3.49.73 \(\int \frac {-135+750 x-81 x^2+477 x^3-150 x^4+(-810 x^2+2655 x^3-1125 x^4+(-162 x^2+531 x^3-225 x^4) \log (\frac {1}{9} (-9 x+25 x^2))) \log (5+\log (\frac {1}{9} (-9 x+25 x^2)))}{-45 x+125 x^2+(-9 x+25 x^2) \log (\frac {1}{9} (-9 x+25 x^2))} \, dx\) [4873]

Optimal. Leaf size=29 \[ -3+3 \left (5-(-3+x) x^2\right ) \log \left (5+\log \left (-x+\frac {25 x^2}{9}\right )\right ) \]

[Out]

3*ln(ln(25/9*x^2-x)+5)*(5-x^2*(-3+x))-3

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Rubi [F]
time = 1.64, 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 {-135+750 x-81 x^2+477 x^3-150 x^4+\left (-810 x^2+2655 x^3-1125 x^4+\left (-162 x^2+531 x^3-225 x^4\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right ) \log \left (5+\log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right )}{-45 x+125 x^2+\left (-9 x+25 x^2\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-135 + 750*x - 81*x^2 + 477*x^3 - 150*x^4 + (-810*x^2 + 2655*x^3 - 1125*x^4 + (-162*x^2 + 531*x^3 - 225*x
^4)*Log[(-9*x + 25*x^2)/9])*Log[5 + Log[(-9*x + 25*x^2)/9]])/(-45*x + 125*x^2 + (-9*x + 25*x^2)*Log[(-9*x + 25
*x^2)/9]),x]

[Out]

(1782*Defer[Int][(5 + Log[(x*(-9 + 25*x))/9])^(-1), x])/625 + 15*Defer[Int][1/(x*(5 + Log[(x*(-9 + 25*x))/9]))
, x] + (423*Defer[Int][x/(5 + Log[(x*(-9 + 25*x))/9]), x])/25 - 6*Defer[Int][x^2/(5 + Log[(x*(-9 + 25*x))/9]),
x] + (250413*Defer[Int][1/((-9 + 25*x)*(5 + Log[(x*(-9 + 25*x))/9])), x])/625 + 18*Defer[Int][x*Log[5 + Log[(
x*(-9 + 25*x))/9]], x] - 9*Defer[Int][x^2*Log[5 + Log[(x*(-9 + 25*x))/9]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (45-250 x+27 x^2-159 x^3+50 x^4+3 x^2 \left (18-59 x+25 x^2\right ) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right ) \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )\right )}{(9-25 x) x \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx\\ &=3 \int \frac {45-250 x+27 x^2-159 x^3+50 x^4+3 x^2 \left (18-59 x+25 x^2\right ) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right ) \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}{(9-25 x) x \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx\\ &=3 \int \left (\frac {250}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}-\frac {45}{x (-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}-\frac {27 x}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}+\frac {159 x^2}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}-\frac {50 x^3}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}-3 (-2+x) x \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )\right ) \, dx\\ &=-\left (9 \int (-2+x) x \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right ) \, dx\right )-81 \int \frac {x}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx-135 \int \frac {1}{x (-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx-150 \int \frac {x^3}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx+477 \int \frac {x^2}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx+750 \int \frac {1}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx\\ &=-\left (9 \int \left (-2 x \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )+x^2 \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )\right ) \, dx\right )-81 \int \left (\frac {1}{25 \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}+\frac {9}{25 (-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}\right ) \, dx-135 \int \left (-\frac {1}{9 x \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}+\frac {25}{9 (-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}\right ) \, dx-150 \int \left (\frac {81}{15625 \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}+\frac {9 x}{625 \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}+\frac {x^2}{25 \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}+\frac {729}{15625 (-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}\right ) \, dx+477 \int \left (\frac {9}{625 \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}+\frac {x}{25 \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}+\frac {81}{625 (-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )}\right ) \, dx+750 \int \frac {1}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx\\ &=-\left (\frac {486}{625} \int \frac {1}{5+\log \left (\frac {1}{9} x (-9+25 x)\right )} \, dx\right )-\frac {54}{25} \int \frac {x}{5+\log \left (\frac {1}{9} x (-9+25 x)\right )} \, dx-\frac {81}{25} \int \frac {1}{5+\log \left (\frac {1}{9} x (-9+25 x)\right )} \, dx-6 \int \frac {x^2}{5+\log \left (\frac {1}{9} x (-9+25 x)\right )} \, dx+\frac {4293}{625} \int \frac {1}{5+\log \left (\frac {1}{9} x (-9+25 x)\right )} \, dx-\frac {4374}{625} \int \frac {1}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx-9 \int x^2 \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right ) \, dx+15 \int \frac {1}{x \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx+18 \int x \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right ) \, dx+\frac {477}{25} \int \frac {x}{5+\log \left (\frac {1}{9} x (-9+25 x)\right )} \, dx-\frac {729}{25} \int \frac {1}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx+\frac {38637}{625} \int \frac {1}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx-375 \int \frac {1}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx+750 \int \frac {1}{(-9+25 x) \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 25, normalized size = 0.86 \begin {gather*} -3 \left (-5+(-3+x) x^2\right ) \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-135 + 750*x - 81*x^2 + 477*x^3 - 150*x^4 + (-810*x^2 + 2655*x^3 - 1125*x^4 + (-162*x^2 + 531*x^3 -
225*x^4)*Log[(-9*x + 25*x^2)/9])*Log[5 + Log[(-9*x + 25*x^2)/9]])/(-45*x + 125*x^2 + (-9*x + 25*x^2)*Log[(-9*
x + 25*x^2)/9]),x]

[Out]

-3*(-5 + (-3 + x)*x^2)*Log[5 + Log[(x*(-9 + 25*x))/9]]

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-225 x^{4}+531 x^{3}-162 x^{2}\right ) \ln \left (\frac {25}{9} x^{2}-x \right )-1125 x^{4}+2655 x^{3}-810 x^{2}\right ) \ln \left (\ln \left (\frac {25}{9} x^{2}-x \right )+5\right )-150 x^{4}+477 x^{3}-81 x^{2}+750 x -135}{\left (25 x^{2}-9 x \right ) \ln \left (\frac {25}{9} x^{2}-x \right )+125 x^{2}-45 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-225*x^4+531*x^3-162*x^2)*ln(25/9*x^2-x)-1125*x^4+2655*x^3-810*x^2)*ln(ln(25/9*x^2-x)+5)-150*x^4+477*x^
3-81*x^2+750*x-135)/((25*x^2-9*x)*ln(25/9*x^2-x)+125*x^2-45*x),x)

[Out]

int((((-225*x^4+531*x^3-162*x^2)*ln(25/9*x^2-x)-1125*x^4+2655*x^3-810*x^2)*ln(ln(25/9*x^2-x)+5)-150*x^4+477*x^
3-81*x^2+750*x-135)/((25*x^2-9*x)*ln(25/9*x^2-x)+125*x^2-45*x),x)

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Maxima [A]
time = 0.59, size = 27, normalized size = 0.93 \begin {gather*} -3 \, {\left (x^{3} - 3 \, x^{2} - 5\right )} \log \left (-2 \, \log \left (3\right ) + \log \left (25 \, x - 9\right ) + \log \left (x\right ) + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-225*x^4+531*x^3-162*x^2)*log(25/9*x^2-x)-1125*x^4+2655*x^3-810*x^2)*log(log(25/9*x^2-x)+5)-150*x
^4+477*x^3-81*x^2+750*x-135)/((25*x^2-9*x)*log(25/9*x^2-x)+125*x^2-45*x),x, algorithm="maxima")

[Out]

-3*(x^3 - 3*x^2 - 5)*log(-2*log(3) + log(25*x - 9) + log(x) + 5)

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Fricas [A]
time = 0.35, size = 25, normalized size = 0.86 \begin {gather*} -3 \, {\left (x^{3} - 3 \, x^{2} - 5\right )} \log \left (\log \left (\frac {25}{9} \, x^{2} - x\right ) + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-225*x^4+531*x^3-162*x^2)*log(25/9*x^2-x)-1125*x^4+2655*x^3-810*x^2)*log(log(25/9*x^2-x)+5)-150*x
^4+477*x^3-81*x^2+750*x-135)/((25*x^2-9*x)*log(25/9*x^2-x)+125*x^2-45*x),x, algorithm="fricas")

[Out]

-3*(x^3 - 3*x^2 - 5)*log(log(25/9*x^2 - x) + 5)

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Sympy [A]
time = 0.37, size = 42, normalized size = 1.45 \begin {gather*} \left (- 3 x^{3} + 9 x^{2} - \frac {54189}{312500}\right ) \log {\left (\log {\left (\frac {25 x^{2}}{9} - x \right )} + 5 \right )} + \frac {4741689 \log {\left (\log {\left (\frac {25 x^{2}}{9} - x \right )} + 5 \right )}}{312500} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-225*x**4+531*x**3-162*x**2)*ln(25/9*x**2-x)-1125*x**4+2655*x**3-810*x**2)*ln(ln(25/9*x**2-x)+5)-
150*x**4+477*x**3-81*x**2+750*x-135)/((25*x**2-9*x)*ln(25/9*x**2-x)+125*x**2-45*x),x)

[Out]

(-3*x**3 + 9*x**2 - 54189/312500)*log(log(25*x**2/9 - x) + 5) + 4741689*log(log(25*x**2/9 - x) + 5)/312500

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Giac [A]
time = 0.47, size = 44, normalized size = 1.52 \begin {gather*} -3 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (\log \left (\frac {25}{9} \, x^{2} - x\right ) + 5\right ) + 15 \, \log \left (-2 \, \log \left (3\right ) + \log \left (25 \, x^{2} - 9 \, x\right ) + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-225*x^4+531*x^3-162*x^2)*log(25/9*x^2-x)-1125*x^4+2655*x^3-810*x^2)*log(log(25/9*x^2-x)+5)-150*x
^4+477*x^3-81*x^2+750*x-135)/((25*x^2-9*x)*log(25/9*x^2-x)+125*x^2-45*x),x, algorithm="giac")

[Out]

-3*(x^3 - 3*x^2)*log(log(25/9*x^2 - x) + 5) + 15*log(-2*log(3) + log(25*x^2 - 9*x) + 5)

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Mupad [B]
time = 3.71, size = 27, normalized size = 0.93 \begin {gather*} 3\,\ln \left (\ln \left (\frac {25\,x^2}{9}-x\right )+5\right )\,\left (-x^3+3\,x^2+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log((25*x^2)/9 - x) + 5)*(log((25*x^2)/9 - x)*(162*x^2 - 531*x^3 + 225*x^4) + 810*x^2 - 2655*x^3 + 11
25*x^4) - 750*x + 81*x^2 - 477*x^3 + 150*x^4 + 135)/(45*x + log((25*x^2)/9 - x)*(9*x - 25*x^2) - 125*x^2),x)

[Out]

3*log(log((25*x^2)/9 - x) + 5)*(3*x^2 - x^3 + 5)

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