3.42.46 \(\int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+(50-20 x+2 x^2) \log (x)+(-25+10 x-x^2) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+(25 x-10 x^2+x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ \log \left (-\frac {x}{5-x}+(-3+x) x+\frac {\log ^2(x)}{x}\right ) \]

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Rubi [F]  time = 11.27, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-80*x^2 + 80*x^3 - 23*x^4 + 2*x^5 + (50 - 20*x + 2*x^2)*Log[x] + (-25 + 10*x - x^2)*Log[x]^2)/(-80*x^3 +
56*x^4 - 13*x^5 + x^6 + (25*x - 10*x^2 + x^3)*Log[x]^2),x]

[Out]

-Log[x] - 5*Defer[Int][((-4 + x)^2*x^2 + (-5 + x)*Log[x]^2)^(-1), x] - 25*Defer[Int][1/((-5 + x)*((-4 + x)^2*x
^2 + (-5 + x)*Log[x]^2)), x] + 31*Defer[Int][x/((-4 + x)^2*x^2 + (-5 + x)*Log[x]^2), x] - 21*Defer[Int][x^2/((
-4 + x)^2*x^2 + (-5 + x)*Log[x]^2), x] + 3*Defer[Int][x^3/((-4 + x)^2*x^2 + (-5 + x)*Log[x]^2), x] + 2*Defer[I
nt][Log[x]/((-4 + x)^2*x^2 + (-5 + x)*Log[x]^2), x] - 10*Defer[Int][Log[x]/((-4 + x)^2*x^3 + (-5 + x)*x*Log[x]
^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {80 x^2-80 x^3+23 x^4-2 x^5-\left (50-20 x+2 x^2\right ) \log (x)-\left (-25+10 x-x^2\right ) \log ^2(x)}{(5-x) x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx\\ &=\int \left (-\frac {1}{x}+\frac {-160 x^2+136 x^3-36 x^4+3 x^5+50 \log (x)-20 x \log (x)+2 x^2 \log (x)}{(-5+x) x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}\right ) \, dx\\ &=-\log (x)+\int \frac {-160 x^2+136 x^3-36 x^4+3 x^5+50 \log (x)-20 x \log (x)+2 x^2 \log (x)}{(-5+x) x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx\\ &=-\log (x)+\int \left (\frac {160 x^2-136 x^3+36 x^4-3 x^5-50 \log (x)+20 x \log (x)-2 x^2 \log (x)}{5 x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {-160 x^2+136 x^3-36 x^4+3 x^5+50 \log (x)-20 x \log (x)+2 x^2 \log (x)}{5 (-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}\right ) \, dx\\ &=-\log (x)+\frac {1}{5} \int \frac {160 x^2-136 x^3+36 x^4-3 x^5-50 \log (x)+20 x \log (x)-2 x^2 \log (x)}{x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+\frac {1}{5} \int \frac {-160 x^2+136 x^3-36 x^4+3 x^5+50 \log (x)-20 x \log (x)+2 x^2 \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx\\ &=-\log (x)+\frac {1}{5} \int \frac {x^2 \left (160-136 x+36 x^2-3 x^3\right )-2 (-5+x)^2 \log (x)}{(-4+x)^2 x^3+(-5+x) x \log ^2(x)} \, dx+\frac {1}{5} \int \left (-\frac {160 x^2}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {136 x^3}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}-\frac {36 x^4}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {3 x^5}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {50 \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}-\frac {20 x \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}+\frac {2 x^2 \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}\right ) \, dx\\ &=-\log (x)+\frac {1}{5} \int \left (\frac {160 x}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}-\frac {136 x^2}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}+\frac {36 x^3}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}-\frac {3 x^4}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}+\frac {20 \log (x)}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}-\frac {50 \log (x)}{x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )}-\frac {2 x \log (x)}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)}\right ) \, dx+\frac {2}{5} \int \frac {x^2 \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+\frac {3}{5} \int \frac {x^5}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx-4 \int \frac {x \log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx-\frac {36}{5} \int \frac {x^4}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+10 \int \frac {\log (x)}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+\frac {136}{5} \int \frac {x^3}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx-32 \int \frac {x^2}{(-5+x) \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx\\ &=-\log (x)+\frac {2}{5} \int \frac {x^2 \log (x)}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx-\frac {2}{5} \int \frac {x \log (x)}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx+\frac {3}{5} \int \frac {x^5}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx-\frac {3}{5} \int \frac {x^4}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx-4 \int \frac {x \log (x)}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx+4 \int \frac {\log (x)}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx-\frac {36}{5} \int \frac {x^4}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx+\frac {36}{5} \int \frac {x^3}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx+10 \int \frac {\log (x)}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx-10 \int \frac {\log (x)}{x \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right )} \, dx+\frac {136}{5} \int \frac {x^3}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx-\frac {136}{5} \int \frac {x^2}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx-32 \int \frac {x^2}{(-5+x) \left ((-4+x)^2 x^2+(-5+x) \log ^2(x)\right )} \, dx+32 \int \frac {x}{16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 40, normalized size = 1.60 \begin {gather*} -\log (5-x)-\log (x)+\log \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-80*x^2 + 80*x^3 - 23*x^4 + 2*x^5 + (50 - 20*x + 2*x^2)*Log[x] + (-25 + 10*x - x^2)*Log[x]^2)/(-80*
x^3 + 56*x^4 - 13*x^5 + x^6 + (25*x - 10*x^2 + x^3)*Log[x]^2),x]

[Out]

-Log[5 - x] - Log[x] + Log[16*x^2 - 8*x^3 + x^4 - 5*Log[x]^2 + x*Log[x]^2]

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fricas [A]  time = 1.22, size = 34, normalized size = 1.36 \begin {gather*} -\log \relax (x) + \log \left (\frac {x^{4} - 8 \, x^{3} + {\left (x - 5\right )} \log \relax (x)^{2} + 16 \, x^{2}}{x - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+10*x-25)*log(x)^2+(2*x^2-20*x+50)*log(x)+2*x^5-23*x^4+80*x^3-80*x^2)/((x^3-10*x^2+25*x)*log(x
)^2+x^6-13*x^5+56*x^4-80*x^3),x, algorithm="fricas")

[Out]

-log(x) + log((x^4 - 8*x^3 + (x - 5)*log(x)^2 + 16*x^2)/(x - 5))

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giac [A]  time = 0.17, size = 38, normalized size = 1.52 \begin {gather*} \log \left (x^{4} - 8 \, x^{3} + x \log \relax (x)^{2} + 16 \, x^{2} - 5 \, \log \relax (x)^{2}\right ) - \log \left (x - 5\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+10*x-25)*log(x)^2+(2*x^2-20*x+50)*log(x)+2*x^5-23*x^4+80*x^3-80*x^2)/((x^3-10*x^2+25*x)*log(x
)^2+x^6-13*x^5+56*x^4-80*x^3),x, algorithm="giac")

[Out]

log(x^4 - 8*x^3 + x*log(x)^2 + 16*x^2 - 5*log(x)^2) - log(x - 5) - log(x)

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maple [A]  time = 0.06, size = 29, normalized size = 1.16

method	result	size
risch	\(-\ln \relax (x )+\ln \left (\ln \relax (x )^{2}+\frac {x^{2} \left (x^{2}-8 x +16\right )}{x -5}\right )\)	\(29\)
norman	\(-\ln \relax (x )-\ln \left (x -5\right )+\ln \left (x^{4}-8 x^{3}+x \ln \relax (x )^{2}+16 x^{2}-5 \ln \relax (x )^{2}\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+10*x-25)*ln(x)^2+(2*x^2-20*x+50)*ln(x)+2*x^5-23*x^4+80*x^3-80*x^2)/((x^3-10*x^2+25*x)*ln(x)^2+x^6-1
3*x^5+56*x^4-80*x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 34, normalized size = 1.36 \begin {gather*} -\log \relax (x) + \log \left (\frac {x^{4} - 8 \, x^{3} + {\left (x - 5\right )} \log \relax (x)^{2} + 16 \, x^{2}}{x - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+10*x-25)*log(x)^2+(2*x^2-20*x+50)*log(x)+2*x^5-23*x^4+80*x^3-80*x^2)/((x^3-10*x^2+25*x)*log(x
)^2+x^6-13*x^5+56*x^4-80*x^3),x, algorithm="maxima")

[Out]

-log(x) + log((x^4 - 8*x^3 + (x - 5)*log(x)^2 + 16*x^2)/(x - 5))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \relax (x)\,\left (2\,x^2-20\,x+50\right )-{\ln \relax (x)}^2\,\left (x^2-10\,x+25\right )-80\,x^2+80\,x^3-23\,x^4+2\,x^5}{56\,x^4-80\,x^3-13\,x^5+x^6+{\ln \relax (x)}^2\,\left (x^3-10\,x^2+25\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(2*x^2 - 20*x + 50) - log(x)^2*(x^2 - 10*x + 25) - 80*x^2 + 80*x^3 - 23*x^4 + 2*x^5)/(56*x^4 - 80*
x^3 - 13*x^5 + x^6 + log(x)^2*(25*x - 10*x^2 + x^3)),x)

[Out]

int((log(x)*(2*x^2 - 20*x + 50) - log(x)^2*(x^2 - 10*x + 25) - 80*x^2 + 80*x^3 - 23*x^4 + 2*x^5)/(56*x^4 - 80*
x^3 - 13*x^5 + x^6 + log(x)^2*(25*x - 10*x^2 + x^3)), x)

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sympy [A]  time = 0.43, size = 26, normalized size = 1.04 \begin {gather*} - \log {\relax (x )} + \log {\left (\log {\relax (x )}^{2} + \frac {x^{4} - 8 x^{3} + 16 x^{2}}{x - 5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+10*x-25)*ln(x)**2+(2*x**2-20*x+50)*ln(x)+2*x**5-23*x**4+80*x**3-80*x**2)/((x**3-10*x**2+25*x
)*ln(x)**2+x**6-13*x**5+56*x**4-80*x**3),x)

[Out]

-log(x) + log(log(x)**2 + (x**4 - 8*x**3 + 16*x**2)/(x - 5))

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