\(\int \frac {8 x^4-4 x^5+(-20 x^4+8 x^5) \log (2 x)+(72-72 x+18 x^2) \log ^3(2 x)}{(36-36 x+9 x^2) \log ^3(2 x)} \, dx\) [4043]

Optimal result

Integrand size = 63, antiderivative size = 25 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=-4+2 x-\frac {2 x^5}{9 (2-x) \log ^2(2 x)} \]

[Out]

2*x-4-2/9*x^5/ln(2*x)^2/(2-x)

Rubi [F]

\[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx \]

[In]

Int[(8*x^4 - 4*x^5 + (-20*x^4 + 8*x^5)*Log[2*x] + (72 - 72*x + 18*x^2)*Log[2*x]^3)/((36 - 36*x + 9*x^2)*Log[2*
x]^3),x]

[Out]

2*x - (4*Defer[Int][x^4/((-2 + x)*Log[2*x]^3), x])/9 + (4*Defer[Int][(x^4*(-5 + 2*x))/((-2 + x)^2*Log[2*x]^2),
x])/9

Rubi steps \begin{align*} \text {integral}& = \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{9 (-2+x)^2 \log ^3(2 x)} \, dx \\ & = \frac {1}{9} \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{(-2+x)^2 \log ^3(2 x)} \, dx \\ & = \frac {1}{9} \int \left (18-\frac {4 x^4}{(-2+x) \log ^3(2 x)}+\frac {4 x^4 (-5+2 x)}{(-2+x)^2 \log ^2(2 x)}\right ) \, dx \\ & = 2 x-\frac {4}{9} \int \frac {x^4}{(-2+x) \log ^3(2 x)} \, dx+\frac {4}{9} \int \frac {x^4 (-5+2 x)}{(-2+x)^2 \log ^2(2 x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=2 x+\frac {2 x^5}{9 (-2+x) \log ^2(2 x)} \]

[In]

Integrate[(8*x^4 - 4*x^5 + (-20*x^4 + 8*x^5)*Log[2*x] + (72 - 72*x + 18*x^2)*Log[2*x]^3)/((36 - 36*x + 9*x^2)*
Log[2*x]^3),x]

[Out]

2*x + (2*x^5)/(9*(-2 + x)*Log[2*x]^2)

Maple [A] (verified)

Time = 491.87 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

method	result	size
risch	\(2 x +\frac {2 x^{5}}{9 \left (-2+x \right ) \ln \left (2 x \right )^{2}}\)	\(21\)
norman	\(\frac {-8 \ln \left (2 x \right )^{2}+\frac {2 x^{5}}{9}+2 x^{2} \ln \left (2 x \right )^{2}}{\left (-2+x \right ) \ln \left (2 x \right )^{2}}\)	\(38\)
parallelrisch	\(\frac {2 x^{5}+18 x^{2} \ln \left (2 x \right )^{2}-72 \ln \left (2 x \right )^{2}}{9 \ln \left (2 x \right )^{2} \left (-2+x \right )}\)	\(39\)
derivativedivides	\(2 x +\frac {32}{9 \ln \left (2 x \right )^{2}}+\frac {16 x}{9 \ln \left (2 x \right )^{2}}+\frac {4 x^{3}}{9 \ln \left (2 x \right )^{2}}+\frac {8 x^{2}}{9 \ln \left (2 x \right )^{2}}+\frac {2 x^{4}}{9 \ln \left (2 x \right )^{2}}+\frac {128}{9 \left (2 x -4\right ) \ln \left (2 x \right )^{2}}\)	\(70\)
default	\(2 x +\frac {32}{9 \ln \left (2 x \right )^{2}}+\frac {16 x}{9 \ln \left (2 x \right )^{2}}+\frac {4 x^{3}}{9 \ln \left (2 x \right )^{2}}+\frac {8 x^{2}}{9 \ln \left (2 x \right )^{2}}+\frac {2 x^{4}}{9 \ln \left (2 x \right )^{2}}+\frac {128}{9 \left (2 x -4\right ) \ln \left (2 x \right )^{2}}\)	\(70\)
parts	\(2 x +\frac {32}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {64}{9 \left (-2+x \right ) \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {2 x^{4}}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {16 x}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {8 x^{2}}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}+\frac {4 x^{3}}{9 \left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}\)	\(74\)

[In]

int(((18*x^2-72*x+72)*ln(2*x)^3+(8*x^5-20*x^4)*ln(2*x)-4*x^5+8*x^4)/(9*x^2-36*x+36)/ln(2*x)^3,x,method=_RETURN
VERBOSE)

[Out]

2*x+2/9*x^5/(-2+x)/ln(2*x)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2 \, {\left (x^{5} + 9 \, {\left (x^{2} - 2 \, x\right )} \log \left (2 \, x\right )^{2}\right )}}{9 \, {\left (x - 2\right )} \log \left (2 \, x\right )^{2}} \]

[In]

integrate(((18*x^2-72*x+72)*log(2*x)^3+(8*x^5-20*x^4)*log(2*x)-4*x^5+8*x^4)/(9*x^2-36*x+36)/log(2*x)^3,x, algo
rithm="fricas")

[Out]

2/9*(x^5 + 9*(x^2 - 2*x)*log(2*x)^2)/((x - 2)*log(2*x)^2)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2 x^{5}}{\left (9 x - 18\right ) \log {\left (2 x \right )}^{2}} + 2 x \]

[In]

integrate(((18*x**2-72*x+72)*ln(2*x)**3+(8*x**5-20*x**4)*ln(2*x)-4*x**5+8*x**4)/(9*x**2-36*x+36)/ln(2*x)**3,x)

[Out]

2*x**5/((9*x - 18)*log(2*x)**2) + 2*x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (21) = 42\).

Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.48 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2 \, {\left (x^{5} + 9 \, x^{2} \log \left (2\right )^{2} - 18 \, x \log \left (2\right )^{2} + 9 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right )^{2} + 18 \, {\left (x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right )\right )} \log \left (x\right )\right )}}{9 \, {\left (x \log \left (2\right )^{2} + {\left (x - 2\right )} \log \left (x\right )^{2} - 2 \, \log \left (2\right )^{2} + 2 \, {\left (x \log \left (2\right ) - 2 \, \log \left (2\right )\right )} \log \left (x\right )\right )}} \]

[In]

integrate(((18*x^2-72*x+72)*log(2*x)^3+(8*x^5-20*x^4)*log(2*x)-4*x^5+8*x^4)/(9*x^2-36*x+36)/log(2*x)^3,x, algo
rithm="maxima")

[Out]

2/9*(x^5 + 9*x^2*log(2)^2 - 18*x*log(2)^2 + 9*(x^2 - 2*x)*log(x)^2 + 18*(x^2*log(2) - 2*x*log(2))*log(x))/(x*l
og(2)^2 + (x - 2)*log(x)^2 - 2*log(2)^2 + 2*(x*log(2) - 2*log(2))*log(x))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2 \, x^{5}}{9 \, {\left (x \log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right )^{2}\right )}} + 2 \, x \]

[In]

integrate(((18*x^2-72*x+72)*log(2*x)^3+(8*x^5-20*x^4)*log(2*x)-4*x^5+8*x^4)/(9*x^2-36*x+36)/log(2*x)^3,x, algo
rithm="giac")

[Out]

2/9*x^5/(x*log(2*x)^2 - 2*log(2*x)^2) + 2*x

Mupad [B] (verification not implemented)

Time = 9.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {8 x^4-4 x^5+\left (-20 x^4+8 x^5\right ) \log (2 x)+\left (72-72 x+18 x^2\right ) \log ^3(2 x)}{\left (36-36 x+9 x^2\right ) \log ^3(2 x)} \, dx=\frac {2\,x^5}{9\,{\ln \left (2\,x\right )}^2\,\left (x-2\right )}+\frac {2\,x\,\left (9\,x-18\right )}{9\,\left (x-2\right )} \]

[In]

int(-(log(2*x)*(20*x^4 - 8*x^5) - log(2*x)^3*(18*x^2 - 72*x + 72) - 8*x^4 + 4*x^5)/(log(2*x)^3*(9*x^2 - 36*x +
36)),x)

[Out]

(2*x^5)/(9*log(2*x)^2*(x - 2)) + (2*x*(9*x - 18))/(9*(x - 2))