∫ −936−1848x+2256x2+640x3+(−480−984x−2408x2−640x3) log(9+3x x )+(480x+640x2+160x3) log2(9+3x x ) __________________________________________________________________________________________________________________________________________

Optimal. Leaf size=26 \[ \left (1-4 \left (\frac {11}{5}-2 x+(1+x) \log \left (\frac {3 (3+x)}{x}\right )\right )\right )^2 \]

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Rubi [F] time = 0.74, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{15 x+5 x^2} \, dx \end {gather*}

Int[(-936 - 1848*x + 2256*x^2 + 640*x^3 + (-480 - 984*x - 2408*x^2 - 640*x^3)*Log[(9 + 3*x)/x] + (480*x + 640*

(-624*x)/5 + 64*x^2 - (312*Log[x])/5 + (488*x*Log[x])/5 + 64*x^2*Log[x] - 32*Log[9]*Log[x] - 128*Log[1 + x/3]*

Log[x] + 16*Log[x]^2 + (336*Log[3 + x])/5 - (488*(3 + x)*Log[3*(3 + x)])/5 + 64*Log[3*(3 + x)]^2 - (488*x*(Log

[x] + Log[(3*(3 + x))/x] - Log[9 + 3*x]))/5 - 64*x^2*(Log[x] + Log[(3*(3 + x))/x] - Log[9 + 3*x]) - 32*Log[x]*

(Log[x] + Log[(3*(3 + x))/x] - Log[9 + 3*x]) + 128*Log[3 + x]*(Log[x] + Log[(3*(3 + x))/x] - Log[9 + 3*x]) - 6

4*x^2*Log[9 + 3*x] - 96*PolyLog[2, -1/3*x] + 32*Defer[Int][(1 + x)*Log[3 + 9/x]^2, x]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-936-1848 x+2256 x^2+640 x^3+\left (-480-984 x-2408 x^2-640 x^3\right ) \log \left (\frac {9+3 x}{x}\right )+\left (480 x+640 x^2+160 x^3\right ) \log ^2\left (\frac {9+3 x}{x}\right )}{x (15+5 x)} \, dx\\ &=\int \left (\frac {8 \left (-117-231 x+282 x^2+80 x^3\right )}{5 x (3+x)}-\frac {8 \left (60+123 x+301 x^2+80 x^3\right ) \log \left (3+\frac {9}{x}\right )}{5 x (3+x)}+32 (1+x) \log ^2\left (3+\frac {9}{x}\right )\right ) \, dx\\ &=\frac {8}{5} \int \frac {-117-231 x+282 x^2+80 x^3}{x (3+x)} \, dx-\frac {8}{5} \int \frac {\left (60+123 x+301 x^2+80 x^3\right ) \log \left (3+\frac {9}{x}\right )}{x (3+x)} \, dx+32 \int (1+x) \log ^2\left (3+\frac {9}{x}\right ) \, dx\\ &=\frac {8}{5} \int \left (42-\frac {39}{x}+80 x-\frac {318}{3+x}\right ) \, dx-\frac {8}{5} \int \frac {\left (60+123 x+301 x^2+80 x^3\right ) \log \left (\frac {9+3 x}{x}\right )}{x (3+x)} \, dx+32 \int (1+x) \log ^2\left (3+\frac {9}{x}\right ) \, dx\\ &=\frac {336 x}{5}+64 x^2-\frac {312 \log (x)}{5}-\frac {2544}{5} \log (3+x)+\frac {8}{5} \int \frac {\left (60+123 x+301 x^2+80 x^3\right ) \log (x)}{x (3+x)} \, dx-\frac {8}{5} \int \frac {\left (60+123 x+301 x^2+80 x^3\right ) \log (9+3 x)}{x (3+x)} \, dx+32 \int (1+x) \log ^2\left (3+\frac {9}{x}\right ) \, dx+\frac {1}{5} \left (8 \left (-\log (x)+\log (9+3 x)-\log \left (\frac {9+3 x}{x}\right )\right )\right ) \int \frac {60+123 x+301 x^2+80 x^3}{x (3+x)} \, dx\\ &=\frac {336 x}{5}+64 x^2-\frac {312 \log (x)}{5}-\frac {2544}{5} \log (3+x)+\frac {8}{5} \int \left (61 \log (x)+\frac {20 \log (x)}{x}+80 x \log (x)-\frac {80 \log (x)}{3+x}\right ) \, dx-\frac {8}{5} \int \left (61 \log (9+3 x)+\frac {20 \log (9+3 x)}{x}+80 x \log (9+3 x)-\frac {80 \log (9+3 x)}{3+x}\right ) \, dx+32 \int (1+x) \log ^2\left (3+\frac {9}{x}\right ) \, dx+\frac {1}{5} \left (8 \left (-\log (x)+\log (9+3 x)-\log \left (\frac {9+3 x}{x}\right )\right )\right ) \int \left (61+\frac {20}{x}+80 x-\frac {80}{3+x}\right ) \, dx\\ &=\frac {336 x}{5}+64 x^2-\frac {312 \log (x)}{5}-\frac {2544}{5} \log (3+x)-\frac {488}{5} x \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-64 x^2 \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-32 \log (x) \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )+128 \log (3+x) \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )+32 \int (1+x) \log ^2\left (3+\frac {9}{x}\right ) \, dx+32 \int \frac {\log (x)}{x} \, dx-32 \int \frac {\log (9+3 x)}{x} \, dx+\frac {488}{5} \int \log (x) \, dx-\frac {488}{5} \int \log (9+3 x) \, dx+128 \int x \log (x) \, dx-128 \int \frac {\log (x)}{3+x} \, dx-128 \int x \log (9+3 x) \, dx+128 \int \frac {\log (9+3 x)}{3+x} \, dx\\ &=-\frac {152 x}{5}+32 x^2-\frac {312 \log (x)}{5}+\frac {488}{5} x \log (x)+64 x^2 \log (x)-32 \log (9) \log (x)-128 \log \left (1+\frac {x}{3}\right ) \log (x)+16 \log ^2(x)-\frac {2544}{5} \log (3+x)-\frac {488}{5} x \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-64 x^2 \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-32 \log (x) \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )+128 \log (3+x) \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-64 x^2 \log (9+3 x)+32 \int (1+x) \log ^2\left (3+\frac {9}{x}\right ) \, dx-32 \int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx-\frac {488}{15} \operatorname {Subst}(\int \log (x) \, dx,x,9+3 x)+\frac {128}{3} \operatorname {Subst}\left (\int \frac {3 \log (x)}{x} \, dx,x,9+3 x\right )+128 \int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx+192 \int \frac {x^2}{9+3 x} \, dx\\ &=\frac {336 x}{5}+32 x^2-\frac {312 \log (x)}{5}+\frac {488}{5} x \log (x)+64 x^2 \log (x)-32 \log (9) \log (x)-128 \log \left (1+\frac {x}{3}\right ) \log (x)+16 \log ^2(x)-\frac {2544}{5} \log (3+x)-\frac {488}{5} (3+x) \log (3 (3+x))-\frac {488}{5} x \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-64 x^2 \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-32 \log (x) \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )+128 \log (3+x) \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-64 x^2 \log (9+3 x)-96 \text {Li}_2\left (-\frac {x}{3}\right )+32 \int (1+x) \log ^2\left (3+\frac {9}{x}\right ) \, dx+128 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,9+3 x\right )+192 \int \left (-1+\frac {x}{3}+\frac {3}{3+x}\right ) \, dx\\ &=-\frac {624 x}{5}+64 x^2-\frac {312 \log (x)}{5}+\frac {488}{5} x \log (x)+64 x^2 \log (x)-32 \log (9) \log (x)-128 \log \left (1+\frac {x}{3}\right ) \log (x)+16 \log ^2(x)+\frac {336}{5} \log (3+x)-\frac {488}{5} (3+x) \log (3 (3+x))+64 \log ^2(3 (3+x))-\frac {488}{5} x \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-64 x^2 \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-32 \log (x) \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )+128 \log (3+x) \left (\log (x)+\log \left (\frac {3 (3+x)}{x}\right )-\log (9+3 x)\right )-64 x^2 \log (9+3 x)-96 \text {Li}_2\left (-\frac {x}{3}\right )+32 \int (1+x) \log ^2\left (3+\frac {9}{x}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [C] time = 0.14, size = 175, normalized size = 6.73 \begin {gather*} \frac {8}{5} \left (-78 x+40 x^2-40 x^2 \log \left (3+\frac {9}{x}\right )+10 x^2 \log ^2\left (3+\frac {9}{x}\right )-42 \log (x)-120 \log (3) \log (x)+42 \log (3+x)-100 \log \left (3+\frac {9}{x}\right ) \log (3+x)+50 \log ^2(3+x)-3 \log \left (\frac {3 (3+x)}{x}\right )-x \log \left (\frac {3 (3+x)}{x}\right )-120 \log \left (-\frac {3}{x}\right ) \log \left (\frac {3 (3+x)}{x}\right )+60 \log ^2\left (\frac {3 (3+x)}{x}\right )+20 x \log ^2\left (\frac {3 (3+x)}{x}\right )-20 \text {Li}_2\left (-\frac {3}{x}\right )+100 \text {Li}_2\left (-\frac {x}{3}\right )-120 \text {Li}_2\left (\frac {3+x}{x}\right )\right ) \end {gather*}

Integrate[(-936 - 1848*x + 2256*x^2 + 640*x^3 + (-480 - 984*x - 2408*x^2 - 640*x^3)*Log[(9 + 3*x)/x] + (480*x

(8*(-78*x + 40*x^2 - 40*x^2*Log[3 + 9/x] + 10*x^2*Log[3 + 9/x]^2 - 42*Log[x] - 120*Log[3]*Log[x] + 42*Log[3 +

x] - 100*Log[3 + 9/x]*Log[3 + x] + 50*Log[3 + x]^2 - 3*Log[(3*(3 + x))/x] - x*Log[(3*(3 + x))/x] - 120*Log[-3/

x]*Log[(3*(3 + x))/x] + 60*Log[(3*(3 + x))/x]^2 + 20*x*Log[(3*(3 + x))/x]^2 - 20*PolyLog[2, -3/x] + 100*PolyLo

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fricas [B] time = 1.41, size = 49, normalized size = 1.88 \begin {gather*} 16 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (\frac {3 \, {\left (x + 3\right )}}{x}\right )^{2} + 64 \, x^{2} - \frac {8}{5} \, {\left (40 \, x^{2} + x - 39\right )} \log \left (\frac {3 \, {\left (x + 3\right )}}{x}\right ) - \frac {624}{5} \, x \end {gather*}

integrate(((160*x^3+640*x^2+480*x)*log((3*x+9)/x)^2+(-640*x^3-2408*x^2-984*x-480)*log((3*x+9)/x)+640*x^3+2256*

16*(x^2 + 2*x + 1)*log(3*(x + 3)/x)^2 + 64*x^2 - 8/5*(40*x^2 + x - 39)*log(3*(x + 3)/x) - 624/5*x

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {8 \, {\left (80 \, x^{3} + 20 \, {\left (x^{3} + 4 \, x^{2} + 3 \, x\right )} \log \left (\frac {3 \, {\left (x + 3\right )}}{x}\right )^{2} + 282 \, x^{2} - {\left (80 \, x^{3} + 301 \, x^{2} + 123 \, x + 60\right )} \log \left (\frac {3 \, {\left (x + 3\right )}}{x}\right ) - 231 \, x - 117\right )}}{5 \, {\left (x^{2} + 3 \, x\right )}}\,{d x} \end {gather*}

integrate(((160*x^3+640*x^2+480*x)*log((3*x+9)/x)^2+(-640*x^3-2408*x^2-984*x-480)*log((3*x+9)/x)+640*x^3+2256*

integrate(8/5*(80*x^3 + 20*(x^3 + 4*x^2 + 3*x)*log(3*(x + 3)/x)^2 + 282*x^2 - (80*x^3 + 301*x^2 + 123*x + 60)*

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method	result	size

norman	\(\frac {312 \ln \left (\frac {3 x +9}{x}\right )}{5}-\frac {624 x}{5}+64 x^{2}+16 \ln \left (\frac {3 x +9}{x}\right )^{2}-\frac {8 x \ln \left (\frac {3 x +9}{x}\right )}{5}+32 x \ln \left (\frac {3 x +9}{x}\right )^{2}-64 x^{2} \ln \left (\frac {3 x +9}{x}\right )+16 x^{2} \ln \left (\frac {3 x +9}{x}\right )^{2}\)	\(96\)

int(((160*x^3+640*x^2+480*x)*ln((3*x+9)/x)^2+(-640*x^3-2408*x^2-984*x-480)*ln((3*x+9)/x)+640*x^3+2256*x^2-1848

312/5*ln((3*x+9)/x)-624/5*x+64*x^2+16*ln((3*x+9)/x)^2-8/5*x*ln((3*x+9)/x)+32*x*ln((3*x+9)/x)^2-64*x^2*ln((3*x+

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maxima [B] time = 0.58, size = 168, normalized size = 6.46 \begin {gather*} 16 \, {\left (\log \relax (3)^{2} - 4 \, \log \relax (3) + 4\right )} x^{2} + 16 \, {\left (x^{2} + 2 \, x\right )} \log \left (x + 3\right )^{2} + 16 \, {\left (x^{2} + 2 \, x\right )} \log \relax (x)^{2} + \frac {8}{5} \, {\left (20 \, \log \relax (3)^{2} - \log \relax (3) - 78\right )} x + \frac {8}{5} \, {\left (20 \, x^{2} {\left (\log \relax (3) - 2\right )} + x {\left (40 \, \log \relax (3) - 1\right )} - 20 \, {\left (x^{2} + 2 \, x\right )} \log \relax (x) + 231\right )} \log \left (x + 3\right ) - 16 \, \log \left (x + 3\right )^{2} - \frac {8}{5} \, {\left (20 \, x^{2} {\left (\log \relax (3) - 2\right )} + x {\left (40 \, \log \relax (3) - 1\right )}\right )} \log \relax (x) + 32 \, \log \left (x + 3\right ) \log \relax (x) - 16 \, \log \relax (x)^{2} + 32 \, {\left (\log \left (x + 3\right ) - \log \relax (x)\right )} \log \left (\frac {9}{x} + 3\right ) - \frac {1536}{5} \, \log \left (x + 3\right ) - \frac {312}{5} \, \log \relax (x) \end {gather*}

integrate(((160*x^3+640*x^2+480*x)*log((3*x+9)/x)^2+(-640*x^3-2408*x^2-984*x-480)*log((3*x+9)/x)+640*x^3+2256*

16*(log(3)^2 - 4*log(3) + 4)*x^2 + 16*(x^2 + 2*x)*log(x + 3)^2 + 16*(x^2 + 2*x)*log(x)^2 + 8/5*(20*log(3)^2 -

log(3) - 78)*x + 8/5*(20*x^2*(log(3) - 2) + x*(40*log(3) - 1) - 20*(x^2 + 2*x)*log(x) + 231)*log(x + 3) - 16*l

og(x + 3)^2 - 8/5*(20*x^2*(log(3) - 2) + x*(40*log(3) - 1))*log(x) + 32*log(x + 3)*log(x) - 16*log(x)^2 + 32*(

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mupad [B] time = 1.85, size = 90, normalized size = 3.46 \begin {gather*} \frac {312\,\ln \left (\frac {3\,x+9}{x}\right )}{5}-x\,\left (-32\,{\ln \left (\frac {3\,x+9}{x}\right )}^2+\frac {8\,\ln \left (\frac {3\,x+9}{x}\right )}{5}+\frac {624}{5}\right )+16\,{\ln \left (\frac {3\,x+9}{x}\right )}^2+x^2\,\left (16\,{\ln \left (\frac {3\,x+9}{x}\right )}^2-64\,\ln \left (\frac {3\,x+9}{x}\right )+64\right ) \end {gather*}

int(-(1848*x - log((3*x + 9)/x)^2*(480*x + 640*x^2 + 160*x^3) - 2256*x^2 - 640*x^3 + log((3*x + 9)/x)*(984*x +

(312*log((3*x + 9)/x))/5 - x*((8*log((3*x + 9)/x))/5 - 32*log((3*x + 9)/x)^2 + 624/5) + 16*log((3*x + 9)/x)^2

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sympy [B] time = 0.24, size = 65, normalized size = 2.50 \begin {gather*} 64 x^{2} - \frac {624 x}{5} + \left (- 64 x^{2} - \frac {8 x}{5}\right ) \log {\left (\frac {3 x + 9}{x} \right )} + \left (16 x^{2} + 32 x + 16\right ) \log {\left (\frac {3 x + 9}{x} \right )}^{2} - \frac {312 \log {\relax (x )}}{5} + \frac {312 \log {\left (x + 3 \right )}}{5} \end {gather*}

integrate(((160*x**3+640*x**2+480*x)*ln((3*x+9)/x)**2+(-640*x**3-2408*x**2-984*x-480)*ln((3*x+9)/x)+640*x**3+2

64*x**2 - 624*x/5 + (-64*x**2 - 8*x/5)*log((3*x + 9)/x) + (16*x**2 + 32*x + 16)*log((3*x + 9)/x)**2 - 312*log(

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3.29.24 \(\int \frac {-936-1848 x+2256 x^2+640 x^3+(-480-984 x-2408 x^2-640 x^3) \log (\frac {9+3 x}{x})+(480 x+640 x^2+160 x^3) \log ^2(\frac {9+3 x}{x})}{15 x+5 x^2} \, dx\)