\(\int \frac {-24-8 x+(-48-24 x) \log (x)+(27 x^3+18 x^4+3 x^5) \log ^2(x)}{(18 x^3+12 x^4+2 x^5) \log ^2(x)} \, dx\) [5580]

Optimal result

Integrand size = 57, antiderivative size = 27 \[ \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx=2 x+\frac {1}{2} \left (-1-x+\frac {8}{x^2 (3+x) \log (x)}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx=\int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx \]

[In]

Int[(-24 - 8*x + (-48 - 24*x)*Log[x] + (27*x^3 + 18*x^4 + 3*x^5)*Log[x]^2)/((18*x^3 + 12*x^4 + 2*x^5)*Log[x]^2
),x]

[Out]

(3*x)/2 - 4*Defer[Int][1/(x^3*(3 + x)*Log[x]^2), x] - 12*Defer[Int][(2 + x)/(x^3*(3 + x)^2*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{x^3 \left (18+12 x+2 x^2\right ) \log ^2(x)} \, dx \\ & = \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{2 x^3 (3+x)^2 \log ^2(x)} \, dx \\ & = \frac {1}{2} \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{x^3 (3+x)^2 \log ^2(x)} \, dx \\ & = \frac {1}{2} \int \left (3-\frac {8}{x^3 (3+x) \log ^2(x)}-\frac {24 (2+x)}{x^3 (3+x)^2 \log (x)}\right ) \, dx \\ & = \frac {3 x}{2}-4 \int \frac {1}{x^3 (3+x) \log ^2(x)} \, dx-12 \int \frac {2+x}{x^3 (3+x)^2 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.87 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx=\frac {3 x}{2}+\frac {4}{x^2 (3+x) \log (x)} \]

[In]

Integrate[(-24 - 8*x + (-48 - 24*x)*Log[x] + (27*x^3 + 18*x^4 + 3*x^5)*Log[x]^2)/((18*x^3 + 12*x^4 + 2*x^5)*Lo
g[x]^2),x]

[Out]

(3*x)/2 + 4/(x^2*(3 + x)*Log[x])

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70

method	result	size
risch	\(\frac {3 x}{2}+\frac {4}{x^{2} \ln \left (x \right ) \left (3+x \right )}\)	\(19\)
norman	\(\frac {4-\frac {27 x^{2} \ln \left (x \right )}{2}+\frac {3 x^{4} \ln \left (x \right )}{2}}{x^{2} \left (3+x \right ) \ln \left (x \right )}\)	\(30\)
parallelrisch	\(\frac {3 x^{4} \ln \left (x \right )+8-27 x^{2} \ln \left (x \right )}{2 x^{2} \left (3+x \right ) \ln \left (x \right )}\)	\(31\)

[In]

int(((3*x^5+18*x^4+27*x^3)*ln(x)^2+(-24*x-48)*ln(x)-8*x-24)/(2*x^5+12*x^4+18*x^3)/ln(x)^2,x,method=_RETURNVERB
OSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (x^{4} + 3 \, x^{3}\right )} \log \left (x\right ) + 8}{2 \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x\right )} \]

[In]

integrate(((3*x^5+18*x^4+27*x^3)*log(x)^2+(-24*x-48)*log(x)-8*x-24)/(2*x^5+12*x^4+18*x^3)/log(x)^2,x, algorith
m="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx=\frac {3 x}{2} + \frac {4}{\left (x^{3} + 3 x^{2}\right ) \log {\left (x \right )}} \]

[In]

integrate(((3*x**5+18*x**4+27*x**3)*ln(x)**2+(-24*x-48)*ln(x)-8*x-24)/(2*x**5+12*x**4+18*x**3)/ln(x)**2,x)

[Out]

3*x/2 + 4/((x**3 + 3*x**2)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (x^{4} + 3 \, x^{3}\right )} \log \left (x\right ) + 8}{2 \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x\right )} \]

[In]

integrate(((3*x^5+18*x^4+27*x^3)*log(x)^2+(-24*x-48)*log(x)-8*x-24)/(2*x^5+12*x^4+18*x^3)/log(x)^2,x, algorith
m="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx=\frac {3}{2} \, x + \frac {4}{x^{3} \log \left (x\right ) + 3 \, x^{2} \log \left (x\right )} \]

[In]

integrate(((3*x^5+18*x^4+27*x^3)*log(x)^2+(-24*x-48)*log(x)-8*x-24)/(2*x^5+12*x^4+18*x^3)/log(x)^2,x, algorith
m="giac")

[Out]

3/2*x + 4/(x^3*log(x) + 3*x^2*log(x))

Mupad [B] (verification not implemented)

Time = 11.49 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {-24-8 x+(-48-24 x) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)}{\left (18 x^3+12 x^4+2 x^5\right ) \log ^2(x)} \, dx=\frac {3\,x}{2}+\frac {4}{x^2\,\ln \left (x\right )\,\left (x+3\right )} \]

[In]

int(-(8*x + log(x)*(24*x + 48) - log(x)^2*(27*x^3 + 18*x^4 + 3*x^5) + 24)/(log(x)^2*(18*x^3 + 12*x^4 + 2*x^5))
,x)

[Out]

(3*x)/2 + 4/(x^2*log(x)*(x + 3))