3.58.56 \(\int \frac {-10 x-x^2+(-75+20 x+17 x^2+2 x^3) \log (3)}{(-5 x^2-x^3+(-75 x-5 x^2+7 x^3+x^4) \log (3)) \log (\frac {-x^2+(-15 x+2 x^2+x^3) \log (3)}{5+x}) \log (5 \log (\frac {-x^2+(-15 x+2 x^2+x^3) \log (3)}{5+x}))} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\log \left (5 \log \left (-\frac {x^2}{5+x}+(-3+x) x \log (3)\right )\right )\right ) \]

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Rubi [F]  time = 7.34, 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 {-10 x-x^2+\left (-75+20 x+17 x^2+2 x^3\right ) \log (3)}{\left (-5 x^2-x^3+\left (-75 x-5 x^2+7 x^3+x^4\right ) \log (3)\right ) \log \left (\frac {-x^2+\left (-15 x+2 x^2+x^3\right ) \log (3)}{5+x}\right ) \log \left (5 \log \left (\frac {-x^2+\left (-15 x+2 x^2+x^3\right ) \log (3)}{5+x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10*x - x^2 + (-75 + 20*x + 17*x^2 + 2*x^3)*Log[3])/((-5*x^2 - x^3 + (-75*x - 5*x^2 + 7*x^3 + x^4)*Log[3]
)*Log[(-x^2 + (-15*x + 2*x^2 + x^3)*Log[3])/(5 + x)]*Log[5*Log[(-x^2 + (-15*x + 2*x^2 + x^3)*Log[3])/(5 + x)]]
),x]

[Out]

Defer[Int][1/((-5 - x)*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]*Log[5*Log[(x*(-15*Log[3] +
x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]]), x] + Defer[Int][1/(x*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9]
)))/(5 + x)]*Log[5*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]]), x] - (1 - (1 - Log[9])/Sqrt[
60*Log[3]^2 + (-1 + Log[9])^2])*(4*Log[3]^2 + Log[9] - Log[9]^2)*Defer[Int][1/((1 - 2*x*Log[3] - Log[9] - Sqrt
[1 + 60*Log[3]^2 - 2*Log[9] + Log[9]^2])*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]*Log[5*Log
[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]]), x] - (1 + (1 - Log[9])/Sqrt[60*Log[3]^2 + (-1 + Lo
g[9])^2])*(4*Log[3]^2 + Log[9] - Log[9]^2)*Defer[Int][1/((1 - 2*x*Log[3] - Log[9] + Sqrt[1 + 60*Log[3]^2 - 2*L
og[9] + Log[9]^2])*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]*Log[5*Log[(x*(-15*Log[3] + x^2*
Log[3] - x*(1 - Log[9])))/(5 + x)]]), x] + (2*Log[3]*(1 + 12*Log[3]^2 - Log[9]*(1 + Log[729]))*Defer[Int][1/((
1 - Log[9] - x*Log[9] + Sqrt[1 + 60*Log[3]^2 - 2*Log[9] + Log[9]^2])*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 -
Log[9])))/(5 + x)]*Log[5*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]]), x])/Sqrt[60*Log[3]^2 +
(-1 + Log[9])^2] + (2*Log[3]*(1 + 12*Log[3]^2 - Log[9]*(1 + Log[729]))*Defer[Int][1/((-1 + Log[9] + x*Log[9]
+ Sqrt[1 + 60*Log[3]^2 - 2*Log[9] + Log[9]^2])*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]*Log
[5*Log[(x*(-15*Log[3] + x^2*Log[3] - x*(1 - Log[9])))/(5 + x)]]), x])/Sqrt[60*Log[3]^2 + (-1 + Log[9])^2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 (1-17 \log (3))+75 \log (3)-2 x^3 \log (3)+10 x (1-\log (9))}{x \left (75 \log (3)-x^3 \log (3)+5 x (1+\log (3))-x^2 (-1+7 \log (3))\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx\\ &=\int \frac {x^2 (1-17 \log (3))+75 \log (3)-2 x^3 \log (3)+10 x (1-\log (9))}{x \left (x^2 (1-7 \log (3))+75 \log (3)-x^3 \log (3)+5 x (1+\log (3))\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx\\ &=\int \left (\frac {1}{(-5-x) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}+\frac {1}{x \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}+\frac {1+12 \log ^2(3)-x \left (4 \log ^2(3)+\log (9)-\log ^2(9)\right )-\log (9) (1+\log (729))}{\left (15 \log (3)-x^2 \log (3)+x (1-\log (9))\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}\right ) \, dx\\ &=\int \frac {1}{(-5-x) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx+\int \frac {1}{x \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx+\int \frac {1+12 \log ^2(3)-x \left (4 \log ^2(3)+\log (9)-\log ^2(9)\right )-\log (9) (1+\log (729))}{\left (15 \log (3)-x^2 \log (3)+x (1-\log (9))\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx\\ &=\int \left (\frac {x \left (-4 \log ^2(3)-\log (9)+\log ^2(9)\right )}{\left (15 \log (3)-x^2 \log (3)+x (1-\log (9))\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}+\frac {1+12 \log ^2(3)-\log (9) (1+\log (729))}{\left (15 \log (3)-x^2 \log (3)+x (1-\log (9))\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}\right ) \, dx+\int \frac {1}{(-5-x) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx+\int \frac {1}{x \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx\\ &=\left (-4 \log ^2(3)-\log (9)+\log ^2(9)\right ) \int \frac {x}{\left (15 \log (3)-x^2 \log (3)+x (1-\log (9))\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx+\left (1+12 \log ^2(3)-\log (9) (1+\log (729))\right ) \int \frac {1}{\left (15 \log (3)-x^2 \log (3)+x (1-\log (9))\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx+\int \frac {1}{(-5-x) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx+\int \frac {1}{x \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx\\ &=\left (-4 \log ^2(3)-\log (9)+\log ^2(9)\right ) \int \left (\frac {1+\frac {-1+\log (9)}{\sqrt {1+60 \log ^2(3)-2 \log (9)+\log ^2(9)}}}{\left (1-2 x \log (3)-\log (9)-\sqrt {1+60 \log ^2(3)-2 \log (9)+\log ^2(9)}\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}+\frac {1-\frac {-1+\log (9)}{\sqrt {1+60 \log ^2(3)-2 \log (9)+\log ^2(9)}}}{\left (1-2 x \log (3)-\log (9)+\sqrt {1+60 \log ^2(3)-2 \log (9)+\log ^2(9)}\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}\right ) \, dx+\left (1+12 \log ^2(3)-\log (9) (1+\log (729))\right ) \int \left (\frac {2 \log (3)}{\sqrt {1+60 \log ^2(3)-2 \log (9)+\log ^2(9)} \left (1-\log (9)-x \log (9)+\sqrt {1+60 \log ^2(3)-2 \log (9)+\log ^2(9)}\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}+\frac {2 \log (3)}{\sqrt {1+60 \log ^2(3)-2 \log (9)+\log ^2(9)} \left (-1+\log (9)+x \log (9)+\sqrt {1+60 \log ^2(3)-2 \log (9)+\log ^2(9)}\right ) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )}\right ) \, dx+\int \frac {1}{(-5-x) \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx+\int \frac {1}{x \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right ) \log \left (5 \log \left (\frac {x \left (-15 \log (3)+x^2 \log (3)-x (1-\log (9))\right )}{5+x}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 4.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-10 x-x^2+\left (-75+20 x+17 x^2+2 x^3\right ) \log (3)}{\left (-5 x^2-x^3+\left (-75 x-5 x^2+7 x^3+x^4\right ) \log (3)\right ) \log \left (\frac {-x^2+\left (-15 x+2 x^2+x^3\right ) \log (3)}{5+x}\right ) \log \left (5 \log \left (\frac {-x^2+\left (-15 x+2 x^2+x^3\right ) \log (3)}{5+x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-10*x - x^2 + (-75 + 20*x + 17*x^2 + 2*x^3)*Log[3])/((-5*x^2 - x^3 + (-75*x - 5*x^2 + 7*x^3 + x^4)*
Log[3])*Log[(-x^2 + (-15*x + 2*x^2 + x^3)*Log[3])/(5 + x)]*Log[5*Log[(-x^2 + (-15*x + 2*x^2 + x^3)*Log[3])/(5
+ x)]]),x]

[Out]

Integrate[(-10*x - x^2 + (-75 + 20*x + 17*x^2 + 2*x^3)*Log[3])/((-5*x^2 - x^3 + (-75*x - 5*x^2 + 7*x^3 + x^4)*
Log[3])*Log[(-x^2 + (-15*x + 2*x^2 + x^3)*Log[3])/(5 + x)]*Log[5*Log[(-x^2 + (-15*x + 2*x^2 + x^3)*Log[3])/(5
+ x)]]), x]

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fricas [A]  time = 0.62, size = 32, normalized size = 1.39 \begin {gather*} \log \left (\log \left (5 \, \log \left (-\frac {x^{2} - {\left (x^{3} + 2 \, x^{2} - 15 \, x\right )} \log \relax (3)}{x + 5}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+20*x-75)*log(3)-x^2-10*x)/((x^4+7*x^3-5*x^2-75*x)*log(3)-x^3-5*x^2)/log(((x^3+2*x^2-1
5*x)*log(3)-x^2)/(5+x))/log(5*log(((x^3+2*x^2-15*x)*log(3)-x^2)/(5+x))),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.41, size = 36, normalized size = 1.57 \begin {gather*} \log \left (\log \left (5 \, \log \left (x^{3} \log \relax (3) + 2 \, x^{2} \log \relax (3) - x^{2} - 15 \, x \log \relax (3)\right ) - 5 \, \log \left (x + 5\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+20*x-75)*log(3)-x^2-10*x)/((x^4+7*x^3-5*x^2-75*x)*log(3)-x^3-5*x^2)/log(((x^3+2*x^2-1
5*x)*log(3)-x^2)/(5+x))/log(5*log(((x^3+2*x^2-15*x)*log(3)-x^2)/(5+x))),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{3}+17 x^{2}+20 x -75\right ) \ln \relax (3)-x^{2}-10 x}{\left (\left (x^{4}+7 x^{3}-5 x^{2}-75 x \right ) \ln \relax (3)-x^{3}-5 x^{2}\right ) \ln \left (\frac {\left (x^{3}+2 x^{2}-15 x \right ) \ln \relax (3)-x^{2}}{5+x}\right ) \ln \left (5 \ln \left (\frac {\left (x^{3}+2 x^{2}-15 x \right ) \ln \relax (3)-x^{2}}{5+x}\right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+17*x^2+20*x-75)*ln(3)-x^2-10*x)/((x^4+7*x^3-5*x^2-75*x)*ln(3)-x^3-5*x^2)/ln(((x^3+2*x^2-15*x)*ln(3
)-x^2)/(5+x))/ln(5*ln(((x^3+2*x^2-15*x)*ln(3)-x^2)/(5+x))),x)

[Out]

int(((2*x^3+17*x^2+20*x-75)*ln(3)-x^2-10*x)/((x^4+7*x^3-5*x^2-75*x)*ln(3)-x^3-5*x^2)/ln(((x^3+2*x^2-15*x)*ln(3
)-x^2)/(5+x))/ln(5*ln(((x^3+2*x^2-15*x)*ln(3)-x^2)/(5+x))),x)

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maxima [A]  time = 0.54, size = 34, normalized size = 1.48 \begin {gather*} \log \left (\log \relax (5) + \log \left (\log \left (x^{2} \log \relax (3) + x {\left (2 \, \log \relax (3) - 1\right )} - 15 \, \log \relax (3)\right ) - \log \left (x + 5\right ) + \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+17*x^2+20*x-75)*log(3)-x^2-10*x)/((x^4+7*x^3-5*x^2-75*x)*log(3)-x^3-5*x^2)/log(((x^3+2*x^2-1
5*x)*log(3)-x^2)/(5+x))/log(5*log(((x^3+2*x^2-15*x)*log(3)-x^2)/(5+x))),x, algorithm="maxima")

[Out]

log(log(5) + log(log(x^2*log(3) + x*(2*log(3) - 1) - 15*log(3)) - log(x + 5) + log(x)))

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mupad [B]  time = 5.54, size = 32, normalized size = 1.39 \begin {gather*} \ln \left (\ln \left (5\,\ln \left (\frac {\ln \relax (3)\,\left (x^3+2\,x^2-15\,x\right )-x^2}{x+5}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x - log(3)*(20*x + 17*x^2 + 2*x^3 - 75) + x^2)/(log(5*log((log(3)*(2*x^2 - 15*x + x^3) - x^2)/(x + 5))
)*log((log(3)*(2*x^2 - 15*x + x^3) - x^2)/(x + 5))*(log(3)*(75*x + 5*x^2 - 7*x^3 - x^4) + 5*x^2 + x^3)),x)

[Out]

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

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sympy [A]  time = 1.46, size = 27, normalized size = 1.17 \begin {gather*} \log {\left (\log {\left (5 \log {\left (\frac {- x^{2} + \left (x^{3} + 2 x^{2} - 15 x\right ) \log {\relax (3 )}}{x + 5} \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+17*x**2+20*x-75)*ln(3)-x**2-10*x)/((x**4+7*x**3-5*x**2-75*x)*ln(3)-x**3-5*x**2)/ln(((x**3+2
*x**2-15*x)*ln(3)-x**2)/(5+x))/ln(5*ln(((x**3+2*x**2-15*x)*ln(3)-x**2)/(5+x))),x)

[Out]

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

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