3.80.16 \(\int \frac {2-80 x-4 x^2-5 x^3+5 x \log (x)}{-80 x-5 x^3+5 x \log (x)} \, dx\)

Optimal. Leaf size=17 \[ -4+x+\frac {2}{5} \log \left (-16-x^2+\log (x)\right ) \]

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Rubi [A]  time = 0.33, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6741, 12, 6742, 6684} \begin {gather*} \frac {2}{5} \log \left (x^2-\log (x)+16\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - 80*x - 4*x^2 - 5*x^3 + 5*x*Log[x])/(-80*x - 5*x^3 + 5*x*Log[x]),x]

[Out]

x + (2*Log[16 + x^2 - Log[x]])/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+80 x+4 x^2+5 x^3-5 x \log (x)}{5 x \left (16+x^2-\log (x)\right )} \, dx\\ &=\frac {1}{5} \int \frac {-2+80 x+4 x^2+5 x^3-5 x \log (x)}{x \left (16+x^2-\log (x)\right )} \, dx\\ &=\frac {1}{5} \int \left (5+\frac {2 \left (-1+2 x^2\right )}{x \left (16+x^2-\log (x)\right )}\right ) \, dx\\ &=x+\frac {2}{5} \int \frac {-1+2 x^2}{x \left (16+x^2-\log (x)\right )} \, dx\\ &=x+\frac {2}{5} \log \left (16+x^2-\log (x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 20, normalized size = 1.18 \begin {gather*} \frac {1}{5} \left (5 x+2 \log \left (16+x^2-\log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 80*x - 4*x^2 - 5*x^3 + 5*x*Log[x])/(-80*x - 5*x^3 + 5*x*Log[x]),x]

[Out]

(5*x + 2*Log[16 + x^2 - Log[x]])/5

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fricas [A]  time = 0.66, size = 14, normalized size = 0.82 \begin {gather*} x + \frac {2}{5} \, \log \left (-x^{2} + \log \relax (x) - 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x*log(x)-5*x^3-4*x^2-80*x+2)/(5*x*log(x)-5*x^3-80*x),x, algorithm="fricas")

[Out]

x + 2/5*log(-x^2 + log(x) - 16)

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giac [A]  time = 0.25, size = 14, normalized size = 0.82 \begin {gather*} x + \frac {2}{5} \, \log \left (-x^{2} + \log \relax (x) - 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x*log(x)-5*x^3-4*x^2-80*x+2)/(5*x*log(x)-5*x^3-80*x),x, algorithm="giac")

[Out]

x + 2/5*log(-x^2 + log(x) - 16)

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maple [A]  time = 0.02, size = 15, normalized size = 0.88

method	result	size
norman	\(x +\frac {2 \ln \left (x^{2}-\ln \relax (x )+16\right )}{5}\)	\(15\)
risch	\(x +\frac {2 \ln \left (\ln \relax (x )-16-x^{2}\right )}{5}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x*ln(x)-5*x^3-4*x^2-80*x+2)/(5*x*ln(x)-5*x^3-80*x),x,method=_RETURNVERBOSE)

[Out]

x+2/5*ln(x^2-ln(x)+16)

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maxima [A]  time = 0.39, size = 14, normalized size = 0.82 \begin {gather*} x + \frac {2}{5} \, \log \left (-x^{2} + \log \relax (x) - 16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x*log(x)-5*x^3-4*x^2-80*x+2)/(5*x*log(x)-5*x^3-80*x),x, algorithm="maxima")

[Out]

x + 2/5*log(-x^2 + log(x) - 16)

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mupad [B]  time = 4.78, size = 14, normalized size = 0.82 \begin {gather*} x+\frac {2\,\ln \left (x^2-\ln \relax (x)+16\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((80*x - 5*x*log(x) + 4*x^2 + 5*x^3 - 2)/(80*x - 5*x*log(x) + 5*x^3),x)

[Out]

x + (2*log(x^2 - log(x) + 16))/5

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sympy [A]  time = 0.14, size = 14, normalized size = 0.82 \begin {gather*} x + \frac {2 \log {\left (- x^{2} + \log {\relax (x )} - 16 \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x*ln(x)-5*x**3-4*x**2-80*x+2)/(5*x*ln(x)-5*x**3-80*x),x)

[Out]

x + 2*log(-x**2 + log(x) - 16)/5

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