2*x)+3*x^2+12*x+8,x,method=_RETURNVERBOSE)

[Out]

x^3+2*x*exp(2*x)+8*x+6*x^2

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maxima [A]  time = 0.38, size = 19, normalized size = 1.06 \begin {gather*} x^{3} + 6 \, x^{2} + 2 \, x e^{\left (2 \, x\right )} + 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+1)*exp(log(2)+2*x)+3*x^2+12*x+8,x, algorithm="maxima")

[Out]

x^3 + 6*x^2 + 2*x*e^(2*x) + 8*x

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mupad [B]  time = 4.22, size = 16, normalized size = 0.89 \begin {gather*} x\,\left (6\,x+2\,{\mathrm {e}}^{2\,x}+x^2+8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(12*x + exp(2*x + log(2))*(2*x + 1) + 3*x^2 + 8,x)

[Out]

x*(6*x + 2*exp(2*x) + x^2 + 8)

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sympy [A]  time = 0.08, size = 19, normalized size = 1.06 \begin {gather*} x^{3} + 6 x^{2} + 2 x e^{2 x} + 8 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+1)*exp(ln(2)+2*x)+3*x**2+12*x+8,x)

[Out]

x**3 + 6*x**2 + 2*x*exp(2*x) + 8*x

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