\(\int \frac {-3 x-7 x^2-4 x^3+(6 x+3 x^2-4 x^4) \log (x)+(x^3-6 x^4) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx\) [5893]

Optimal result

Integrand size = 72, antiderivative size = 26 \[ \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx=\frac {x \left (3 x+x \left (x+x^2 \log ^2(x)\right )\right )}{-2 x+\log (x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx=\int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx \]

[In]

Int[(-3*x - 7*x^2 - 4*x^3 + (6*x + 3*x^2 - 4*x^4)*Log[x] + (x^3 - 6*x^4)*Log[x]^2 + 4*x^3*Log[x]^3)/(4*x^2 - 4
*x*Log[x] + Log[x]^2),x]

[Out]

2*x^5 + x^4*Log[x] - 3*Defer[Int][x/(2*x - Log[x])^2, x] + 5*Defer[Int][x^2/(2*x - Log[x])^2, x] + 2*Defer[Int
][x^3/(2*x - Log[x])^2, x] - 4*Defer[Int][x^5/(2*x - Log[x])^2, x] + 8*Defer[Int][x^6/(2*x - Log[x])^2, x] - 6
*Defer[Int][x/(2*x - Log[x]), x] - 3*Defer[Int][x^2/(2*x - Log[x]), x] - 24*Defer[Int][x^5/(2*x - Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{(2 x-\log (x))^2} \, dx \\ & = \int \left (x^3 (1+10 x)+\frac {x \left (-3+5 x+2 x^2-4 x^4+8 x^5\right )}{(2 x-\log (x))^2}-\frac {3 x \left (2+x+8 x^4\right )}{2 x-\log (x)}+4 x^3 \log (x)\right ) \, dx \\ & = -\left (3 \int \frac {x \left (2+x+8 x^4\right )}{2 x-\log (x)} \, dx\right )+4 \int x^3 \log (x) \, dx+\int x^3 (1+10 x) \, dx+\int \frac {x \left (-3+5 x+2 x^2-4 x^4+8 x^5\right )}{(2 x-\log (x))^2} \, dx \\ & = -\frac {x^4}{4}+x^4 \log (x)-3 \int \left (\frac {2 x}{2 x-\log (x)}+\frac {x^2}{2 x-\log (x)}+\frac {8 x^5}{2 x-\log (x)}\right ) \, dx+\int \left (x^3+10 x^4\right ) \, dx+\int \left (-\frac {3 x}{(2 x-\log (x))^2}+\frac {5 x^2}{(2 x-\log (x))^2}+\frac {2 x^3}{(2 x-\log (x))^2}-\frac {4 x^5}{(2 x-\log (x))^2}+\frac {8 x^6}{(2 x-\log (x))^2}\right ) \, dx \\ & = 2 x^5+x^4 \log (x)+2 \int \frac {x^3}{(2 x-\log (x))^2} \, dx-3 \int \frac {x}{(2 x-\log (x))^2} \, dx-3 \int \frac {x^2}{2 x-\log (x)} \, dx-4 \int \frac {x^5}{(2 x-\log (x))^2} \, dx+5 \int \frac {x^2}{(2 x-\log (x))^2} \, dx-6 \int \frac {x}{2 x-\log (x)} \, dx+8 \int \frac {x^6}{(2 x-\log (x))^2} \, dx-24 \int \frac {x^5}{2 x-\log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx=-\frac {x^2 \left (3+x+x^2 \log ^2(x)\right )}{2 x-\log (x)} \]

[In]

Integrate[(-3*x - 7*x^2 - 4*x^3 + (6*x + 3*x^2 - 4*x^4)*Log[x] + (x^3 - 6*x^4)*Log[x]^2 + 4*x^3*Log[x]^3)/(4*x
^2 - 4*x*Log[x] + Log[x]^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04

method	result	size
default	\(\frac {x^{3}+x^{4} \ln \left (x \right )^{2}+3 x^{2}}{\ln \left (x \right )-2 x}\)	\(27\)
norman	\(\frac {-3 x^{2}-x^{3}-x^{4} \ln \left (x \right )^{2}}{2 x -\ln \left (x \right )}\)	\(32\)
parallelrisch	\(\frac {-3 x^{2}-x^{3}-x^{4} \ln \left (x \right )^{2}}{2 x -\ln \left (x \right )}\)	\(32\)
risch	\(2 x^{5}+x^{4} \ln \left (x \right )-\frac {\left (4 x^{4}+x +3\right ) x^{2}}{2 x -\ln \left (x \right )}\)	\(36\)

[In]

int((4*x^3*ln(x)^3+(-6*x^4+x^3)*ln(x)^2+(-4*x^4+3*x^2+6*x)*ln(x)-4*x^3-7*x^2-3*x)/(ln(x)^2-4*x*ln(x)+4*x^2),x,
method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx=-\frac {x^{4} \log \left (x\right )^{2} + x^{3} + 3 \, x^{2}}{2 \, x - \log \left (x\right )} \]

[In]

integrate((4*x^3*log(x)^3+(-6*x^4+x^3)*log(x)^2+(-4*x^4+3*x^2+6*x)*log(x)-4*x^3-7*x^2-3*x)/(log(x)^2-4*x*log(x
)+4*x^2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx=2 x^{5} + x^{4} \log {\left (x \right )} + \frac {4 x^{6} + x^{3} + 3 x^{2}}{- 2 x + \log {\left (x \right )}} \]

[In]

integrate((4*x**3*ln(x)**3+(-6*x**4+x**3)*ln(x)**2+(-4*x**4+3*x**2+6*x)*ln(x)-4*x**3-7*x**2-3*x)/(ln(x)**2-4*x
*ln(x)+4*x**2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx=-\frac {x^{4} \log \left (x\right )^{2} + x^{3} + 3 \, x^{2}}{2 \, x - \log \left (x\right )} \]

[In]

integrate((4*x^3*log(x)^3+(-6*x^4+x^3)*log(x)^2+(-4*x^4+3*x^2+6*x)*log(x)-4*x^3-7*x^2-3*x)/(log(x)^2-4*x*log(x
)+4*x^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx=2 \, x^{5} + x^{4} \log \left (x\right ) - \frac {4 \, x^{6} + x^{3} + 3 \, x^{2}}{2 \, x - \log \left (x\right )} \]

[In]

integrate((4*x^3*log(x)^3+(-6*x^4+x^3)*log(x)^2+(-4*x^4+3*x^2+6*x)*log(x)-4*x^3-7*x^2-3*x)/(log(x)^2-4*x*log(x
)+4*x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.88 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-3 x-7 x^2-4 x^3+\left (6 x+3 x^2-4 x^4\right ) \log (x)+\left (x^3-6 x^4\right ) \log ^2(x)+4 x^3 \log ^3(x)}{4 x^2-4 x \log (x)+\log ^2(x)} \, dx=-\frac {x^4\,{\ln \left (x\right )}^2+x^3+3\,x^2}{2\,x-\ln \left (x\right )} \]

[In]

int(-(3*x - log(x)^2*(x^3 - 6*x^4) - 4*x^3*log(x)^3 + 7*x^2 + 4*x^3 - log(x)*(6*x + 3*x^2 - 4*x^4))/(log(x)^2
- 4*x*log(x) + 4*x^2),x)

[Out]

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