3.74.63 \(\int \frac {-20+18 x-112 x^2-18 x^3+36 x^4+(2-2 x+12 x^2) \log (x)}{x} \, dx\)

Optimal. Leaf size=15 \[ (10+(1-3 x) x-\log (x))^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 39, normalized size of antiderivative = 2.60, number of steps used = 9, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {14, 2357, 2295, 2301, 2304} \begin {gather*} 9 x^4-6 x^3-59 x^2+6 x^2 \log (x)+20 x+\log ^2(x)-2 x \log (x)-20 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20 + 18*x - 112*x^2 - 18*x^3 + 36*x^4 + (2 - 2*x + 12*x^2)*Log[x])/x,x]

[Out]

20*x - 59*x^2 - 6*x^3 + 9*x^4 - 20*Log[x] - 2*x*Log[x] + 6*x^2*Log[x] + Log[x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{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 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (-10+9 x-56 x^2-9 x^3+18 x^4\right )}{x}+\frac {2 \left (1-x+6 x^2\right ) \log (x)}{x}\right ) \, dx\\ &=2 \int \frac {-10+9 x-56 x^2-9 x^3+18 x^4}{x} \, dx+2 \int \frac {\left (1-x+6 x^2\right ) \log (x)}{x} \, dx\\ &=2 \int \left (9-\frac {10}{x}-56 x-9 x^2+18 x^3\right ) \, dx+2 \int \left (-\log (x)+\frac {\log (x)}{x}+6 x \log (x)\right ) \, dx\\ &=18 x-56 x^2-6 x^3+9 x^4-20 \log (x)-2 \int \log (x) \, dx+2 \int \frac {\log (x)}{x} \, dx+12 \int x \log (x) \, dx\\ &=20 x-59 x^2-6 x^3+9 x^4-20 \log (x)-2 x \log (x)+6 x^2 \log (x)+\log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.00, size = 39, normalized size = 2.60 \begin {gather*} 20 x-59 x^2-6 x^3+9 x^4-20 \log (x)-2 x \log (x)+6 x^2 \log (x)+\log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20 + 18*x - 112*x^2 - 18*x^3 + 36*x^4 + (2 - 2*x + 12*x^2)*Log[x])/x,x]

[Out]

20*x - 59*x^2 - 6*x^3 + 9*x^4 - 20*Log[x] - 2*x*Log[x] + 6*x^2*Log[x] + Log[x]^2

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fricas [B]  time = 0.56, size = 37, normalized size = 2.47 \begin {gather*} 9 \, x^{4} - 6 \, x^{3} - 59 \, x^{2} + 2 \, {\left (3 \, x^{2} - x - 10\right )} \log \relax (x) + \log \relax (x)^{2} + 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-2*x+2)*log(x)+36*x^4-18*x^3-112*x^2+18*x-20)/x,x, algorithm="fricas")

[Out]

9*x^4 - 6*x^3 - 59*x^2 + 2*(3*x^2 - x - 10)*log(x) + log(x)^2 + 20*x

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giac [B]  time = 0.13, size = 40, normalized size = 2.67 \begin {gather*} 9 \, x^{4} - 6 \, x^{3} - 59 \, x^{2} + 2 \, {\left (3 \, x^{2} - x\right )} \log \relax (x) + \log \relax (x)^{2} + 20 \, x - 20 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-2*x+2)*log(x)+36*x^4-18*x^3-112*x^2+18*x-20)/x,x, algorithm="giac")

[Out]

9*x^4 - 6*x^3 - 59*x^2 + 2*(3*x^2 - x)*log(x) + log(x)^2 + 20*x - 20*log(x)

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maple [B]  time = 0.02, size = 40, normalized size = 2.67

method	result	size
default	\(9 x^{4}+6 x^{2} \ln \relax (x )-59 x^{2}-6 x^{3}-2 x \ln \relax (x )+20 x +\ln \relax (x )^{2}-20 \ln \relax (x )\)	\(40\)
norman	\(9 x^{4}+6 x^{2} \ln \relax (x )-59 x^{2}-6 x^{3}-2 x \ln \relax (x )+20 x +\ln \relax (x )^{2}-20 \ln \relax (x )\)	\(40\)
risch	\(\ln \relax (x )^{2}+\left (6 x^{2}-2 x \right ) \ln \relax (x )+9 x^{4}-6 x^{3}-59 x^{2}+20 x -20 \ln \relax (x )\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^2-2*x+2)*ln(x)+36*x^4-18*x^3-112*x^2+18*x-20)/x,x,method=_RETURNVERBOSE)

[Out]

9*x^4+6*x^2*ln(x)-59*x^2-6*x^3-2*x*ln(x)+20*x+ln(x)^2-20*ln(x)

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maxima [B]  time = 0.36, size = 39, normalized size = 2.60 \begin {gather*} 9 \, x^{4} - 6 \, x^{3} + 6 \, x^{2} \log \relax (x) - 59 \, x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2} + 20 \, x - 20 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-2*x+2)*log(x)+36*x^4-18*x^3-112*x^2+18*x-20)/x,x, algorithm="maxima")

[Out]

9*x^4 - 6*x^3 + 6*x^2*log(x) - 59*x^2 - 2*x*log(x) + log(x)^2 + 20*x - 20*log(x)

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mupad [B]  time = 4.68, size = 25, normalized size = 1.67 \begin {gather*} -\left (\ln \relax (x)-x+3\,x^2\right )\,\left (x-\ln \relax (x)-3\,x^2+20\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x + log(x)*(12*x^2 - 2*x + 2) - 112*x^2 - 18*x^3 + 36*x^4 - 20)/x,x)

[Out]

-(log(x) - x + 3*x^2)*(x - log(x) - 3*x^2 + 20)

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sympy [B]  time = 0.13, size = 39, normalized size = 2.60 \begin {gather*} 9 x^{4} - 6 x^{3} - 59 x^{2} + 20 x + \left (6 x^{2} - 2 x\right ) \log {\relax (x )} + \log {\relax (x )}^{2} - 20 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**2-2*x+2)*ln(x)+36*x**4-18*x**3-112*x**2+18*x-20)/x,x)

[Out]

9*x**4 - 6*x**3 - 59*x**2 + 20*x + (6*x**2 - 2*x)*log(x) + log(x)**2 - 20*log(x)

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