3.47.60 \(\int (1+10750 x-10752 x^2 \log (x)-16128 x^2 \log ^2(x)) \, dx\)

Optimal. Leaf size=21 \[ x-x^2+5376 x \left (x-x^2 \log ^2(x)\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2304, 2305} \begin {gather*} -5376 x^3 \log ^2(x)+5375 x^2+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + 10750*x - 10752*x^2*Log[x] - 16128*x^2*Log[x]^2,x]

[Out]

x + 5375*x^2 - 5376*x^3*Log[x]^2

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} &=x+5375 x^2-10752 \int x^2 \log (x) \, dx-16128 \int x^2 \log ^2(x) \, dx\\ &=x+5375 x^2+\frac {3584 x^3}{3}-3584 x^3 \log (x)-5376 x^3 \log ^2(x)+10752 \int x^2 \log (x) \, dx\\ &=x+5375 x^2-5376 x^3 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.76 \begin {gather*} x+5375 x^2-5376 x^3 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + 10750*x - 10752*x^2*Log[x] - 16128*x^2*Log[x]^2,x]

[Out]

x + 5375*x^2 - 5376*x^3*Log[x]^2

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fricas [A]  time = 0.73, size = 16, normalized size = 0.76 \begin {gather*} -5376 \, x^{3} \log \relax (x)^{2} + 5375 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-16128*x^2*log(x)^2-10752*x^2*log(x)+10750*x+1,x, algorithm="fricas")

[Out]

-5376*x^3*log(x)^2 + 5375*x^2 + x

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giac [A]  time = 0.14, size = 16, normalized size = 0.76 \begin {gather*} -5376 \, x^{3} \log \relax (x)^{2} + 5375 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-16128*x^2*log(x)^2-10752*x^2*log(x)+10750*x+1,x, algorithm="giac")

[Out]

-5376*x^3*log(x)^2 + 5375*x^2 + x

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maple [A]  time = 0.02, size = 17, normalized size = 0.81

method	result	size
default	\(5375 x^{2}+x -5376 x^{3} \ln \relax (x )^{2}\)	\(17\)
norman	\(5375 x^{2}+x -5376 x^{3} \ln \relax (x )^{2}\)	\(17\)
risch	\(5375 x^{2}+x -5376 x^{3} \ln \relax (x )^{2}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-16128*x^2*ln(x)^2-10752*x^2*ln(x)+10750*x+1,x,method=_RETURNVERBOSE)

[Out]

5375*x^2+x-5376*x^3*ln(x)^2

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maxima [A]  time = 0.37, size = 36, normalized size = 1.71 \begin {gather*} -\frac {1792}{3} \, {\left (9 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 2\right )} x^{3} - 3584 \, x^{3} \log \relax (x) + \frac {3584}{3} \, x^{3} + 5375 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-16128*x^2*log(x)^2-10752*x^2*log(x)+10750*x+1,x, algorithm="maxima")

[Out]

-1792/3*(9*log(x)^2 - 6*log(x) + 2)*x^3 - 3584*x^3*log(x) + 3584/3*x^3 + 5375*x^2 + x

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mupad [B]  time = 3.09, size = 16, normalized size = 0.76 \begin {gather*} -5376\,x^3\,{\ln \relax (x)}^2+5375\,x^2+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10750*x - 10752*x^2*log(x) - 16128*x^2*log(x)^2 + 1,x)

[Out]

x - 5376*x^3*log(x)^2 + 5375*x^2

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sympy [A]  time = 0.09, size = 15, normalized size = 0.71 \begin {gather*} - 5376 x^{3} \log {\relax (x )}^{2} + 5375 x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-16128*x**2*ln(x)**2-10752*x**2*ln(x)+10750*x+1,x)

[Out]

-5376*x**3*log(x)**2 + 5375*x**2 + x

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