3.73.21 \(\int \frac {1}{2} e^{-e^4} (2 x+e^{e^4} (3+11 e^{\frac {11 x^3}{6}} x^2)) \, dx\)

Optimal. Leaf size=26 \[ e^{\frac {11 x^3}{6}}+\frac {1}{2} x \left (3+e^{-e^4} x\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {12, 2209} \begin {gather*} e^{\frac {11 x^3}{6}}+\frac {1}{2} e^{-e^4} x^2+\frac {3 x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x + E^E^4*(3 + 11*E^((11*x^3)/6)*x^2))/(2*E^E^4),x]

[Out]

E^((11*x^3)/6) + (3*x)/2 + x^2/(2*E^E^4)

Rule 12

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} e^{-e^4} \int \left (2 x+e^{e^4} \left (3+11 e^{\frac {11 x^3}{6}} x^2\right )\right ) \, dx\\ &=\frac {1}{2} e^{-e^4} x^2+\frac {1}{2} \int \left (3+11 e^{\frac {11 x^3}{6}} x^2\right ) \, dx\\ &=\frac {3 x}{2}+\frac {1}{2} e^{-e^4} x^2+\frac {11}{2} \int e^{\frac {11 x^3}{6}} x^2 \, dx\\ &=e^{\frac {11 x^3}{6}}+\frac {3 x}{2}+\frac {1}{2} e^{-e^4} x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.12 \begin {gather*} e^{\frac {11 x^3}{6}}+\frac {3 x}{2}+\frac {1}{2} e^{-e^4} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x + E^E^4*(3 + 11*E^((11*x^3)/6)*x^2))/(2*E^E^4),x]

[Out]

E^((11*x^3)/6) + (3*x)/2 + x^2/(2*E^E^4)

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fricas [A]  time = 0.75, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + {\left (3 \, x + 2 \, e^{\left (\frac {11}{6} \, x^{3}\right )}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((11*x^2*exp(11/6*x^3)+3)*exp(exp(4))+2*x)/exp(exp(4)),x, algorithm="fricas")

[Out]

1/2*(x^2 + (3*x + 2*e^(11/6*x^3))*e^(e^4))*e^(-e^4)

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giac [A]  time = 0.18, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + {\left (3 \, x + 2 \, e^{\left (\frac {11}{6} \, x^{3}\right )}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((11*x^2*exp(11/6*x^3)+3)*exp(exp(4))+2*x)/exp(exp(4)),x, algorithm="giac")

[Out]

1/2*(x^2 + (3*x + 2*e^(11/6*x^3))*e^(e^4))*e^(-e^4)

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maple [A]  time = 0.03, size = 21, normalized size = 0.81

method	result	size
norman	\(\frac {3 x}{2}+\frac {{\mathrm e}^{-{\mathrm e}^{4}} x^{2}}{2}+{\mathrm e}^{\frac {11 x^{3}}{6}}\)	\(21\)
risch	\(\frac {3 x}{2}+\frac {{\mathrm e}^{-{\mathrm e}^{4}} x^{2}}{2}+{\mathrm e}^{\frac {11 x^{3}}{6}}\)	\(21\)
default	\(\frac {{\mathrm e}^{-{\mathrm e}^{4}} \left ({\mathrm e}^{{\mathrm e}^{4}} \left (3 x +2 \,{\mathrm e}^{\frac {11 x^{3}}{6}}\right )+x^{2}\right )}{2}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((11*x^2*exp(11/6*x^3)+3)*exp(exp(4))+2*x)/exp(exp(4)),x,method=_RETURNVERBOSE)

[Out]

3/2*x+1/2/exp(exp(4))*x^2+exp(11/6*x^3)

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maxima [A]  time = 0.35, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + {\left (3 \, x + 2 \, e^{\left (\frac {11}{6} \, x^{3}\right )}\right )} e^{\left (e^{4}\right )}\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((11*x^2*exp(11/6*x^3)+3)*exp(exp(4))+2*x)/exp(exp(4)),x, algorithm="maxima")

[Out]

1/2*(x^2 + (3*x + 2*e^(11/6*x^3))*e^(e^4))*e^(-e^4)

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mupad [B]  time = 0.07, size = 20, normalized size = 0.77 \begin {gather*} \frac {3\,x}{2}+{\mathrm {e}}^{\frac {11\,x^3}{6}}+\frac {x^2\,{\mathrm {e}}^{-{\mathrm {e}}^4}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-exp(4))*(x + (exp(exp(4))*(11*x^2*exp((11*x^3)/6) + 3))/2),x)

[Out]

(3*x)/2 + exp((11*x^3)/6) + (x^2*exp(-exp(4)))/2

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sympy [A]  time = 0.12, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{2 e^{e^{4}}} + \frac {3 x}{2} + e^{\frac {11 x^{3}}{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((11*x**2*exp(11/6*x**3)+3)*exp(exp(4))+2*x)/exp(exp(4)),x)

[Out]

x**2*exp(-exp(4))/2 + 3*x/2 + exp(11*x**3/6)

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