\(\int \frac {9-3 x-3 x^2+(-3 x-6 x^2) \log (x)+(144 x^2-96 x^3-80 x^4+32 x^5+16 x^6) \log ^2(x)}{(180 x-120 x^2-100 x^3+40 x^4+20 x^5) \log ^2(x)} \, dx\) [7303]

Optimal result

Integrand size = 84, antiderivative size = 26 \[ \int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx=\frac {2}{5} x \left (x+\frac {3}{8 x \left (-3+x+x^2\right ) \log (x)}\right ) \]

[Out]

2/5*x*(3/8/x/ln(x)/(x^2+x-3)+x)

Rubi [F]

\[ \int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx=\int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx \]

[In]

Int[(9 - 3*x - 3*x^2 + (-3*x - 6*x^2)*Log[x] + (144*x^2 - 96*x^3 - 80*x^4 + 32*x^5 + 16*x^6)*Log[x]^2)/((180*x
- 120*x^2 - 100*x^3 + 40*x^4 + 20*x^5)*Log[x]^2),x]

[Out]

(2*x^2)/5 - (3*Defer[Int][1/(x*(-3 + x + x^2)*Log[x]^2), x])/20 - (3*Defer[Int][(1 + 2*x)/((-3 + x + x^2)^2*Lo
g[x]), x])/20

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{20} \left (16 x-\frac {3}{x \left (-3+x+x^2\right ) \log ^2(x)}-\frac {3 (1+2 x)}{\left (-3+x+x^2\right )^2 \log (x)}\right ) \, dx \\ & = \frac {1}{20} \int \left (16 x-\frac {3}{x \left (-3+x+x^2\right ) \log ^2(x)}-\frac {3 (1+2 x)}{\left (-3+x+x^2\right )^2 \log (x)}\right ) \, dx \\ & = \frac {2 x^2}{5}-\frac {3}{20} \int \frac {1}{x \left (-3+x+x^2\right ) \log ^2(x)} \, dx-\frac {3}{20} \int \frac {1+2 x}{\left (-3+x+x^2\right )^2 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx=\frac {1}{20} \left (8 x^2+\frac {3}{\left (-3+x+x^2\right ) \log (x)}\right ) \]

[In]

Integrate[(9 - 3*x - 3*x^2 + (-3*x - 6*x^2)*Log[x] + (144*x^2 - 96*x^3 - 80*x^4 + 32*x^5 + 16*x^6)*Log[x]^2)/(
(180*x - 120*x^2 - 100*x^3 + 40*x^4 + 20*x^5)*Log[x]^2),x]

[Out]

(8*x^2 + 3/((-3 + x + x^2)*Log[x]))/20

Maple [A] (verified)

Time = 7.83 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81

method	result	size
risch	\(\frac {2 x^{2}}{5}+\frac {3}{20 \left (x^{2}+x -3\right ) \ln \left (x \right )}\)	\(21\)
norman	\(\frac {\frac {3}{20}-\frac {18 \ln \left (x \right )}{5}+\frac {6 x \ln \left (x \right )}{5}+\frac {2 x^{3} \ln \left (x \right )}{5}+\frac {2 x^{4} \ln \left (x \right )}{5}}{\left (x^{2}+x -3\right ) \ln \left (x \right )}\)	\(39\)
default	\(\frac {3-72 \ln \left (x \right )+24 x \ln \left (x \right )+8 x^{3} \ln \left (x \right )+8 x^{4} \ln \left (x \right )}{20 \left (x^{2}+x -3\right ) \ln \left (x \right )}\)	\(40\)
parallelrisch	\(\frac {3-72 \ln \left (x \right )+24 x \ln \left (x \right )+8 x^{3} \ln \left (x \right )+8 x^{4} \ln \left (x \right )}{20 \left (x^{2}+x -3\right ) \ln \left (x \right )}\)	\(40\)

[In]

int(((16*x^6+32*x^5-80*x^4-96*x^3+144*x^2)*ln(x)^2+(-6*x^2-3*x)*ln(x)-3*x^2-3*x+9)/(20*x^5+40*x^4-100*x^3-120*
x^2+180*x)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

2/5*x^2+3/20/(x^2+x-3)/ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx=\frac {8 \, {\left (x^{4} + x^{3} - 3 \, x^{2}\right )} \log \left (x\right ) + 3}{20 \, {\left (x^{2} + x - 3\right )} \log \left (x\right )} \]

[In]

integrate(((16*x^6+32*x^5-80*x^4-96*x^3+144*x^2)*log(x)^2+(-6*x^2-3*x)*log(x)-3*x^2-3*x+9)/(20*x^5+40*x^4-100*
x^3-120*x^2+180*x)/log(x)^2,x, algorithm="fricas")

[Out]

1/20*(8*(x^4 + x^3 - 3*x^2)*log(x) + 3)/((x^2 + x - 3)*log(x))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx=\frac {2 x^{2}}{5} + \frac {3}{\left (20 x^{2} + 20 x - 60\right ) \log {\left (x \right )}} \]

[In]

integrate(((16*x**6+32*x**5-80*x**4-96*x**3+144*x**2)*ln(x)**2+(-6*x**2-3*x)*ln(x)-3*x**2-3*x+9)/(20*x**5+40*x
**4-100*x**3-120*x**2+180*x)/ln(x)**2,x)

[Out]

2*x**2/5 + 3/((20*x**2 + 20*x - 60)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx=\frac {8 \, {\left (x^{4} + x^{3} - 3 \, x^{2}\right )} \log \left (x\right ) + 3}{20 \, {\left (x^{2} + x - 3\right )} \log \left (x\right )} \]

[In]

integrate(((16*x^6+32*x^5-80*x^4-96*x^3+144*x^2)*log(x)^2+(-6*x^2-3*x)*log(x)-3*x^2-3*x+9)/(20*x^5+40*x^4-100*
x^3-120*x^2+180*x)/log(x)^2,x, algorithm="maxima")

[Out]

1/20*(8*(x^4 + x^3 - 3*x^2)*log(x) + 3)/((x^2 + x - 3)*log(x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx=\frac {2}{5} \, x^{2} + \frac {3}{20 \, {\left (x^{2} \log \left (x\right ) + x \log \left (x\right ) - 3 \, \log \left (x\right )\right )}} \]

[In]

integrate(((16*x^6+32*x^5-80*x^4-96*x^3+144*x^2)*log(x)^2+(-6*x^2-3*x)*log(x)-3*x^2-3*x+9)/(20*x^5+40*x^4-100*
x^3-120*x^2+180*x)/log(x)^2,x, algorithm="giac")

[Out]

2/5*x^2 + 3/20/(x^2*log(x) + x*log(x) - 3*log(x))

Mupad [B] (verification not implemented)

Time = 13.46 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {9-3 x-3 x^2+\left (-3 x-6 x^2\right ) \log (x)+\left (144 x^2-96 x^3-80 x^4+32 x^5+16 x^6\right ) \log ^2(x)}{\left (180 x-120 x^2-100 x^3+40 x^4+20 x^5\right ) \log ^2(x)} \, dx=\frac {\frac {2\,x^4}{5}+\frac {2\,x^3}{5}-\frac {6\,x^2}{5}}{x^2+x-3}+\frac {3}{20\,\ln \left (x\right )\,\left (x^2+x-3\right )} \]

[In]

int(-(3*x - log(x)^2*(144*x^2 - 96*x^3 - 80*x^4 + 32*x^5 + 16*x^6) + log(x)*(3*x + 6*x^2) + 3*x^2 - 9)/(log(x)
^2*(180*x - 120*x^2 - 100*x^3 + 40*x^4 + 20*x^5)),x)

[Out]

((2*x^3)/5 - (6*x^2)/5 + (2*x^4)/5)/(x + x^2 - 3) + 3/(20*log(x)*(x + x^2 - 3))