3.13.77 \(\int \frac {30+10 x-3 x^2+(-10+2 x) \log (5-x)}{15 x+2 x^2-x^3+(-5 x+x^2) \log (5-x)} \, dx\)

Optimal. Leaf size=17 \[ \log \left (50 x^2 (-3-x+\log (5-x))\right ) \]

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Rubi [A]  time = 0.44, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6741, 6742, 6684} \begin {gather*} 2 \log (x)+\log (x-\log (5-x)+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(30 + 10*x - 3*x^2 + (-10 + 2*x)*Log[5 - x])/(15*x + 2*x^2 - x^3 + (-5*x + x^2)*Log[5 - x]),x]

[Out]

2*Log[x] + Log[3 + x - Log[5 - 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 {30+10 x-3 x^2+(-10+2 x) \log (5-x)}{(5-x) x (3+x-\log (5-x))} \, dx\\ &=\int \left (\frac {2}{x}+\frac {-6+x}{(-5+x) (3+x-\log (5-x))}\right ) \, dx\\ &=2 \log (x)+\int \frac {-6+x}{(-5+x) (3+x-\log (5-x))} \, dx\\ &=2 \log (x)+\log (3+x-\log (5-x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 17, normalized size = 1.00 \begin {gather*} 2 \log (x)+\log (3+x-\log (5-x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(30 + 10*x - 3*x^2 + (-10 + 2*x)*Log[5 - x])/(15*x + 2*x^2 - x^3 + (-5*x + x^2)*Log[5 - x]),x]

[Out]

2*Log[x] + Log[3 + x - Log[5 - x]]

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fricas [A]  time = 0.70, size = 17, normalized size = 1.00 \begin {gather*} 2 \, \log \relax (x) + \log \left (-x + \log \left (-x + 5\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-10)*log(5-x)-3*x^2+10*x+30)/((x^2-5*x)*log(5-x)-x^3+2*x^2+15*x),x, algorithm="fricas")

[Out]

2*log(x) + log(-x + log(-x + 5) - 3)

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giac [A]  time = 0.21, size = 17, normalized size = 1.00 \begin {gather*} 2 \, \log \relax (x) + \log \left (-x + \log \left (-x + 5\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-10)*log(5-x)-3*x^2+10*x+30)/((x^2-5*x)*log(5-x)-x^3+2*x^2+15*x),x, algorithm="giac")

[Out]

2*log(x) + log(-x + log(-x + 5) - 3)

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maple [A]  time = 0.05, size = 18, normalized size = 1.06

method	result	size
norman	\(2 \ln \relax (x )+\ln \left (x -\ln \left (5-x \right )+3\right )\)	\(18\)
risch	\(2 \ln \relax (x )+\ln \left (\ln \left (5-x \right )-x -3\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-10)*ln(5-x)-3*x^2+10*x+30)/((x^2-5*x)*ln(5-x)-x^3+2*x^2+15*x),x,method=_RETURNVERBOSE)

[Out]

2*ln(x)+ln(x-ln(5-x)+3)

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maxima [A]  time = 0.50, size = 17, normalized size = 1.00 \begin {gather*} 2 \, \log \relax (x) + \log \left (-x + \log \left (-x + 5\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-10)*log(5-x)-3*x^2+10*x+30)/((x^2-5*x)*log(5-x)-x^3+2*x^2+15*x),x, algorithm="maxima")

[Out]

2*log(x) + log(-x + log(-x + 5) - 3)

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mupad [B]  time = 0.96, size = 17, normalized size = 1.00 \begin {gather*} \ln \left (x-\ln \left (5-x\right )+3\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x + log(5 - x)*(2*x - 10) - 3*x^2 + 30)/(15*x - log(5 - x)*(5*x - x^2) + 2*x^2 - x^3),x)

[Out]

log(x - log(5 - x) + 3) + 2*log(x)

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sympy [A]  time = 0.16, size = 14, normalized size = 0.82 \begin {gather*} 2 \log {\relax (x )} + \log {\left (- x + \log {\left (5 - x \right )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-10)*ln(5-x)-3*x**2+10*x+30)/((x**2-5*x)*ln(5-x)-x**3+2*x**2+15*x),x)

[Out]

2*log(x) + log(-x + log(5 - x) - 3)

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