\(\int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+(2 x-4 x^2+10 x^3-6 x^4) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+(-14 x^2+24 x^3-6 x^4-4 x^5) \log (x)+(1-2 x+x^2) \log ^2(x)} \, dx\) [8732]

Optimal result

Integrand size = 117, antiderivative size = 28 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {2 x+\frac {1+x}{-1+x}}{7+2 x-\frac {\log (x)}{x^2}} \]

[Out]

(2*x+(1+x)/(-1+x))/(2*x+7-ln(x)/x^2)

Rubi [F]

\[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx \]

[In]

Int[(-x + 2*x^2 - 3*x^3 + 4*x^4 - 32*x^5 + 12*x^6 + (2*x - 4*x^2 + 10*x^3 - 6*x^4)*Log[x])/(49*x^4 - 70*x^5 -
3*x^6 + 20*x^7 + 4*x^8 + (-14*x^2 + 24*x^3 - 6*x^4 - 4*x^5)*Log[x] + (1 - 2*x + x^2)*Log[x]^2),x]

[Out]

-38*Defer[Int][(7*x^2 + 2*x^3 - Log[x])^(-2), x] - 38*Defer[Int][1/((-1 + x)*(7*x^2 + 2*x^3 - Log[x])^2), x] -
39*Defer[Int][x/(7*x^2 + 2*x^3 - Log[x])^2, x] - 38*Defer[Int][x^2/(7*x^2 + 2*x^3 - Log[x])^2, x] - 26*Defer[
Int][x^3/(7*x^2 + 2*x^3 - Log[x])^2, x] - 34*Defer[Int][x^4/(7*x^2 + 2*x^3 - Log[x])^2, x] - 12*Defer[Int][x^5
/(7*x^2 + 2*x^3 - Log[x])^2, x] + 2*Defer[Int][(7*x^2 + 2*x^3 - Log[x])^(-1), x] - 2*Defer[Int][1/((-1 + x)^2*
(7*x^2 + 2*x^3 - Log[x])), x] + 2*Defer[Int][x/(7*x^2 + 2*x^3 - Log[x]), x] + 6*Defer[Int][x^2/(7*x^2 + 2*x^3
- Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-1+2 x-3 x^2+4 x^3-32 x^4+12 x^5+\left (2-4 x+10 x^2-6 x^3\right ) \log (x)\right )}{(1-x)^2 \left (x^2 (7+2 x)-\log (x)\right )^2} \, dx \\ & = \int \left (-\frac {x \left (-1+x+12 x^2-8 x^3+22 x^4+12 x^5\right )}{(-1+x) \left (7 x^2+2 x^3-\log (x)\right )^2}+\frac {2 x \left (-1+2 x-5 x^2+3 x^3\right )}{(-1+x)^2 \left (7 x^2+2 x^3-\log (x)\right )}\right ) \, dx \\ & = 2 \int \frac {x \left (-1+2 x-5 x^2+3 x^3\right )}{(-1+x)^2 \left (7 x^2+2 x^3-\log (x)\right )} \, dx-\int \frac {x \left (-1+x+12 x^2-8 x^3+22 x^4+12 x^5\right )}{(-1+x) \left (7 x^2+2 x^3-\log (x)\right )^2} \, dx \\ & = 2 \int \left (\frac {1}{7 x^2+2 x^3-\log (x)}-\frac {1}{(-1+x)^2 \left (7 x^2+2 x^3-\log (x)\right )}+\frac {x}{7 x^2+2 x^3-\log (x)}+\frac {3 x^2}{7 x^2+2 x^3-\log (x)}\right ) \, dx-\int \left (\frac {38}{\left (7 x^2+2 x^3-\log (x)\right )^2}+\frac {38}{(-1+x) \left (7 x^2+2 x^3-\log (x)\right )^2}+\frac {39 x}{\left (7 x^2+2 x^3-\log (x)\right )^2}+\frac {38 x^2}{\left (7 x^2+2 x^3-\log (x)\right )^2}+\frac {26 x^3}{\left (7 x^2+2 x^3-\log (x)\right )^2}+\frac {34 x^4}{\left (7 x^2+2 x^3-\log (x)\right )^2}+\frac {12 x^5}{\left (7 x^2+2 x^3-\log (x)\right )^2}\right ) \, dx \\ & = 2 \int \frac {1}{7 x^2+2 x^3-\log (x)} \, dx-2 \int \frac {1}{(-1+x)^2 \left (7 x^2+2 x^3-\log (x)\right )} \, dx+2 \int \frac {x}{7 x^2+2 x^3-\log (x)} \, dx+6 \int \frac {x^2}{7 x^2+2 x^3-\log (x)} \, dx-12 \int \frac {x^5}{\left (7 x^2+2 x^3-\log (x)\right )^2} \, dx-26 \int \frac {x^3}{\left (7 x^2+2 x^3-\log (x)\right )^2} \, dx-34 \int \frac {x^4}{\left (7 x^2+2 x^3-\log (x)\right )^2} \, dx-38 \int \frac {1}{\left (7 x^2+2 x^3-\log (x)\right )^2} \, dx-38 \int \frac {1}{(-1+x) \left (7 x^2+2 x^3-\log (x)\right )^2} \, dx-38 \int \frac {x^2}{\left (7 x^2+2 x^3-\log (x)\right )^2} \, dx-39 \int \frac {x}{\left (7 x^2+2 x^3-\log (x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=-\frac {x^2 \left (1-x+2 x^2\right )}{(-1+x) \left (-7 x^2-2 x^3+\log (x)\right )} \]

[In]

Integrate[(-x + 2*x^2 - 3*x^3 + 4*x^4 - 32*x^5 + 12*x^6 + (2*x - 4*x^2 + 10*x^3 - 6*x^4)*Log[x])/(49*x^4 - 70*
x^5 - 3*x^6 + 20*x^7 + 4*x^8 + (-14*x^2 + 24*x^3 - 6*x^4 - 4*x^5)*Log[x] + (1 - 2*x + x^2)*Log[x]^2),x]

[Out]

-((x^2*(1 - x + 2*x^2))/((-1 + x)*(-7*x^2 - 2*x^3 + Log[x])))

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32

method	result	size
default	\(-\frac {2 x^{4}-x^{3}+x^{2}}{\left (-1+x \right ) \left (-2 x^{3}-7 x^{2}+\ln \left (x \right )\right )}\)	\(37\)
risch	\(\frac {\left (2 x^{2}-x +1\right ) x^{2}}{\left (-1+x \right ) \left (2 x^{3}+7 x^{2}-\ln \left (x \right )\right )}\)	\(37\)
parallelrisch	\(\frac {2 x^{4}-x^{3}+x^{2}}{2 x^{4}+5 x^{3}-7 x^{2}-x \ln \left (x \right )+\ln \left (x \right )}\)	\(41\)

[In]

int(((-6*x^4+10*x^3-4*x^2+2*x)*ln(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^2-x)/((x^2-2*x+1)*ln(x)^2+(-4*x^5-6*x^4+24*
x^3-14*x^2)*ln(x)+4*x^8+20*x^7-3*x^6-70*x^5+49*x^4),x,method=_RETURNVERBOSE)

[Out]

-(2*x^4-x^3+x^2)/(-1+x)/(-2*x^3-7*x^2+ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {2 \, x^{4} - x^{3} + x^{2}}{2 \, x^{4} + 5 \, x^{3} - 7 \, x^{2} - {\left (x - 1\right )} \log \left (x\right )} \]

[In]

integrate(((-6*x^4+10*x^3-4*x^2+2*x)*log(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^2-x)/((x^2-2*x+1)*log(x)^2+(-4*x^5-6
*x^4+24*x^3-14*x^2)*log(x)+4*x^8+20*x^7-3*x^6-70*x^5+49*x^4),x, algorithm="fricas")

[Out]

(2*x^4 - x^3 + x^2)/(2*x^4 + 5*x^3 - 7*x^2 - (x - 1)*log(x))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {- 2 x^{4} + x^{3} - x^{2}}{- 2 x^{4} - 5 x^{3} + 7 x^{2} + \left (x - 1\right ) \log {\left (x \right )}} \]

[In]

integrate(((-6*x**4+10*x**3-4*x**2+2*x)*ln(x)+12*x**6-32*x**5+4*x**4-3*x**3+2*x**2-x)/((x**2-2*x+1)*ln(x)**2+(
-4*x**5-6*x**4+24*x**3-14*x**2)*ln(x)+4*x**8+20*x**7-3*x**6-70*x**5+49*x**4),x)

[Out]

(-2*x**4 + x**3 - x**2)/(-2*x**4 - 5*x**3 + 7*x**2 + (x - 1)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {2 \, x^{4} - x^{3} + x^{2}}{2 \, x^{4} + 5 \, x^{3} - 7 \, x^{2} - {\left (x - 1\right )} \log \left (x\right )} \]

[In]

integrate(((-6*x^4+10*x^3-4*x^2+2*x)*log(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^2-x)/((x^2-2*x+1)*log(x)^2+(-4*x^5-6
*x^4+24*x^3-14*x^2)*log(x)+4*x^8+20*x^7-3*x^6-70*x^5+49*x^4),x, algorithm="maxima")

[Out]

(2*x^4 - x^3 + x^2)/(2*x^4 + 5*x^3 - 7*x^2 - (x - 1)*log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {2 \, x^{4} - x^{3} + x^{2}}{2 \, x^{4} + 5 \, x^{3} - 7 \, x^{2} - x \log \left (x\right ) + \log \left (x\right )} \]

[In]

integrate(((-6*x^4+10*x^3-4*x^2+2*x)*log(x)+12*x^6-32*x^5+4*x^4-3*x^3+2*x^2-x)/((x^2-2*x+1)*log(x)^2+(-4*x^5-6
*x^4+24*x^3-14*x^2)*log(x)+4*x^8+20*x^7-3*x^6-70*x^5+49*x^4),x, algorithm="giac")

[Out]

(2*x^4 - x^3 + x^2)/(2*x^4 + 5*x^3 - 7*x^2 - x*log(x) + log(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {-x+2 x^2-3 x^3+4 x^4-32 x^5+12 x^6+\left (2 x-4 x^2+10 x^3-6 x^4\right ) \log (x)}{49 x^4-70 x^5-3 x^6+20 x^7+4 x^8+\left (-14 x^2+24 x^3-6 x^4-4 x^5\right ) \log (x)+\left (1-2 x+x^2\right ) \log ^2(x)} \, dx=\int \frac {\ln \left (x\right )\,\left (-6\,x^4+10\,x^3-4\,x^2+2\,x\right )-x+2\,x^2-3\,x^3+4\,x^4-32\,x^5+12\,x^6}{{\ln \left (x\right )}^2\,\left (x^2-2\,x+1\right )-\ln \left (x\right )\,\left (4\,x^5+6\,x^4-24\,x^3+14\,x^2\right )+49\,x^4-70\,x^5-3\,x^6+20\,x^7+4\,x^8} \,d x \]

[In]

int((log(x)*(2*x - 4*x^2 + 10*x^3 - 6*x^4) - x + 2*x^2 - 3*x^3 + 4*x^4 - 32*x^5 + 12*x^6)/(log(x)^2*(x^2 - 2*x
+ 1) - log(x)*(14*x^2 - 24*x^3 + 6*x^4 + 4*x^5) + 49*x^4 - 70*x^5 - 3*x^6 + 20*x^7 + 4*x^8),x)

[Out]

int((log(x)*(2*x - 4*x^2 + 10*x^3 - 6*x^4) - x + 2*x^2 - 3*x^3 + 4*x^4 - 32*x^5 + 12*x^6)/(log(x)^2*(x^2 - 2*x
+ 1) - log(x)*(14*x^2 - 24*x^3 + 6*x^4 + 4*x^5) + 49*x^4 - 70*x^5 - 3*x^6 + 20*x^7 + 4*x^8), x)