\(\int \frac {-160+24 x^2+16 x^3+(-160-24 x^2-8 x^3) \log (\frac {20+3 x^2+x^3}{x^2})}{20+3 x^2+x^3+(-40-6 x^2-2 x^3) \log (\frac {20+3 x^2+x^3}{x^2})+(20+3 x^2+x^3) \log ^2(\frac {20+3 x^2+x^3}{x^2})} \, dx\) [7516]

Optimal result

Integrand size = 109, antiderivative size = 27 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=\frac {8 x}{1-\log \left (5+x-\frac {-\frac {20}{x}+2 x}{x}\right )} \]

[Out]

8*x/(1-ln(x-(2*x-20/x)/x+5))

Rubi [F]

\[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=\int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx \]

[In]

Int[(-160 + 24*x^2 + 16*x^3 + (-160 - 24*x^2 - 8*x^3)*Log[(20 + 3*x^2 + x^3)/x^2])/(20 + 3*x^2 + x^3 + (-40 -
6*x^2 - 2*x^3)*Log[(20 + 3*x^2 + x^3)/x^2] + (20 + 3*x^2 + x^3)*Log[(20 + 3*x^2 + x^3)/x^2]^2),x]

[Out]

8*Defer[Int][(1 - Log[3 + 20/x^2 + x])^(-1), x] + 8*Defer[Int][(-1 + Log[3 + 20/x^2 + x])^(-2), x] - 480*Defer
[Int][1/((20 + 3*x^2 + x^3)*(-1 + Log[3 + 20/x^2 + x])^2), x] - 24*Defer[Int][x^2/((20 + 3*x^2 + x^3)*(-1 + Lo
g[3 + 20/x^2 + x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {8 \left (-20+3 x^2+2 x^3-\left (20+3 x^2+x^3\right ) \log \left (3+\frac {20}{x^2}+x\right )\right )}{\left (20+3 x^2+x^3\right ) \left (1-\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx \\ & = 8 \int \frac {-20+3 x^2+2 x^3-\left (20+3 x^2+x^3\right ) \log \left (3+\frac {20}{x^2}+x\right )}{\left (20+3 x^2+x^3\right ) \left (1-\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx \\ & = 8 \int \left (\frac {1}{1-\log \left (3+\frac {20}{x^2}+x\right )}+\frac {-40+x^3}{\left (20+3 x^2+x^3\right ) \left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2}\right ) \, dx \\ & = 8 \int \frac {1}{1-\log \left (3+\frac {20}{x^2}+x\right )} \, dx+8 \int \frac {-40+x^3}{\left (20+3 x^2+x^3\right ) \left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx \\ & = 8 \int \left (\frac {1}{\left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2}-\frac {3 \left (20+x^2\right )}{\left (20+3 x^2+x^3\right ) \left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2}\right ) \, dx+8 \int \frac {1}{1-\log \left (3+\frac {20}{x^2}+x\right )} \, dx \\ & = 8 \int \frac {1}{1-\log \left (3+\frac {20}{x^2}+x\right )} \, dx+8 \int \frac {1}{\left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx-24 \int \frac {20+x^2}{\left (20+3 x^2+x^3\right ) \left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx \\ & = 8 \int \frac {1}{1-\log \left (3+\frac {20}{x^2}+x\right )} \, dx+8 \int \frac {1}{\left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx-24 \int \left (\frac {20}{\left (20+3 x^2+x^3\right ) \left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2}+\frac {x^2}{\left (20+3 x^2+x^3\right ) \left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2}\right ) \, dx \\ & = 8 \int \frac {1}{1-\log \left (3+\frac {20}{x^2}+x\right )} \, dx+8 \int \frac {1}{\left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx-24 \int \frac {x^2}{\left (20+3 x^2+x^3\right ) \left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx-480 \int \frac {1}{\left (20+3 x^2+x^3\right ) \left (-1+\log \left (3+\frac {20}{x^2}+x\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 x}{-1+\log \left (3+\frac {20}{x^2}+x\right )} \]

[In]

Integrate[(-160 + 24*x^2 + 16*x^3 + (-160 - 24*x^2 - 8*x^3)*Log[(20 + 3*x^2 + x^3)/x^2])/(20 + 3*x^2 + x^3 + (
-40 - 6*x^2 - 2*x^3)*Log[(20 + 3*x^2 + x^3)/x^2] + (20 + 3*x^2 + x^3)*Log[(20 + 3*x^2 + x^3)/x^2]^2),x]

[Out]

(-8*x)/(-1 + Log[3 + 20/x^2 + x])

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
norman	\(-\frac {8 x}{\ln \left (\frac {x^{3}+3 x^{2}+20}{x^{2}}\right )-1}\)	\(23\)
risch	\(-\frac {8 x}{\ln \left (\frac {x^{3}+3 x^{2}+20}{x^{2}}\right )-1}\)	\(23\)
parallelrisch	\(-\frac {8 x}{\ln \left (\frac {x^{3}+3 x^{2}+20}{x^{2}}\right )-1}\)	\(23\)

[In]

int(((-8*x^3-24*x^2-160)*ln((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/((x^3+3*x^2+20)*ln((x^3+3*x^2+20)/x^2)^2+(-
2*x^3-6*x^2-40)*ln((x^3+3*x^2+20)/x^2)+x^3+3*x^2+20),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 \, x}{\log \left (\frac {x^{3} + 3 \, x^{2} + 20}{x^{2}}\right ) - 1} \]

[In]

integrate(((-8*x^3-24*x^2-160)*log((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/((x^3+3*x^2+20)*log((x^3+3*x^2+20)/x
^2)^2+(-2*x^3-6*x^2-40)*log((x^3+3*x^2+20)/x^2)+x^3+3*x^2+20),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=- \frac {8 x}{\log {\left (\frac {x^{3} + 3 x^{2} + 20}{x^{2}} \right )} - 1} \]

[In]

integrate(((-8*x**3-24*x**2-160)*ln((x**3+3*x**2+20)/x**2)+16*x**3+24*x**2-160)/((x**3+3*x**2+20)*ln((x**3+3*x
**2+20)/x**2)**2+(-2*x**3-6*x**2-40)*ln((x**3+3*x**2+20)/x**2)+x**3+3*x**2+20),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 \, x}{\log \left (x^{3} + 3 \, x^{2} + 20\right ) - 2 \, \log \left (x\right ) - 1} \]

[In]

integrate(((-8*x^3-24*x^2-160)*log((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/((x^3+3*x^2+20)*log((x^3+3*x^2+20)/x
^2)^2+(-2*x^3-6*x^2-40)*log((x^3+3*x^2+20)/x^2)+x^3+3*x^2+20),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 \, x}{\log \left (\frac {x^{3} + 3 \, x^{2} + 20}{x^{2}}\right ) - 1} \]

[In]

integrate(((-8*x^3-24*x^2-160)*log((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/((x^3+3*x^2+20)*log((x^3+3*x^2+20)/x
^2)^2+(-2*x^3-6*x^2-40)*log((x^3+3*x^2+20)/x^2)+x^3+3*x^2+20),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8\,\left (x-3\,\ln \left (\frac {x^3+3\,x^2+20}{x^2}\right )+3\right )}{\ln \left (\frac {x^3+3\,x^2+20}{x^2}\right )-1} \]

[In]

int(-(log((3*x^2 + x^3 + 20)/x^2)*(24*x^2 + 8*x^3 + 160) - 24*x^2 - 16*x^3 + 160)/(log((3*x^2 + x^3 + 20)/x^2)
^2*(3*x^2 + x^3 + 20) - log((3*x^2 + x^3 + 20)/x^2)*(6*x^2 + 2*x^3 + 40) + 3*x^2 + x^3 + 20),x)

[Out]

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