\(\int \frac {-9+28 x+11 x^2+(-9+6 x+3 x^2) \log (2 x)}{27 x+16 x^2+3 x^3+(9 x+6 x^2+x^3) \log (2 x)} \, dx\) [5555]

Optimal result

Integrand size = 59, antiderivative size = 29 \[ \int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx=\frac {1}{2} \log \left (x^2 \left (4-\frac {2 (3+x)^2 (3+\log (2 x))}{x}\right )^4\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx=\int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx \]

[In]

Int[(-9 + 28*x + 11*x^2 + (-9 + 6*x + 3*x^2)*Log[2*x])/(27*x + 16*x^2 + 3*x^3 + (9*x + 6*x^2 + x^3)*Log[2*x]),
x]

[Out]

-Log[x] + 4*Log[3 + x] + 16*Defer[Int][(27 + 16*x + 3*x^2 + (3 + x)^2*Log[2*x])^(-1), x] + 18*Defer[Int][1/(x*
(27 + 16*x + 3*x^2 + (3 + x)^2*Log[2*x])), x] + 2*Defer[Int][x/(27 + 16*x + 3*x^2 + (3 + x)^2*Log[2*x]), x] -
24*Defer[Int][1/((3 + x)*(27 + 16*x + 3*x^2 + (3 + x)^2*Log[2*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 (-1+x)}{x (3+x)}+\frac {2 \left (27+21 x+11 x^2+x^3\right )}{x (3+x) \left (27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)\right )}\right ) \, dx \\ & = 2 \int \frac {27+21 x+11 x^2+x^3}{x (3+x) \left (27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)\right )} \, dx+3 \int \frac {-1+x}{x (3+x)} \, dx \\ & = 2 \int \frac {27+21 x+11 x^2+x^3}{x (3+x) \left (27+16 x+3 x^2+(3+x)^2 \log (2 x)\right )} \, dx+3 \int \left (-\frac {1}{3 x}+\frac {4}{3 (3+x)}\right ) \, dx \\ & = -\log (x)+4 \log (3+x)+2 \int \left (\frac {8}{27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)}+\frac {9}{x \left (27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)\right )}+\frac {x}{27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)}-\frac {12}{(3+x) \left (27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)\right )}\right ) \, dx \\ & = -\log (x)+4 \log (3+x)+2 \int \frac {x}{27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)} \, dx+16 \int \frac {1}{27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)} \, dx+18 \int \frac {1}{x \left (27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)\right )} \, dx-24 \int \frac {1}{(3+x) \left (27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)\right )} \, dx \\ & = -\log (x)+4 \log (3+x)+2 \int \frac {x}{27+16 x+3 x^2+(3+x)^2 \log (2 x)} \, dx+16 \int \frac {1}{27+16 x+3 x^2+(3+x)^2 \log (2 x)} \, dx+18 \int \frac {1}{x \left (27+16 x+3 x^2+(3+x)^2 \log (2 x)\right )} \, dx-24 \int \frac {1}{(3+x) \left (27+16 x+3 x^2+(3+x)^2 \log (2 x)\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx=-\log (x)+2 \log \left (27+16 x+3 x^2+9 \log (2 x)+6 x \log (2 x)+x^2 \log (2 x)\right ) \]

[In]

Integrate[(-9 + 28*x + 11*x^2 + (-9 + 6*x + 3*x^2)*Log[2*x])/(27*x + 16*x^2 + 3*x^3 + (9*x + 6*x^2 + x^3)*Log[
2*x]),x]

[Out]

-Log[x] + 2*Log[27 + 16*x + 3*x^2 + 9*Log[2*x] + 6*x*Log[2*x] + x^2*Log[2*x]]

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41

method	result	size
risch	\(-\ln \left (x \right )+4 \ln \left (3+x \right )+2 \ln \left (\ln \left (2 x \right )+\frac {3 x^{2}+16 x +27}{x^{2}+6 x +9}\right )\)	\(41\)
norman	\(-\ln \left (2 x \right )+2 \ln \left (x^{2} \ln \left (2 x \right )+6 x \ln \left (2 x \right )+3 x^{2}+9 \ln \left (2 x \right )+16 x +27\right )\)	\(42\)
parallelrisch	\(-\ln \left (2 x \right )+2 \ln \left (x^{2} \ln \left (2 x \right )+6 x \ln \left (2 x \right )+3 x^{2}+9 \ln \left (2 x \right )+16 x +27\right )\)	\(42\)
derivativedivides	\(-\ln \left (2 x \right )+2 \ln \left (4 x^{2} \ln \left (2 x \right )+24 x \ln \left (2 x \right )+12 x^{2}+36 \ln \left (2 x \right )+64 x +108\right )\)	\(43\)
default	\(-\ln \left (2 x \right )+2 \ln \left (4 x^{2} \ln \left (2 x \right )+24 x \ln \left (2 x \right )+12 x^{2}+36 \ln \left (2 x \right )+64 x +108\right )\)	\(43\)

[In]

int(((3*x^2+6*x-9)*ln(2*x)+11*x^2+28*x-9)/((x^3+6*x^2+9*x)*ln(2*x)+3*x^3+16*x^2+27*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+4*ln(3+x)+2*ln(ln(2*x)+(3*x^2+16*x+27)/(x^2+6*x+9))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx=4 \, \log \left (x + 3\right ) - \log \left (x\right ) + 2 \, \log \left (\frac {3 \, x^{2} + {\left (x^{2} + 6 \, x + 9\right )} \log \left (2 \, x\right ) + 16 \, x + 27}{x^{2} + 6 \, x + 9}\right ) \]

[In]

integrate(((3*x^2+6*x-9)*log(2*x)+11*x^2+28*x-9)/((x^3+6*x^2+9*x)*log(2*x)+3*x^3+16*x^2+27*x),x, algorithm="fr
icas")

[Out]

4*log(x + 3) - log(x) + 2*log((3*x^2 + (x^2 + 6*x + 9)*log(2*x) + 16*x + 27)/(x^2 + 6*x + 9))

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx=- \log {\left (x \right )} + 4 \log {\left (x + 3 \right )} + 2 \log {\left (\log {\left (2 x \right )} + \frac {3 x^{2} + 16 x + 27}{x^{2} + 6 x + 9} \right )} \]

[In]

integrate(((3*x**2+6*x-9)*ln(2*x)+11*x**2+28*x-9)/((x**3+6*x**2+9*x)*ln(2*x)+3*x**3+16*x**2+27*x),x)

[Out]

-log(x) + 4*log(x + 3) + 2*log(log(2*x) + (3*x**2 + 16*x + 27)/(x**2 + 6*x + 9))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).

Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx=4 \, \log \left (x + 3\right ) - \log \left (x\right ) + 2 \, \log \left (\frac {x^{2} {\left (\log \left (2\right ) + 3\right )} + 2 \, x {\left (3 \, \log \left (2\right ) + 8\right )} + {\left (x^{2} + 6 \, x + 9\right )} \log \left (x\right ) + 9 \, \log \left (2\right ) + 27}{x^{2} + 6 \, x + 9}\right ) \]

[In]

integrate(((3*x^2+6*x-9)*log(2*x)+11*x^2+28*x-9)/((x^3+6*x^2+9*x)*log(2*x)+3*x^3+16*x^2+27*x),x, algorithm="ma
xima")

[Out]

4*log(x + 3) - log(x) + 2*log((x^2*(log(2) + 3) + 2*x*(3*log(2) + 8) + (x^2 + 6*x + 9)*log(x) + 9*log(2) + 27)
/(x^2 + 6*x + 9))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx=2 \, \log \left (x^{2} \log \left (2 \, x\right ) + 3 \, x^{2} + 6 \, x \log \left (2 \, x\right ) + 16 \, x + 9 \, \log \left (2 \, x\right ) + 27\right ) - \log \left (x\right ) \]

[In]

integrate(((3*x^2+6*x-9)*log(2*x)+11*x^2+28*x-9)/((x^3+6*x^2+9*x)*log(2*x)+3*x^3+16*x^2+27*x),x, algorithm="gi
ac")

[Out]

2*log(x^2*log(2*x) + 3*x^2 + 6*x*log(2*x) + 16*x + 9*log(2*x) + 27) - log(x)

Mupad [B] (verification not implemented)

Time = 11.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-9+28 x+11 x^2+\left (-9+6 x+3 x^2\right ) \log (2 x)}{27 x+16 x^2+3 x^3+\left (9 x+6 x^2+x^3\right ) \log (2 x)} \, dx=2\,\ln \left (16\,x+9\,\ln \left (2\,x\right )+6\,x\,\ln \left (2\,x\right )+x^2\,\ln \left (2\,x\right )+3\,x^2+27\right )-\ln \left (x\right ) \]

[In]

int((28*x + log(2*x)*(6*x + 3*x^2 - 9) + 11*x^2 - 9)/(27*x + 16*x^2 + 3*x^3 + log(2*x)*(9*x + 6*x^2 + x^3)),x)

[Out]

2*log(16*x + 9*log(2*x) + 6*x*log(2*x) + x^2*log(2*x) + 3*x^2 + 27) - log(x)