3.22.84 \(\int \frac {480-240 x+(-480+480 x) \log (x)+576 \log ^2(x)}{100 x^2-100 x^3+25 x^4+(480 x-480 x^2+120 x^3) \log (x)+(576-576 x+144 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {12}{(2-x) \left (3+\frac {5 x}{4 \log (x)}\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 0.50, 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 {480-240 x+(-480+480 x) \log (x)+576 \log ^2(x)}{100 x^2-100 x^3+25 x^4+\left (480 x-480 x^2+120 x^3\right ) \log (x)+\left (576-576 x+144 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(480 - 240*x + (-480 + 480*x)*Log[x] + 576*Log[x]^2)/(100*x^2 - 100*x^3 + 25*x^4 + (480*x - 480*x^2 + 120*
x^3)*Log[x] + (576 - 576*x + 144*x^2)*Log[x]^2),x]

[Out]

4/(2 - x) - 100*Defer[Int][(5*x + 12*Log[x])^(-2), x] - 440*Defer[Int][1/((-2 + x)*(5*x + 12*Log[x])^2), x] -
40*Defer[Int][1/((-2 + x)^2*(5*x + 12*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {48 \left (-5 (-2+x)+10 (-1+x) \log (x)+12 \log ^2(x)\right )}{(2-x)^2 (5 x+12 \log (x))^2} \, dx\\ &=48 \int \frac {-5 (-2+x)+10 (-1+x) \log (x)+12 \log ^2(x)}{(2-x)^2 (5 x+12 \log (x))^2} \, dx\\ &=48 \int \left (\frac {1}{12 (-2+x)^2}-\frac {5 (12+5 x)}{12 (-2+x) (5 x+12 \log (x))^2}-\frac {5}{6 (-2+x)^2 (5 x+12 \log (x))}\right ) \, dx\\ &=\frac {4}{2-x}-20 \int \frac {12+5 x}{(-2+x) (5 x+12 \log (x))^2} \, dx-40 \int \frac {1}{(-2+x)^2 (5 x+12 \log (x))} \, dx\\ &=\frac {4}{2-x}-20 \int \left (\frac {5}{(5 x+12 \log (x))^2}+\frac {22}{(-2+x) (5 x+12 \log (x))^2}\right ) \, dx-40 \int \frac {1}{(-2+x)^2 (5 x+12 \log (x))} \, dx\\ &=\frac {4}{2-x}-40 \int \frac {1}{(-2+x)^2 (5 x+12 \log (x))} \, dx-100 \int \frac {1}{(5 x+12 \log (x))^2} \, dx-440 \int \frac {1}{(-2+x) (5 x+12 \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 19, normalized size = 0.86 \begin {gather*} -\frac {48 \log (x)}{(-2+x) (5 x+12 \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(480 - 240*x + (-480 + 480*x)*Log[x] + 576*Log[x]^2)/(100*x^2 - 100*x^3 + 25*x^4 + (480*x - 480*x^2
+ 120*x^3)*Log[x] + (576 - 576*x + 144*x^2)*Log[x]^2),x]

[Out]

(-48*Log[x])/((-2 + x)*(5*x + 12*Log[x]))

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 22, normalized size = 1.00 \begin {gather*} -\frac {48 \, \log \relax (x)}{5 \, x^{2} + 12 \, {\left (x - 2\right )} \log \relax (x) - 10 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((576*log(x)^2+(480*x-480)*log(x)-240*x+480)/((144*x^2-576*x+576)*log(x)^2+(120*x^3-480*x^2+480*x)*lo
g(x)+25*x^4-100*x^3+100*x^2),x, algorithm="fricas")

[Out]

-48*log(x)/(5*x^2 + 12*(x - 2)*log(x) - 10*x)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 31, normalized size = 1.41 \begin {gather*} \frac {20 \, x}{5 \, x^{2} + 12 \, x \log \relax (x) - 10 \, x - 24 \, \log \relax (x)} - \frac {4}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((576*log(x)^2+(480*x-480)*log(x)-240*x+480)/((144*x^2-576*x+576)*log(x)^2+(120*x^3-480*x^2+480*x)*lo
g(x)+25*x^4-100*x^3+100*x^2),x, algorithm="giac")

[Out]

20*x/(5*x^2 + 12*x*log(x) - 10*x - 24*log(x)) - 4/(x - 2)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 25, normalized size = 1.14

method	result	size
norman	\(-\frac {48 \ln \relax (x )}{12 x \ln \relax (x )+5 x^{2}-24 \ln \relax (x )-10 x}\)	\(25\)
risch	\(-\frac {4}{x -2}+\frac {20 x}{\left (x -2\right ) \left (5 x +12 \ln \relax (x )\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((576*ln(x)^2+(480*x-480)*ln(x)-240*x+480)/((144*x^2-576*x+576)*ln(x)^2+(120*x^3-480*x^2+480*x)*ln(x)+25*x^
4-100*x^3+100*x^2),x,method=_RETURNVERBOSE)

[Out]

-48*ln(x)/(12*x*ln(x)+5*x^2-24*ln(x)-10*x)

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 22, normalized size = 1.00 \begin {gather*} -\frac {48 \, \log \relax (x)}{5 \, x^{2} + 12 \, {\left (x - 2\right )} \log \relax (x) - 10 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((576*log(x)^2+(480*x-480)*log(x)-240*x+480)/((144*x^2-576*x+576)*log(x)^2+(120*x^3-480*x^2+480*x)*lo
g(x)+25*x^4-100*x^3+100*x^2),x, algorithm="maxima")

[Out]

-48*log(x)/(5*x^2 + 12*(x - 2)*log(x) - 10*x)

________________________________________________________________________________________

mupad [B]  time = 1.54, size = 19, normalized size = 0.86 \begin {gather*} -\frac {48\,\ln \relax (x)}{\left (5\,x+12\,\ln \relax (x)\right )\,\left (x-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((576*log(x)^2 - 240*x + log(x)*(480*x - 480) + 480)/(log(x)^2*(144*x^2 - 576*x + 576) + 100*x^2 - 100*x^3
+ 25*x^4 + log(x)*(480*x - 480*x^2 + 120*x^3)),x)

[Out]

-(48*log(x))/((5*x + 12*log(x))*(x - 2))

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 27, normalized size = 1.23 \begin {gather*} \frac {5 x}{\frac {5 x^{2}}{4} - \frac {5 x}{2} + \left (3 x - 6\right ) \log {\relax (x )}} - \frac {4}{x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((576*ln(x)**2+(480*x-480)*ln(x)-240*x+480)/((144*x**2-576*x+576)*ln(x)**2+(120*x**3-480*x**2+480*x)*
ln(x)+25*x**4-100*x**3+100*x**2),x)

[Out]

5*x/(5*x**2/4 - 5*x/2 + (3*x - 6)*log(x)) - 4/(x - 2)

________________________________________________________________________________________