3.89.76 \(\int \frac {1}{3} (-3+3 e^x+8 x+7 x \log (x)+x \log ^2(x)) \, dx\)

Optimal. Leaf size=23 \[ -1+e^x-x+\frac {1}{6} x (5+\log (x)) (x+x \log (x)) \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {12, 2194, 2304, 2305} \begin {gather*} \frac {5 x^2}{6}+\frac {1}{6} x^2 \log ^2(x)+x^2 \log (x)-x+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 3*E^x + 8*x + 7*x*Log[x] + x*Log[x]^2)/3,x]

[Out]

E^x - x + (5*x^2)/6 + x^2*Log[x] + (x^2*Log[x]^2)/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 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (-3+3 e^x+8 x+7 x \log (x)+x \log ^2(x)\right ) \, dx\\ &=-x+\frac {4 x^2}{3}+\frac {1}{3} \int x \log ^2(x) \, dx+\frac {7}{3} \int x \log (x) \, dx+\int e^x \, dx\\ &=e^x-x+\frac {3 x^2}{4}+\frac {7}{6} x^2 \log (x)+\frac {1}{6} x^2 \log ^2(x)-\frac {1}{3} \int x \log (x) \, dx\\ &=e^x-x+\frac {5 x^2}{6}+x^2 \log (x)+\frac {1}{6} x^2 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 38, normalized size = 1.65 \begin {gather*} \frac {1}{3} \left (3 e^x-3 x+\frac {5 x^2}{2}+3 x^2 \log (x)+\frac {1}{2} x^2 \log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 3*E^x + 8*x + 7*x*Log[x] + x*Log[x]^2)/3,x]

[Out]

(3*E^x - 3*x + (5*x^2)/2 + 3*x^2*Log[x] + (x^2*Log[x]^2)/2)/3

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fricas [A]  time = 0.48, size = 26, normalized size = 1.13 \begin {gather*} \frac {1}{6} \, x^{2} \log \relax (x)^{2} + x^{2} \log \relax (x) + \frac {5}{6} \, x^{2} - x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*x*log(x)^2+7/3*x*log(x)+exp(x)+8/3*x-1,x, algorithm="fricas")

[Out]

1/6*x^2*log(x)^2 + x^2*log(x) + 5/6*x^2 - x + e^x

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giac [A]  time = 0.22, size = 26, normalized size = 1.13 \begin {gather*} \frac {1}{6} \, x^{2} \log \relax (x)^{2} + x^{2} \log \relax (x) + \frac {5}{6} \, x^{2} - x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*x*log(x)^2+7/3*x*log(x)+exp(x)+8/3*x-1,x, algorithm="giac")

[Out]

1/6*x^2*log(x)^2 + x^2*log(x) + 5/6*x^2 - x + e^x

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maple [A]  time = 0.03, size = 27, normalized size = 1.17

method	result	size
default	\(-x +\frac {5 x^{2}}{6}+x^{2} \ln \relax (x )+\frac {x^{2} \ln \relax (x )^{2}}{6}+{\mathrm e}^{x}\)	\(27\)
norman	\(-x +\frac {5 x^{2}}{6}+x^{2} \ln \relax (x )+\frac {x^{2} \ln \relax (x )^{2}}{6}+{\mathrm e}^{x}\)	\(27\)
risch	\(-x +\frac {5 x^{2}}{6}+x^{2} \ln \relax (x )+\frac {x^{2} \ln \relax (x )^{2}}{6}+{\mathrm e}^{x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*x*ln(x)^2+7/3*x*ln(x)+exp(x)+8/3*x-1,x,method=_RETURNVERBOSE)

[Out]

-x+5/6*x^2+x^2*ln(x)+1/6*x^2*ln(x)^2+exp(x)

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maxima [A]  time = 0.36, size = 35, normalized size = 1.52 \begin {gather*} \frac {1}{12} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + \frac {7}{6} \, x^{2} \log \relax (x) + \frac {3}{4} \, x^{2} - x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*x*log(x)^2+7/3*x*log(x)+exp(x)+8/3*x-1,x, algorithm="maxima")

[Out]

1/12*(2*log(x)^2 - 2*log(x) + 1)*x^2 + 7/6*x^2*log(x) + 3/4*x^2 - x + e^x

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mupad [B]  time = 5.12, size = 26, normalized size = 1.13 \begin {gather*} {\mathrm {e}}^x-x+x^2\,\ln \relax (x)+\frac {x^2\,{\ln \relax (x)}^2}{6}+\frac {5\,x^2}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x)/3 + exp(x) + (x*log(x)^2)/3 + (7*x*log(x))/3 - 1,x)

[Out]

exp(x) - x + x^2*log(x) + (x^2*log(x)^2)/6 + (5*x^2)/6

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sympy [A]  time = 0.27, size = 27, normalized size = 1.17 \begin {gather*} \frac {x^{2} \log {\relax (x )}^{2}}{6} + x^{2} \log {\relax (x )} + \frac {5 x^{2}}{6} - x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*x*ln(x)**2+7/3*x*ln(x)+exp(x)+8/3*x-1,x)

[Out]

x**2*log(x)**2/6 + x**2*log(x) + 5*x**2/6 - x + exp(x)

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