∫ 56x2−8x3+(−160x−8x2−16x3−32x log(2)) log(x)+(100+20x+30x2+(40+4x+6x2) log(2)+4 log2(2)) log2(x)_____________________________________________________________________________ 16x2+(−40x−8x log(2)) log(x)+(25+10 log(2)+log2(2)) log2(x) dx

Optimal. Leaf size=27 \[ 4 \left (x+\frac {x^2 (1+x)}{2 \left (5+\log (2)-\frac {4 x}{\log (x)}\right )}\right ) \]

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Rubi [F] time = 1.43, 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 {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx \end {gather*}

Int[(56*x^2 - 8*x^3 + (-160*x - 8*x^2 - 16*x^3 - 32*x*Log[2])*Log[x] + (100 + 20*x + 30*x^2 + (40 + 4*x + 6*x^

2)*Log[2] + 4*Log[2]^2)*Log[x]^2)/(16*x^2 + (-40*x - 8*x*Log[2])*Log[x] + (25 + 10*Log[2] + Log[2]^2)*Log[x]^2

4*x + (x^2*(10 + Log[4]))/(5 + Log[2])^2 + (2*x^3*(15 + Log[8]))/(3*(5 + Log[2])^2) - (8*(25 - 15*Log[2]^2 + L

og[16]^2 + Log[1024])*Defer[Int][x^2/(4*x - 5*(1 + Log[2]/5)*Log[x])^2, x])/(5 + Log[2])^2 - (8*(5 + Log[2]^2

+ Log[64])*Defer[Int][x^3/(4*x - 5*(1 + Log[2]/5)*Log[x])^2, x])/(5 + Log[2])^2 + (32*Defer[Int][x^4/(4*x - 5*

(1 + Log[2]/5)*Log[x])^2, x])/(5 + Log[2]) - (8*(15 + Log[8])*Defer[Int][x^2/(4*x - 5*(1 + Log[2]/5)*Log[x]),

x])/(5 + Log[2])^2 - (8*(20 + Log[16])*Defer[Int][x^3/(4*x - 5*(1 + Log[2]/5)*Log[x]), x])/(5 + Log[2])^2

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8 (-7+x) x^2-8 x \left (20+x+2 x^2+\log (16)\right ) \log (x)+2 \left (2 (5+\log (2))^2+x (10+\log (4))+x^2 (15+\log (8))\right ) \log ^2(x)}{(4 x-(5+\log (2)) \log (x))^2} \, dx\\ &=\int \left (\frac {2 \left (2 (5+\log (2))^2+x (10+\log (4))+x^2 (15+\log (8))\right )}{(5+\log (2))^2}+\frac {8 x^2 \left (-25+15 \log ^2(2)+4 x^2 (5+\log (2))-\log ^2(16)-x \left (5+\log ^2(2)+\log (64)\right )-\log (1024)\right )}{(5+\log (2))^2 \left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}+\frac {8 x^2 (-15-\log (8)-x (20+\log (16)))}{(5+\log (2))^2 \left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )}\right ) \, dx\\ &=\frac {2 \int \left (2 (5+\log (2))^2+x (10+\log (4))+x^2 (15+\log (8))\right ) \, dx}{(5+\log (2))^2}+\frac {8 \int \frac {x^2 \left (-25+15 \log ^2(2)+4 x^2 (5+\log (2))-\log ^2(16)-x \left (5+\log ^2(2)+\log (64)\right )-\log (1024)\right )}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2} \, dx}{(5+\log (2))^2}+\frac {8 \int \frac {x^2 (-15-\log (8)-x (20+\log (16)))}{4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)} \, dx}{(5+\log (2))^2}\\ &=4 x+\frac {x^2 (10+\log (4))}{(5+\log (2))^2}+\frac {2 x^3 (15+\log (8))}{3 (5+\log (2))^2}+\frac {8 \int \left (\frac {4 x^4 (5+\log (2))}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}+\frac {x^3 \left (-5-\log ^2(2)-\log (64)\right )}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}+\frac {x^2 \left (-25+15 \log ^2(2)-\log ^2(16)-\log (1024)\right )}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}\right ) \, dx}{(5+\log (2))^2}+\frac {8 \int \left (\frac {x^2 (-15-\log (8))}{4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)}+\frac {x^3 (-20-\log (16))}{4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)}\right ) \, dx}{(5+\log (2))^2}\\ &=4 x+\frac {x^2 (10+\log (4))}{(5+\log (2))^2}+\frac {2 x^3 (15+\log (8))}{3 (5+\log (2))^2}+\frac {32 \int \frac {x^4}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2} \, dx}{5+\log (2)}-\frac {(8 (15+\log (8))) \int \frac {x^2}{4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)} \, dx}{(5+\log (2))^2}-\frac {(8 (20+\log (16))) \int \frac {x^3}{4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)} \, dx}{(5+\log (2))^2}-\frac {\left (8 \left (5+\log ^2(2)+\log (64)\right )\right ) \int \frac {x^3}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2} \, dx}{(5+\log (2))^2}-\frac {\left (8 \left (25-15 \log ^2(2)+\log ^2(16)+\log (1024)\right )\right ) \int \frac {x^2}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2} \, dx}{(5+\log (2))^2}\\ \end {aligned} \end {gather*}

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Mathematica [B] time = 0.78, size = 104, normalized size = 3.85 \begin {gather*} 4 x+\frac {x^2 (10+\log (4))}{(5+\log (2))^2}+\frac {2 x^3 (15+\log (8))}{3 (5+\log (2))^2}+\frac {8 x^3 \left (25-15 \log ^2(2)-4 x^2 (5+\log (2))+\log ^2(16)+x \left (5+\log ^2(2)+\log (64)\right )+\log (1024)\right )}{(-5+4 x-\log (2)) (5+\log (2))^2 (4 x-(5+\log (2)) \log (x))} \end {gather*}

Integrate[(56*x^2 - 8*x^3 + (-160*x - 8*x^2 - 16*x^3 - 32*x*Log[2])*Log[x] + (100 + 20*x + 30*x^2 + (40 + 4*x

+ 6*x^2)*Log[2] + 4*Log[2]^2)*Log[x]^2)/(16*x^2 + (-40*x - 8*x*Log[2])*Log[x] + (25 + 10*Log[2] + Log[2]^2)*Lo

4*x + (x^2*(10 + Log[4]))/(5 + Log[2])^2 + (2*x^3*(15 + Log[8]))/(3*(5 + Log[2])^2) + (8*x^3*(25 - 15*Log[2]^2

- 4*x^2*(5 + Log[2]) + Log[16]^2 + x*(5 + Log[2]^2 + Log[64]) + Log[1024]))/((-5 + 4*x - Log[2])*(5 + Log[2])

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fricas [A] time = 1.06, size = 40, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (8 \, x^{2} - {\left (x^{3} + x^{2} + 2 \, x \log \relax (2) + 10 \, x\right )} \log \relax (x)\right )}}{{\left (\log \relax (2) + 5\right )} \log \relax (x) - 4 \, x} \end {gather*}

integrate(((4*log(2)^2+(6*x^2+4*x+40)*log(2)+30*x^2+20*x+100)*log(x)^2+(-32*x*log(2)-16*x^3-8*x^2-160*x)*log(x

)-8*x^3+56*x^2)/((log(2)^2+10*log(2)+25)*log(x)^2+(-8*x*log(2)-40*x)*log(x)+16*x^2),x, algorithm="fricas")

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giac [B] time = 0.28, size = 63, normalized size = 2.33 \begin {gather*} \frac {2 \, x^{3}}{\log \relax (2) + 5} + 4 \, x + \frac {2 \, x^{2}}{\log \relax (2) + 5} + \frac {8 \, {\left (x^{4} + x^{3}\right )}}{\log \relax (2)^{2} \log \relax (x) - 4 \, x \log \relax (2) + 10 \, \log \relax (2) \log \relax (x) - 20 \, x + 25 \, \log \relax (x)} \end {gather*}

integrate(((4*log(2)^2+(6*x^2+4*x+40)*log(2)+30*x^2+20*x+100)*log(x)^2+(-32*x*log(2)-16*x^3-8*x^2-160*x)*log(x

)-8*x^3+56*x^2)/((log(2)^2+10*log(2)+25)*log(x)^2+(-8*x*log(2)-40*x)*log(x)+16*x^2),x, algorithm="giac")

2*x^3/(log(2) + 5) + 4*x + 2*x^2/(log(2) + 5) + 8*(x^4 + x^3)/(log(2)^2*log(x) - 4*x*log(2) + 10*log(2)*log(x)

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method	result	size

norman	\(\frac {\left (4 \ln \relax (2)+20\right ) x \ln \relax (x )-16 x^{2}+2 x^{2} \ln \relax (x )+2 x^{3} \ln \relax (x )}{\ln \relax (2) \ln \relax (x )+5 \ln \relax (x )-4 x}\)	\(47\)
risch	\(\frac {2 x \left (x^{2}+2 \ln \relax (2)+x +10\right )}{\ln \relax (2)+5}+\frac {8 \left (x +1\right ) x^{3}}{\left (\ln \relax (2)+5\right ) \left (\ln \relax (2) \ln \relax (x )+5 \ln \relax (x )-4 x \right )}\)	\(50\)

int(((4*ln(2)^2+(6*x^2+4*x+40)*ln(2)+30*x^2+20*x+100)*ln(x)^2+(-32*x*ln(2)-16*x^3-8*x^2-160*x)*ln(x)-8*x^3+56*

x^2)/((ln(2)^2+10*ln(2)+25)*ln(x)^2+(-8*x*ln(2)-40*x)*ln(x)+16*x^2),x,method=_RETURNVERBOSE)

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maxima [A] time = 0.53, size = 39, normalized size = 1.44 \begin {gather*} -\frac {2 \, {\left (8 \, x^{2} - {\left (x^{3} + x^{2} + 2 \, x {\left (\log \relax (2) + 5\right )}\right )} \log \relax (x)\right )}}{{\left (\log \relax (2) + 5\right )} \log \relax (x) - 4 \, x} \end {gather*}

integrate(((4*log(2)^2+(6*x^2+4*x+40)*log(2)+30*x^2+20*x+100)*log(x)^2+(-32*x*log(2)-16*x^3-8*x^2-160*x)*log(x

)-8*x^3+56*x^2)/((log(2)^2+10*log(2)+25)*log(x)^2+(-8*x*log(2)-40*x)*log(x)+16*x^2),x, algorithm="maxima")

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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {\ln \relax (x)\,\left (160\,x+32\,x\,\ln \relax (2)+8\,x^2+16\,x^3\right )-{\ln \relax (x)}^2\,\left (20\,x+\ln \relax (2)\,\left (6\,x^2+4\,x+40\right )+4\,{\ln \relax (2)}^2+30\,x^2+100\right )-56\,x^2+8\,x^3}{{\ln \relax (x)}^2\,\left (10\,\ln \relax (2)+{\ln \relax (2)}^2+25\right )-\ln \relax (x)\,\left (40\,x+8\,x\,\ln \relax (2)\right )+16\,x^2} \,d x \end {gather*}

int(-(log(x)*(160*x + 32*x*log(2) + 8*x^2 + 16*x^3) - log(x)^2*(20*x + log(2)*(4*x + 6*x^2 + 40) + 4*log(2)^2

+ 30*x^2 + 100) - 56*x^2 + 8*x^3)/(log(x)^2*(10*log(2) + log(2)^2 + 25) - log(x)*(40*x + 8*x*log(2)) + 16*x^2)

-int((log(x)*(160*x + 32*x*log(2) + 8*x^2 + 16*x^3) - log(x)^2*(20*x + log(2)*(4*x + 6*x^2 + 40) + 4*log(2)^2

+ 30*x^2 + 100) - 56*x^2 + 8*x^3)/(log(x)^2*(10*log(2) + log(2)^2 + 25) - log(x)*(40*x + 8*x*log(2)) + 16*x^2)

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sympy [B] time = 0.18, size = 58, normalized size = 2.15 \begin {gather*} \frac {2 x^{3}}{\log {\relax (2 )} + 5} + \frac {2 x^{2}}{\log {\relax (2 )} + 5} + 4 x + \frac {8 x^{4} + 8 x^{3}}{- 20 x - 4 x \log {\relax (2 )} + \left (\log {\relax (2 )}^{2} + 10 \log {\relax (2 )} + 25\right ) \log {\relax (x )}} \end {gather*}

integrate(((4*ln(2)**2+(6*x**2+4*x+40)*ln(2)+30*x**2+20*x+100)*ln(x)**2+(-32*x*ln(2)-16*x**3-8*x**2-160*x)*ln(

x)-8*x**3+56*x**2)/((ln(2)**2+10*ln(2)+25)*ln(x)**2+(-8*x*ln(2)-40*x)*ln(x)+16*x**2),x)

2*x**3/(log(2) + 5) + 2*x**2/(log(2) + 5) + 4*x + (8*x**4 + 8*x**3)/(-20*x - 4*x*log(2) + (log(2)**2 + 10*log(

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3.23.34 \(\int \frac {56 x^2-8 x^3+(-160 x-8 x^2-16 x^3-32 x \log (2)) \log (x)+(100+20 x+30 x^2+(40+4 x+6 x^2) \log (2)+4 \log ^2(2)) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+(25+10 \log (2)+\log ^2(2)) \log ^2(x)} \, dx\)