∫ 9−24x−30x2+(5+9x) log(2x)__ −x−10x2−6x3+(5x+3x2) log(2x) dx

Optimal. Leaf size=23 \[ \log \left (x (1+(5+3 x) (2 x-\log (2 x)))^2\right ) \]

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Rubi [F] time = 0.84, 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 {9-24 x-30 x^2+(5+9 x) \log (2 x)}{-x-10 x^2-6 x^3+\left (5 x+3 x^2\right ) \log (2 x)} \, dx \end {gather*}

Int[(9 - 24*x - 30*x^2 + (5 + 9*x)*Log[2*x])/(-x - 10*x^2 - 6*x^3 + (5*x + 3*x^2)*Log[2*x]),x]

Log[x] + 2*Log[5 + 3*x] + 14*Defer[Int][(1 + 10*x + 6*x^2 - 5*Log[2*x] - 3*x*Log[2*x])^(-1), x] - 10*Defer[Int

][1/(x*(1 + 10*x + 6*x^2 - 5*Log[2*x] - 3*x*Log[2*x])), x] + 12*Defer[Int][x/(1 + 10*x + 6*x^2 - 5*Log[2*x] -

3*x*Log[2*x]), x] - 6*Defer[Int][1/((5 + 3*x)*(1 + 10*x + 6*x^2 - 5*Log[2*x] - 3*x*Log[2*x])), x]

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5+9 x}{x (5+3 x)}+\frac {2 \left (-25+17 x+51 x^2+18 x^3\right )}{x (5+3 x) \left (1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)\right )}\right ) \, dx\\ &=2 \int \frac {-25+17 x+51 x^2+18 x^3}{x (5+3 x) \left (1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)\right )} \, dx+\int \frac {5+9 x}{x (5+3 x)} \, dx\\ &=2 \int \left (\frac {7}{1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)}-\frac {5}{x \left (1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)\right )}+\frac {6 x}{1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)}-\frac {3}{(5+3 x) \left (1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)\right )}\right ) \, dx+\int \left (\frac {1}{x}+\frac {6}{5+3 x}\right ) \, dx\\ &=\log (x)+2 \log (5+3 x)-6 \int \frac {1}{(5+3 x) \left (1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)\right )} \, dx-10 \int \frac {1}{x \left (1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)\right )} \, dx+12 \int \frac {x}{1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)} \, dx+14 \int \frac {1}{1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.36, size = 29, normalized size = 1.26 \begin {gather*} \log (x)+2 \log \left (1+10 x+6 x^2-5 \log (2 x)-3 x \log (2 x)\right ) \end {gather*}

Integrate[(9 - 24*x - 30*x^2 + (5 + 9*x)*Log[2*x])/(-x - 10*x^2 - 6*x^3 + (5*x + 3*x^2)*Log[2*x]),x]

Log[x] + 2*Log[1 + 10*x + 6*x^2 - 5*Log[2*x] - 3*x*Log[2*x]]

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fricas [A] time = 0.76, size = 44, normalized size = 1.91 \begin {gather*} 2 \, \log \left (3 \, x + 5\right ) + \log \relax (x) + 2 \, \log \left (-\frac {6 \, x^{2} - {\left (3 \, x + 5\right )} \log \left (2 \, x\right ) + 10 \, x + 1}{3 \, x + 5}\right ) \end {gather*}

integrate(((9*x+5)*log(2*x)-30*x^2-24*x+9)/((3*x^2+5*x)*log(2*x)-6*x^3-10*x^2-x),x, algorithm="fricas")

2*log(3*x + 5) + log(x) + 2*log(-(6*x^2 - (3*x + 5)*log(2*x) + 10*x + 1)/(3*x + 5))

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giac [A] time = 0.31, size = 29, normalized size = 1.26 \begin {gather*} 2 \, \log \left (-6 \, x^{2} + 3 \, x \log \left (2 \, x\right ) - 10 \, x + 5 \, \log \left (2 \, x\right ) - 1\right ) + \log \relax (x) \end {gather*}

integrate(((9*x+5)*log(2*x)-30*x^2-24*x+9)/((3*x^2+5*x)*log(2*x)-6*x^3-10*x^2-x),x, algorithm="giac")

2*log(-6*x^2 + 3*x*log(2*x) - 10*x + 5*log(2*x) - 1) + log(x)

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method	result	size

norman	\(\ln \left (2 x \right )+2 \ln \left (6 x^{2}-3 x \ln \left (2 x \right )+10 x -5 \ln \left (2 x \right )+1\right )\)	\(32\)
risch	\(2 \ln \left (3 x +5\right )+\ln \relax (x )+2 \ln \left (\ln \left (2 x \right )-\frac {6 x^{2}+10 x +1}{3 x +5}\right )\)	\(39\)

int(((9*x+5)*ln(2*x)-30*x^2-24*x+9)/((3*x^2+5*x)*ln(2*x)-6*x^3-10*x^2-x),x,method=_RETURNVERBOSE)

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maxima [B] time = 0.69, size = 52, normalized size = 2.26 \begin {gather*} 2 \, \log \left (3 \, x + 5\right ) + \log \relax (x) + 2 \, \log \left (-\frac {6 \, x^{2} - x {\left (3 \, \log \relax (2) - 10\right )} - {\left (3 \, x + 5\right )} \log \relax (x) - 5 \, \log \relax (2) + 1}{3 \, x + 5}\right ) \end {gather*}

integrate(((9*x+5)*log(2*x)-30*x^2-24*x+9)/((3*x^2+5*x)*log(2*x)-6*x^3-10*x^2-x),x, algorithm="maxima")

2*log(3*x + 5) + log(x) + 2*log(-(6*x^2 - x*(3*log(2) - 10) - (3*x + 5)*log(x) - 5*log(2) + 1)/(3*x + 5))

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mupad [B] time = 0.95, size = 29, normalized size = 1.26 \begin {gather*} 2\,\ln \left (10\,x-5\,\ln \left (2\,x\right )-3\,x\,\ln \left (2\,x\right )+6\,x^2+1\right )+\ln \relax (x) \end {gather*}

int((24*x + 30*x^2 - log(2*x)*(9*x + 5) - 9)/(x - log(2*x)*(5*x + 3*x^2) + 10*x^2 + 6*x^3),x)

2*log(10*x - 5*log(2*x) - 3*x*log(2*x) + 6*x^2 + 1) + log(x)

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sympy [A] time = 0.39, size = 36, normalized size = 1.57 \begin {gather*} \log {\relax (x )} + 2 \log {\left (x + \frac {5}{3} \right )} + 2 \log {\left (\log {\left (2 x \right )} + \frac {- 6 x^{2} - 10 x - 1}{3 x + 5} \right )} \end {gather*}

integrate(((9*x+5)*ln(2*x)-30*x**2-24*x+9)/((3*x**2+5*x)*ln(2*x)-6*x**3-10*x**2-x),x)

log(x) + 2*log(x + 5/3) + 2*log(log(2*x) + (-6*x**2 - 10*x - 1)/(3*x + 5))

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3.8.96 \(\int \frac {9-24 x-30 x^2+(5+9 x) \log (2 x)}{-x-10 x^2-6 x^3+(5 x+3 x^2) \log (2 x)} \, dx\)