3.65.5 \(\int \frac {1}{3} (-3+2 x+150 x \log (x)+150 x \log ^2(x)) \, dx\)

Optimal. Leaf size=21 \[ 7-x+\frac {x^2}{3}+25 x^2 \log ^2(x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2304, 2305} \begin {gather*} \frac {x^2}{3}+25 x^2 \log ^2(x)-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 2*x + 150*x*Log[x] + 150*x*Log[x]^2)/3,x]

[Out]

-x + x^2/3 + 25*x^2*Log[x]^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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+2 x+150 x \log (x)+150 x \log ^2(x)\right ) \, dx\\ &=-x+\frac {x^2}{3}+50 \int x \log (x) \, dx+50 \int x \log ^2(x) \, dx\\ &=-x-\frac {73 x^2}{6}+25 x^2 \log (x)+25 x^2 \log ^2(x)-50 \int x \log (x) \, dx\\ &=-x+\frac {x^2}{3}+25 x^2 \log ^2(x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 20, normalized size = 0.95 \begin {gather*} -x+\frac {x^2}{3}+25 x^2 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 2*x + 150*x*Log[x] + 150*x*Log[x]^2)/3,x]

[Out]

-x + x^2/3 + 25*x^2*Log[x]^2

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 18, normalized size = 0.86 \begin {gather*} 25 \, x^{2} \log \relax (x)^{2} + \frac {1}{3} \, x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50*x*log(x)^2+50*x*log(x)+2/3*x-1,x, algorithm="fricas")

[Out]

25*x^2*log(x)^2 + 1/3*x^2 - x

________________________________________________________________________________________

giac [A]  time = 1.08, size = 18, normalized size = 0.86 \begin {gather*} 25 \, x^{2} \log \relax (x)^{2} + \frac {1}{3} \, x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50*x*log(x)^2+50*x*log(x)+2/3*x-1,x, algorithm="giac")

[Out]

25*x^2*log(x)^2 + 1/3*x^2 - x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 19, normalized size = 0.90

method	result	size
default	\(\frac {x^{2}}{3}-x +25 x^{2} \ln \relax (x )^{2}\)	\(19\)
norman	\(\frac {x^{2}}{3}-x +25 x^{2} \ln \relax (x )^{2}\)	\(19\)
risch	\(\frac {x^{2}}{3}-x +25 x^{2} \ln \relax (x )^{2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(50*x*ln(x)^2+50*x*ln(x)+2/3*x-1,x,method=_RETURNVERBOSE)

[Out]

1/3*x^2-x+25*x^2*ln(x)^2

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 33, normalized size = 1.57 \begin {gather*} \frac {25}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + 25 \, x^{2} \log \relax (x) - \frac {73}{6} \, x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50*x*log(x)^2+50*x*log(x)+2/3*x-1,x, algorithm="maxima")

[Out]

25/2*(2*log(x)^2 - 2*log(x) + 1)*x^2 + 25*x^2*log(x) - 73/6*x^2 - x

________________________________________________________________________________________

mupad [B]  time = 4.20, size = 13, normalized size = 0.62 \begin {gather*} \frac {x\,\left (75\,x\,{\ln \relax (x)}^2+x-3\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x)/3 + 50*x*log(x)^2 + 50*x*log(x) - 1,x)

[Out]

(x*(x + 75*x*log(x)^2 - 3))/3

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 15, normalized size = 0.71 \begin {gather*} 25 x^{2} \log {\relax (x )}^{2} + \frac {x^{2}}{3} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50*x*ln(x)**2+50*x*ln(x)+2/3*x-1,x)

[Out]

25*x**2*log(x)**2 + x**2/3 - x

________________________________________________________________________________________