3.4.21 \(\int \frac {-4-7 x-4 x^2-60 x^4-x^5+6 x^6+(2+x+16 x^4-5 x^5) \log (x)}{x} \, dx\)

Optimal. Leaf size=23 \[ (-4-x+\log (x)) \left (2 x+(4-x) x^4+\log (x)\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 43, normalized size of antiderivative = 1.87, number of steps used = 10, number of rules used = 5, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {14, 2357, 2295, 2301, 2304} \begin {gather*} x^6-x^5 \log (x)-16 x^4+4 x^4 \log (x)-2 x^2-8 x+\log ^2(x)+x \log (x)-4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 - 7*x - 4*x^2 - 60*x^4 - x^5 + 6*x^6 + (2 + x + 16*x^4 - 5*x^5)*Log[x])/x,x]

[Out]

-8*x - 2*x^2 - 16*x^4 + x^6 - 4*Log[x] + x*Log[x] + 4*x^4*Log[x] - x^5*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 {-4-7 x-4 x^2-60 x^4-x^5+6 x^6}{x}-\frac {\left (-2-x-16 x^4+5 x^5\right ) \log (x)}{x}\right ) \, dx\\ &=\int \frac {-4-7 x-4 x^2-60 x^4-x^5+6 x^6}{x} \, dx-\int \frac {\left (-2-x-16 x^4+5 x^5\right ) \log (x)}{x} \, dx\\ &=\int \left (-7-\frac {4}{x}-4 x-60 x^3-x^4+6 x^5\right ) \, dx-\int \left (-\log (x)-\frac {2 \log (x)}{x}-16 x^3 \log (x)+5 x^4 \log (x)\right ) \, dx\\ &=-7 x-2 x^2-15 x^4-\frac {x^5}{5}+x^6-4 \log (x)+2 \int \frac {\log (x)}{x} \, dx-5 \int x^4 \log (x) \, dx+16 \int x^3 \log (x) \, dx+\int \log (x) \, dx\\ &=-8 x-2 x^2-16 x^4+x^6-4 \log (x)+x \log (x)+4 x^4 \log (x)-x^5 \log (x)+\log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 43, normalized size = 1.87 \begin {gather*} -8 x-2 x^2-16 x^4+x^6-4 \log (x)+x \log (x)+4 x^4 \log (x)-x^5 \log (x)+\log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 7*x - 4*x^2 - 60*x^4 - x^5 + 6*x^6 + (2 + x + 16*x^4 - 5*x^5)*Log[x])/x,x]

[Out]

-8*x - 2*x^2 - 16*x^4 + x^6 - 4*Log[x] + x*Log[x] + 4*x^4*Log[x] - x^5*Log[x] + Log[x]^2

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fricas [A]  time = 0.75, size = 38, normalized size = 1.65 \begin {gather*} x^{6} - 16 \, x^{4} - 2 \, x^{2} - {\left (x^{5} - 4 \, x^{4} - x + 4\right )} \log \relax (x) + \log \relax (x)^{2} - 8 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^5+16*x^4+x+2)*log(x)+6*x^6-x^5-60*x^4-4*x^2-7*x-4)/x,x, algorithm="fricas")

[Out]

x^6 - 16*x^4 - 2*x^2 - (x^5 - 4*x^4 - x + 4)*log(x) + log(x)^2 - 8*x

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giac [A]  time = 0.31, size = 41, normalized size = 1.78 \begin {gather*} x^{6} - 16 \, x^{4} - 2 \, x^{2} - {\left (x^{5} - 4 \, x^{4} - x\right )} \log \relax (x) + \log \relax (x)^{2} - 8 \, x - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^5+16*x^4+x+2)*log(x)+6*x^6-x^5-60*x^4-4*x^2-7*x-4)/x,x, algorithm="giac")

[Out]

x^6 - 16*x^4 - 2*x^2 - (x^5 - 4*x^4 - x)*log(x) + log(x)^2 - 8*x - 4*log(x)

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maple [A]  time = 0.02, size = 41, normalized size = 1.78

method	result	size
risch	\(\ln \relax (x )^{2}+\left (-x^{5}+4 x^{4}+x \right ) \ln \relax (x )+x^{6}-16 x^{4}-2 x^{2}-8 x -4 \ln \relax (x )\)	\(41\)
default	\(-x^{5} \ln \relax (x )+x^{6}+4 x^{4} \ln \relax (x )-16 x^{4}+x \ln \relax (x )-8 x -2 x^{2}+\ln \relax (x )^{2}-4 \ln \relax (x )\)	\(44\)
norman	\(-x^{5} \ln \relax (x )+x^{6}+4 x^{4} \ln \relax (x )-16 x^{4}+x \ln \relax (x )-8 x -2 x^{2}+\ln \relax (x )^{2}-4 \ln \relax (x )\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x^5+16*x^4+x+2)*ln(x)+6*x^6-x^5-60*x^4-4*x^2-7*x-4)/x,x,method=_RETURNVERBOSE)

[Out]

ln(x)^2+(-x^5+4*x^4+x)*ln(x)+x^6-16*x^4-2*x^2-8*x-4*ln(x)

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maxima [A]  time = 0.47, size = 43, normalized size = 1.87 \begin {gather*} x^{6} - x^{5} \log \relax (x) + 4 \, x^{4} \log \relax (x) - 16 \, x^{4} - 2 \, x^{2} + x \log \relax (x) + \log \relax (x)^{2} - 8 \, x - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^5+16*x^4+x+2)*log(x)+6*x^6-x^5-60*x^4-4*x^2-7*x-4)/x,x, algorithm="maxima")

[Out]

x^6 - x^5*log(x) + 4*x^4*log(x) - 16*x^4 - 2*x^2 + x*log(x) + log(x)^2 - 8*x - 4*log(x)

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mupad [B]  time = 0.39, size = 25, normalized size = 1.09 \begin {gather*} -\left (x-\ln \relax (x)+4\right )\,\left (2\,x+\ln \relax (x)+4\,x^4-x^5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7*x - log(x)*(x + 16*x^4 - 5*x^5 + 2) + 4*x^2 + 60*x^4 + x^5 - 6*x^6 + 4)/x,x)

[Out]

-(x - log(x) + 4)*(2*x + log(x) + 4*x^4 - x^5)

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sympy [B]  time = 0.14, size = 39, normalized size = 1.70 \begin {gather*} x^{6} - 16 x^{4} - 2 x^{2} - 8 x + \left (- x^{5} + 4 x^{4} + x\right ) \log {\relax (x )} + \log {\relax (x )}^{2} - 4 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x**5+16*x**4+x+2)*ln(x)+6*x**6-x**5-60*x**4-4*x**2-7*x-4)/x,x)

[Out]

x**6 - 16*x**4 - 2*x**2 - 8*x + (-x**5 + 4*x**4 + x)*log(x) + log(x)**2 - 4*log(x)

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