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

Optimal. Leaf size=29 \[ \frac {1}{2} \log \left (x^2 \left (4-\frac {2 (3+x)^2 (3+\log (2 x))}{x}\right )^4\right ) \]

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Rubi [F]  time = 1.21, 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+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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 39, normalized size = 1.34 \begin {gather*} -\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 ) \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 1.10, size = 48, normalized size = 1.66 \begin {gather*} 4 \, \log \left (x + 3\right ) - \log \relax (x) + 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.13, size = 39, normalized size = 1.34 \begin {gather*} 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 \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.05, size = 41, normalized size = 1.41

method	result	size
risch	\(4 \ln \left (3+x \right )-\ln \relax (x )+2 \ln \left (\ln \left (2 x \right )+\frac {3 x^{2}+16 x +27}{x^{2}+6 x +9}\right )\)	\(41\)
norman	\(-\ln \left (2 x \right )+2 \ln \left (x^{2} \ln \left (2 x \right )+6 x \ln \left (2 x \right )+3 x^{2}+9 \ln \left (2 x \right )+16 x +27\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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maxima [B]  time = 0.49, size = 59, normalized size = 2.03 \begin {gather*} 4 \, \log \left (x + 3\right ) - \log \relax (x) + 2 \, \log \left (\frac {x^{2} {\left (\log \relax (2) + 3\right )} + 2 \, x {\left (3 \, \log \relax (2) + 8\right )} + {\left (x^{2} + 6 \, x + 9\right )} \log \relax (x) + 9 \, \log \relax (2) + 27}{x^{2} + 6 \, x + 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 3.81, size = 39, normalized size = 1.34 \begin {gather*} 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 \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.42, size = 36, normalized size = 1.24 \begin {gather*} - \log {\relax (x )} + 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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