3.52.66 \(\int \frac {-640+256 x-32 x^2+(320-144 x+16 x^2) \log (\frac {-5 x+x^2}{-4+x})}{(79380-35721 x+3969 x^2) \log ^3(\frac {-5 x+x^2}{-4+x})} \, dx\)

Optimal. Leaf size=18 \[ \frac {16 x}{3969 \log ^2\left (x-\frac {x}{-4+x}\right )} \]

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Rubi [F]  time = 0.42, 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 {-640+256 x-32 x^2+\left (320-144 x+16 x^2\right ) \log \left (\frac {-5 x+x^2}{-4+x}\right )}{\left (79380-35721 x+3969 x^2\right ) \log ^3\left (\frac {-5 x+x^2}{-4+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-640 + 256*x - 32*x^2 + (320 - 144*x + 16*x^2)*Log[(-5*x + x^2)/(-4 + x)])/((79380 - 35721*x + 3969*x^2)*
Log[(-5*x + x^2)/(-4 + x)]^3),x]

[Out]

(-32*Defer[Int][Log[((-5 + x)*x)/(-4 + x)]^(-3), x])/3969 - (160*Defer[Int][1/((-5 + x)*Log[((-5 + x)*x)/(-4 +
x)]^3), x])/3969 + (128*Defer[Int][1/((-4 + x)*Log[((-5 + x)*x)/(-4 + x)]^3), x])/3969 + (16*Defer[Int][Log[(
(-5 + x)*x)/(-4 + x)]^(-2), x])/3969

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {32 \left (20-8 x+x^2\right )}{3969 (-5+x) (-4+x) \log ^3\left (\frac {(-5+x) x}{-4+x}\right )}+\frac {16}{3969 \log ^2\left (\frac {(-5+x) x}{-4+x}\right )}\right ) \, dx\\ &=\frac {16 \int \frac {1}{\log ^2\left (\frac {(-5+x) x}{-4+x}\right )} \, dx}{3969}-\frac {32 \int \frac {20-8 x+x^2}{(-5+x) (-4+x) \log ^3\left (\frac {(-5+x) x}{-4+x}\right )} \, dx}{3969}\\ &=\frac {16 \int \frac {1}{\log ^2\left (\frac {(-5+x) x}{-4+x}\right )} \, dx}{3969}-\frac {32 \int \left (\frac {1}{\log ^3\left (\frac {(-5+x) x}{-4+x}\right )}+\frac {5}{(-5+x) \log ^3\left (\frac {(-5+x) x}{-4+x}\right )}-\frac {4}{(-4+x) \log ^3\left (\frac {(-5+x) x}{-4+x}\right )}\right ) \, dx}{3969}\\ &=\frac {16 \int \frac {1}{\log ^2\left (\frac {(-5+x) x}{-4+x}\right )} \, dx}{3969}-\frac {32 \int \frac {1}{\log ^3\left (\frac {(-5+x) x}{-4+x}\right )} \, dx}{3969}+\frac {128 \int \frac {1}{(-4+x) \log ^3\left (\frac {(-5+x) x}{-4+x}\right )} \, dx}{3969}-\frac {160 \int \frac {1}{(-5+x) \log ^3\left (\frac {(-5+x) x}{-4+x}\right )} \, dx}{3969}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} \frac {16 x}{3969 \log ^2\left (\frac {(-5+x) x}{-4+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-640 + 256*x - 32*x^2 + (320 - 144*x + 16*x^2)*Log[(-5*x + x^2)/(-4 + x)])/((79380 - 35721*x + 3969
*x^2)*Log[(-5*x + x^2)/(-4 + x)]^3),x]

[Out]

(16*x)/(3969*Log[((-5 + x)*x)/(-4 + x)]^2)

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fricas [A]  time = 0.49, size = 19, normalized size = 1.06 \begin {gather*} \frac {16 \, x}{3969 \, \log \left (\frac {x^{2} - 5 \, x}{x - 4}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-144*x+320)*log((x^2-5*x)/(x-4))-32*x^2+256*x-640)/(3969*x^2-35721*x+79380)/log((x^2-5*x)/(x
-4))^3,x, algorithm="fricas")

[Out]

16/3969*x/log((x^2 - 5*x)/(x - 4))^2

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giac [B]  time = 0.27, size = 74, normalized size = 4.11 \begin {gather*} \frac {16 \, {\left (x^{3} - 8 \, x^{2} + 20 \, x\right )}}{3969 \, {\left (x^{2} \log \left (\frac {x^{2} - 5 \, x}{x - 4}\right )^{2} - 8 \, x \log \left (\frac {x^{2} - 5 \, x}{x - 4}\right )^{2} + 20 \, \log \left (\frac {x^{2} - 5 \, x}{x - 4}\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-144*x+320)*log((x^2-5*x)/(x-4))-32*x^2+256*x-640)/(3969*x^2-35721*x+79380)/log((x^2-5*x)/(x
-4))^3,x, algorithm="giac")

[Out]

16/3969*(x^3 - 8*x^2 + 20*x)/(x^2*log((x^2 - 5*x)/(x - 4))^2 - 8*x*log((x^2 - 5*x)/(x - 4))^2 + 20*log((x^2 -
5*x)/(x - 4))^2)

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maple [A]  time = 0.25, size = 20, normalized size = 1.11

method	result	size
norman	\(\frac {16 x}{3969 \ln \left (\frac {x^{2}-5 x}{x -4}\right )^{2}}\)	\(20\)
risch	\(\frac {16 x}{3969 \ln \left (\frac {x^{2}-5 x}{x -4}\right )^{2}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^2-144*x+320)*ln((x^2-5*x)/(x-4))-32*x^2+256*x-640)/(3969*x^2-35721*x+79380)/ln((x^2-5*x)/(x-4))^3,x
,method=_RETURNVERBOSE)

[Out]

16/3969*x/ln((x^2-5*x)/(x-4))^2

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maxima [B]  time = 0.40, size = 49, normalized size = 2.72 \begin {gather*} -\frac {16 \, x}{3969 \, {\left (2 \, {\left (\log \left (x - 5\right ) + \log \relax (x)\right )} \log \left (x - 4\right ) - \log \left (x - 4\right )^{2} - \log \left (x - 5\right )^{2} - 2 \, \log \left (x - 5\right ) \log \relax (x) - \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2-144*x+320)*log((x^2-5*x)/(x-4))-32*x^2+256*x-640)/(3969*x^2-35721*x+79380)/log((x^2-5*x)/(x
-4))^3,x, algorithm="maxima")

[Out]

-16/3969*x/(2*(log(x - 5) + log(x))*log(x - 4) - log(x - 4)^2 - log(x - 5)^2 - 2*log(x - 5)*log(x) - log(x)^2)

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mupad [B]  time = 3.48, size = 219, normalized size = 12.17 \begin {gather*} \frac {8\,x}{3969}+\frac {\frac {8\,x\,\left (x^2-9\,x+20\right )}{3969\,\left (x^2-8\,x+20\right )}-\frac {8\,x\,\ln \left (-\frac {5\,x-x^2}{x-4}\right )\,\left (x^2-9\,x+20\right )\,\left (x^4-16\,x^3+112\,x^2-360\,x+400\right )}{3969\,{\left (x^2-8\,x+20\right )}^3}}{\ln \left (-\frac {5\,x-x^2}{x-4}\right )}+\frac {\frac {16\,x}{3969}-\frac {8\,x\,\ln \left (-\frac {5\,x-x^2}{x-4}\right )\,\left (x^2-9\,x+20\right )}{3969\,\left (x^2-8\,x+20\right )}}{{\ln \left (-\frac {5\,x-x^2}{x-4}\right )}^2}+\frac {\frac {32\,x^4}{441}-\frac {5056\,x^3}{3969}+\frac {10240\,x^2}{1323}-\frac {25600\,x}{1323}+\frac {64000}{3969}}{x^6-24\,x^5+252\,x^4-1472\,x^3+5040\,x^2-9600\,x+8000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((256*x + log(-(5*x - x^2)/(x - 4))*(16*x^2 - 144*x + 320) - 32*x^2 - 640)/(log(-(5*x - x^2)/(x - 4))^3*(39
69*x^2 - 35721*x + 79380)),x)

[Out]

(8*x)/3969 + ((8*x*(x^2 - 9*x + 20))/(3969*(x^2 - 8*x + 20)) - (8*x*log(-(5*x - x^2)/(x - 4))*(x^2 - 9*x + 20)
*(112*x^2 - 360*x - 16*x^3 + x^4 + 400))/(3969*(x^2 - 8*x + 20)^3))/log(-(5*x - x^2)/(x - 4)) + ((16*x)/3969 -
(8*x*log(-(5*x - x^2)/(x - 4))*(x^2 - 9*x + 20))/(3969*(x^2 - 8*x + 20)))/log(-(5*x - x^2)/(x - 4))^2 + ((102
40*x^2)/1323 - (25600*x)/1323 - (5056*x^3)/3969 + (32*x^4)/441 + 64000/3969)/(5040*x^2 - 9600*x - 1472*x^3 + 2
52*x^4 - 24*x^5 + x^6 + 8000)

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sympy [A]  time = 0.18, size = 17, normalized size = 0.94 \begin {gather*} \frac {16 x}{3969 \log {\left (\frac {x^{2} - 5 x}{x - 4} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**2-144*x+320)*ln((x**2-5*x)/(x-4))-32*x**2+256*x-640)/(3969*x**2-35721*x+79380)/ln((x**2-5*x)
/(x-4))**3,x)

[Out]

16*x/(3969*log((x**2 - 5*x)/(x - 4))**2)

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