3.61.18 \(\int \frac {1}{6} (1+18 e^{3 x}+6 e^{6 x}+4 e^{2 x^2} x+e^{x^2} (12 x+e^{3 x} (6+4 x))) \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{2} \left (-e^4+\frac {1}{3} \left (\left (3+e^{3 x}+e^{x^2}\right )^2+x\right )\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 49, normalized size of antiderivative = 1.63, number of steps used = 9, number of rules used = 5, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {12, 2194, 2209, 6742, 2236} \begin {gather*} e^{x^2}+\frac {e^{2 x^2}}{6}+\frac {1}{3} e^{x^2+3 x}+\frac {x}{6}+e^{3 x}+\frac {e^{6 x}}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 18*E^(3*x) + 6*E^(6*x) + 4*E^(2*x^2)*x + E^x^2*(12*x + E^(3*x)*(6 + 4*x)))/6,x]

[Out]

E^(3*x) + E^(6*x)/6 + E^x^2 + E^(2*x^2)/6 + E^(3*x + x^2)/3 + x/6

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 49, normalized size = 1.63 \begin {gather*} e^{3 x}+\frac {e^{6 x}}{6}+e^{x^2}+\frac {e^{2 x^2}}{6}+\frac {1}{3} e^{3 x+x^2}+\frac {x}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 18*E^(3*x) + 6*E^(6*x) + 4*E^(2*x^2)*x + E^x^2*(12*x + E^(3*x)*(6 + 4*x)))/6,x]

[Out]

E^(3*x) + E^(6*x)/6 + E^x^2 + E^(2*x^2)/6 + E^(3*x + x^2)/3 + x/6

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fricas [A]  time = 0.61, size = 34, normalized size = 1.13 \begin {gather*} \frac {1}{3} \, {\left (e^{\left (3 \, x\right )} + 3\right )} e^{\left (x^{2}\right )} + \frac {1}{6} \, x + \frac {1}{6} \, e^{\left (2 \, x^{2}\right )} + \frac {1}{6} \, e^{\left (6 \, x\right )} + e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*x*exp(x^2)^2+1/6*((4*x+6)*exp(3*x)+12*x)*exp(x^2)+exp(3*x)^2+3*exp(3*x)+1/6,x, algorithm="fricas
")

[Out]

1/3*(e^(3*x) + 3)*e^(x^2) + 1/6*x + 1/6*e^(2*x^2) + 1/6*e^(6*x) + e^(3*x)

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giac [A]  time = 1.10, size = 36, normalized size = 1.20 \begin {gather*} \frac {1}{6} \, x + \frac {1}{6} \, e^{\left (2 \, x^{2}\right )} + \frac {1}{3} \, e^{\left (x^{2} + 3 \, x\right )} + e^{\left (x^{2}\right )} + \frac {1}{6} \, e^{\left (6 \, x\right )} + e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*x*exp(x^2)^2+1/6*((4*x+6)*exp(3*x)+12*x)*exp(x^2)+exp(3*x)^2+3*exp(3*x)+1/6,x, algorithm="giac")

[Out]

1/6*x + 1/6*e^(2*x^2) + 1/3*e^(x^2 + 3*x) + e^(x^2) + 1/6*e^(6*x) + e^(3*x)

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maple [A]  time = 0.04, size = 35, normalized size = 1.17

method	result	size
risch	\(\frac {x}{6}+\frac {{\mathrm e}^{2 x^{2}}}{6}+\frac {{\mathrm e}^{6 x}}{6}+\frac {{\mathrm e}^{\left (3+x \right ) x}}{3}+{\mathrm e}^{x^{2}}+{\mathrm e}^{3 x}\)	\(35\)
default	\(\frac {x}{6}+\frac {{\mathrm e}^{6 x}}{6}+\frac {{\mathrm e}^{2 x^{2}}}{6}+{\mathrm e}^{x^{2}}+\frac {{\mathrm e}^{x^{2}+3 x}}{3}+{\mathrm e}^{3 x}\)	\(39\)
norman	\(\frac {x}{6}+\frac {{\mathrm e}^{2 x^{2}}}{6}+\frac {{\mathrm e}^{6 x}}{6}+\frac {{\mathrm e}^{x^{2}} {\mathrm e}^{3 x}}{3}+{\mathrm e}^{x^{2}}+{\mathrm e}^{3 x}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/3*x*exp(x^2)^2+1/6*((4*x+6)*exp(3*x)+12*x)*exp(x^2)+exp(3*x)^2+3*exp(3*x)+1/6,x,method=_RETURNVERBOSE)

[Out]

1/6*x+1/6*exp(2*x^2)+1/6*exp(6*x)+1/3*exp((3+x)*x)+exp(x^2)+exp(3*x)

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maxima [A]  time = 0.35, size = 34, normalized size = 1.13 \begin {gather*} \frac {1}{3} \, {\left (e^{\left (3 \, x\right )} + 3\right )} e^{\left (x^{2}\right )} + \frac {1}{6} \, x + \frac {1}{6} \, e^{\left (2 \, x^{2}\right )} + \frac {1}{6} \, e^{\left (6 \, x\right )} + e^{\left (3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*x*exp(x^2)^2+1/6*((4*x+6)*exp(3*x)+12*x)*exp(x^2)+exp(3*x)^2+3*exp(3*x)+1/6,x, algorithm="maxima
")

[Out]

1/3*(e^(3*x) + 3)*e^(x^2) + 1/6*x + 1/6*e^(2*x^2) + 1/6*e^(6*x) + e^(3*x)

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mupad [B]  time = 0.10, size = 36, normalized size = 1.20 \begin {gather*} \frac {x}{6}+{\mathrm {e}}^{3\,x}+\frac {{\mathrm {e}}^{6\,x}}{6}+{\mathrm {e}}^{x^2}+\frac {{\mathrm {e}}^{x^2+3\,x}}{3}+\frac {{\mathrm {e}}^{2\,x^2}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*exp(3*x) + exp(6*x) + (2*x*exp(2*x^2))/3 + (exp(x^2)*(12*x + exp(3*x)*(4*x + 6)))/6 + 1/6,x)

[Out]

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

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sympy [A]  time = 0.22, size = 37, normalized size = 1.23 \begin {gather*} \frac {x}{6} + \frac {\left (6 e^{3 x} + 18\right ) e^{x^{2}}}{18} + \frac {e^{6 x}}{6} + e^{3 x} + \frac {e^{2 x^{2}}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*x*exp(x**2)**2+1/6*((4*x+6)*exp(3*x)+12*x)*exp(x**2)+exp(3*x)**2+3*exp(3*x)+1/6,x)

[Out]

x/6 + (6*exp(3*x) + 18)*exp(x**2)/18 + exp(6*x)/6 + exp(3*x) + exp(2*x**2)/6

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