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

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} $$\begin{array}{c}\frac{5x^2}{6}+\frac{1}{6}{x}^2{\log }^2(x)+x^2\log (x)-x+e^x\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{3}\int (-3+3e^x+8x+7x\log (x)+x{\log }^2(x))dx\\ & =-x+\frac{4x^2}{3}+\frac{1}{3}\int x{\log }^2(x)dx+\frac{7}{3}\int x\log (x)dx+\int e^{x}dx\\ & =e^x-x+\frac{3x^2}{4}+\frac{7}{6}{x}^2\log (x)+\frac{1}{6}{x}^2{\log }^2(x)-\frac{1}{3}\int x\log (x)dx\\ & =e^x-x+\frac{5x^2}{6}+x^2\log (x)+\frac{1}{6}{x}^2{\log }^2(x)\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{1}{6}{x}^2\log (x{)}^2+x^2\log (x)+\frac{5}{6}{x}^2-x+e^x\end{array}$$

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 $$\begin{array}{c}\frac{1}{6}{x}^2\log (x{)}^2+x^2\log (x)+\frac{5}{6}{x}^2-x+e^x\end{array}$$

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	$-x+\frac{5x^2}{6}+x^2\ln (x)+\frac{x^2\ln (x{)}^2}{6}+{\mathrm{e}}^x$	$27$
norman	$-x+\frac{5x^2}{6}+x^2\ln (x)+\frac{x^2\ln (x{)}^2}{6}+{\mathrm{e}}^x$	$27$
risch	$-x+\frac{5x^2}{6}+x^2\ln (x)+\frac{x^2\ln (x{)}^2}{6}+{\mathrm{e}}^x$	$27$

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 $$\begin{array}{c}\frac{1}{12}(2\log (x{)}^2-2\log (x)+1)x^2+\frac{7}{6}{x}^2\log (x)+\frac{3}{4}{x}^2-x+e^x\end{array}$$

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 $$\begin{array}{c}{\mathrm{e}}^x-x+x^2\ln (x)+\frac{x^2{\ln (x)}^2}{6}+\frac{5x^2}{6}\end{array}$$

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 $$\begin{array}{c}\frac{x^2\log {(x)}^2}{6}+x^2\log (x)+\frac{5x^2}{6}-x+e^x\end{array}$$

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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