3.58.42 \(\int \frac {-800-31920 x+4798 x^2-240 x^3+4 x^4+(2+80 x-4 x^2) \log (x)}{x} \, dx\)

Optimal. Leaf size=14 \[ \left ((20-x)^2-\log (x)\right )^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 37, normalized size of antiderivative = 2.64, number of steps used = 9, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {14, 893, 2357, 2295, 2301, 2304} \begin {gather*} x^4-80 x^3+2400 x^2-2 x^2 \log (x)-32000 x+\log ^2(x)+80 x \log (x)-800 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-800 - 31920*x + 4798*x^2 - 240*x^3 + 4*x^4 + (2 + 80*x - 4*x^2)*Log[x])/x,x]

[Out]

-32000*x + 2400*x^2 - 80*x^3 + x^4 - 800*Log[x] + 80*x*Log[x] - 2*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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

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 (-20+x)^2 \left (-1-40 x+2 x^2\right )}{x}-\frac {2 \left (-1-40 x+2 x^2\right ) \log (x)}{x}\right ) \, dx\\ &=2 \int \frac {(-20+x)^2 \left (-1-40 x+2 x^2\right )}{x} \, dx-2 \int \frac {\left (-1-40 x+2 x^2\right ) \log (x)}{x} \, dx\\ &=2 \int \left (-15960-\frac {400}{x}+2399 x-120 x^2+2 x^3\right ) \, dx-2 \int \left (-40 \log (x)-\frac {\log (x)}{x}+2 x \log (x)\right ) \, dx\\ &=-31920 x+2399 x^2-80 x^3+x^4-800 \log (x)+2 \int \frac {\log (x)}{x} \, dx-4 \int x \log (x) \, dx+80 \int \log (x) \, dx\\ &=-32000 x+2400 x^2-80 x^3+x^4-800 \log (x)+80 x \log (x)-2 x^2 \log (x)+\log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 37, normalized size = 2.64 \begin {gather*} -32000 x+2400 x^2-80 x^3+x^4-800 \log (x)+80 x \log (x)-2 x^2 \log (x)+\log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-800 - 31920*x + 4798*x^2 - 240*x^3 + 4*x^4 + (2 + 80*x - 4*x^2)*Log[x])/x,x]

[Out]

-32000*x + 2400*x^2 - 80*x^3 + x^4 - 800*Log[x] + 80*x*Log[x] - 2*x^2*Log[x] + Log[x]^2

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fricas [B]  time = 0.75, size = 33, normalized size = 2.36 \begin {gather*} x^{4} - 80 \, x^{3} + 2400 \, x^{2} - 2 \, {\left (x^{2} - 40 \, x + 400\right )} \log \relax (x) + \log \relax (x)^{2} - 32000 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+80*x+2)*log(x)+4*x^4-240*x^3+4798*x^2-31920*x-800)/x,x, algorithm="fricas")

[Out]

x^4 - 80*x^3 + 2400*x^2 - 2*(x^2 - 40*x + 400)*log(x) + log(x)^2 - 32000*x

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giac [B]  time = 0.26, size = 36, normalized size = 2.57 \begin {gather*} x^{4} - 80 \, x^{3} + 2400 \, x^{2} - 2 \, {\left (x^{2} - 40 \, x\right )} \log \relax (x) + \log \relax (x)^{2} - 32000 \, x - 800 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+80*x+2)*log(x)+4*x^4-240*x^3+4798*x^2-31920*x-800)/x,x, algorithm="giac")

[Out]

x^4 - 80*x^3 + 2400*x^2 - 2*(x^2 - 40*x)*log(x) + log(x)^2 - 32000*x - 800*log(x)

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maple [B]  time = 0.04, size = 38, normalized size = 2.71

method	result	size
default	\(x^{4}-2 x^{2} \ln \relax (x )+2400 x^{2}-80 x^{3}+80 x \ln \relax (x )-32000 x +\ln \relax (x )^{2}-800 \ln \relax (x )\)	\(38\)
norman	\(x^{4}-2 x^{2} \ln \relax (x )+2400 x^{2}-80 x^{3}+80 x \ln \relax (x )-32000 x +\ln \relax (x )^{2}-800 \ln \relax (x )\)	\(38\)
risch	\(\ln \relax (x )^{2}+\left (-2 x^{2}+80 x \right ) \ln \relax (x )+x^{4}-80 x^{3}+2400 x^{2}-32000 x -800 \ln \relax (x )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2+80*x+2)*ln(x)+4*x^4-240*x^3+4798*x^2-31920*x-800)/x,x,method=_RETURNVERBOSE)

[Out]

x^4-2*x^2*ln(x)+2400*x^2-80*x^3+80*x*ln(x)-32000*x+ln(x)^2-800*ln(x)

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maxima [B]  time = 0.34, size = 37, normalized size = 2.64 \begin {gather*} x^{4} - 80 \, x^{3} - 2 \, x^{2} \log \relax (x) + 2400 \, x^{2} + 80 \, x \log \relax (x) + \log \relax (x)^{2} - 32000 \, x - 800 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+80*x+2)*log(x)+4*x^4-240*x^3+4798*x^2-31920*x-800)/x,x, algorithm="maxima")

[Out]

x^4 - 80*x^3 - 2*x^2*log(x) + 2400*x^2 + 80*x*log(x) + log(x)^2 - 32000*x - 800*log(x)

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mupad [B]  time = 3.63, size = 24, normalized size = 1.71 \begin {gather*} \left (40\,x+\ln \relax (x)-x^2\right )\,\left (40\,x+\ln \relax (x)-x^2-800\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(31920*x - log(x)*(80*x - 4*x^2 + 2) - 4798*x^2 + 240*x^3 - 4*x^4 + 800)/x,x)

[Out]

(40*x + log(x) - x^2)*(40*x + log(x) - x^2 - 800)

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sympy [B]  time = 0.13, size = 37, normalized size = 2.64 \begin {gather*} x^{4} - 80 x^{3} + 2400 x^{2} - 32000 x + \left (- 2 x^{2} + 80 x\right ) \log {\relax (x )} + \log {\relax (x )}^{2} - 800 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2+80*x+2)*ln(x)+4*x**4-240*x**3+4798*x**2-31920*x-800)/x,x)

[Out]

x**4 - 80*x**3 + 2400*x**2 - 32000*x + (-2*x**2 + 80*x)*log(x) + log(x)**2 - 800*log(x)

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