\(\int \frac {-2 x^4+3 x^5+4 x^6+(5 x^4+12 x^5) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx\) [8415]

Optimal result

Integrand size = 73, antiderivative size = 14 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=x^4 \left (1+\frac {x}{(x+\log (x))^2}\right ) \]

[Out]

x^4*(x/(x+ln(x))^2+1)

Rubi [F]

\[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=\int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx \]

[In]

Int[(-2*x^4 + 3*x^5 + 4*x^6 + (5*x^4 + 12*x^5)*Log[x] + 12*x^4*Log[x]^2 + 4*x^3*Log[x]^3)/(x^3 + 3*x^2*Log[x]
+ 3*x*Log[x]^2 + Log[x]^3),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \left (x \left (-2+3 x+4 x^2\right )+x (5+12 x) \log (x)+12 x \log ^2(x)+4 \log ^3(x)\right )}{(x+\log (x))^3} \, dx \\ & = \int \left (4 x^3-\frac {2 x^4 (1+x)}{(x+\log (x))^3}+\frac {5 x^4}{(x+\log (x))^2}\right ) \, dx \\ & = x^4-2 \int \frac {x^4 (1+x)}{(x+\log (x))^3} \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx \\ & = x^4-2 \int \left (\frac {x^4}{(x+\log (x))^3}+\frac {x^5}{(x+\log (x))^3}\right ) \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx \\ & = x^4-2 \int \frac {x^4}{(x+\log (x))^3} \, dx-2 \int \frac {x^5}{(x+\log (x))^3} \, dx+5 \int \frac {x^4}{(x+\log (x))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=x^4+\frac {x^5}{(x+\log (x))^2} \]

[In]

Integrate[(-2*x^4 + 3*x^5 + 4*x^6 + (5*x^4 + 12*x^5)*Log[x] + 12*x^4*Log[x]^2 + 4*x^3*Log[x]^3)/(x^3 + 3*x^2*L
og[x] + 3*x*Log[x]^2 + Log[x]^3),x]

[Out]

x^4 + x^5/(x + Log[x])^2

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07

method	result	size
risch	\(x^{4}+\frac {x^{5}}{\left (x +\ln \left (x \right )\right )^{2}}\)	\(15\)
default	\(-\frac {-x^{5}-x^{6}-2 x^{5} \ln \left (x \right )-x^{4} \ln \left (x \right )^{2}}{\left (x +\ln \left (x \right )\right )^{2}}\)	\(36\)
parallelrisch	\(\frac {x^{6}+2 x^{5} \ln \left (x \right )+x^{4} \ln \left (x \right )^{2}+x^{5}}{\ln \left (x \right )^{2}+2 x \ln \left (x \right )+x^{2}}\)	\(39\)

[In]

int((4*x^3*ln(x)^3+12*x^4*ln(x)^2+(12*x^5+5*x^4)*ln(x)+4*x^6+3*x^5-2*x^4)/(ln(x)^3+3*x*ln(x)^2+3*x^2*ln(x)+x^3
),x,method=_RETURNVERBOSE)

[Out]

x^4+x^5/(x+ln(x))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.71 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=\frac {x^{6} + 2 \, x^{5} \log \left (x\right ) + x^{4} \log \left (x\right )^{2} + x^{5}}{x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]

[In]

integrate((4*x^3*log(x)^3+12*x^4*log(x)^2+(12*x^5+5*x^4)*log(x)+4*x^6+3*x^5-2*x^4)/(log(x)^3+3*x*log(x)^2+3*x^
2*log(x)+x^3),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=\frac {x^{5}}{x^{2} + 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} + x^{4} \]

[In]

integrate((4*x**3*ln(x)**3+12*x**4*ln(x)**2+(12*x**5+5*x**4)*ln(x)+4*x**6+3*x**5-2*x**4)/(ln(x)**3+3*x*ln(x)**
2+3*x**2*ln(x)+x**3),x)

[Out]

x**5/(x**2 + 2*x*log(x) + log(x)**2) + x**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).

Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.71 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=\frac {x^{6} + 2 \, x^{5} \log \left (x\right ) + x^{4} \log \left (x\right )^{2} + x^{5}}{x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]

[In]

integrate((4*x^3*log(x)^3+12*x^4*log(x)^2+(12*x^5+5*x^4)*log(x)+4*x^6+3*x^5-2*x^4)/(log(x)^3+3*x*log(x)^2+3*x^
2*log(x)+x^3),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (14) = 28\).

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 3.07 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=x^{4} + \frac {x^{6} + x^{5}}{x^{3} + 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]

[In]

integrate((4*x^3*log(x)^3+12*x^4*log(x)^2+(12*x^5+5*x^4)*log(x)+4*x^6+3*x^5-2*x^4)/(log(x)^3+3*x*log(x)^2+3*x^
2*log(x)+x^3),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {-2 x^4+3 x^5+4 x^6+\left (5 x^4+12 x^5\right ) \log (x)+12 x^4 \log ^2(x)+4 x^3 \log ^3(x)}{x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)} \, dx=x^4-\frac {x^6-x^4\,\left (x^2+x\right )}{{\left (x+\ln \left (x\right )\right )}^2} \]

[In]

int((log(x)*(5*x^4 + 12*x^5) + 4*x^3*log(x)^3 + 12*x^4*log(x)^2 - 2*x^4 + 3*x^5 + 4*x^6)/(3*x*log(x)^2 + 3*x^2
*log(x) + log(x)^3 + x^3),x)

[Out]

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