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

Optimal. Leaf size=25 \[ -4+2 x-\frac {2 x^5}{9 (2-x) \log ^2(2 x)} \]

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Rubi [F]  time = 0.28, 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 {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 \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [A]  time = 1.28, size = 22, normalized size = 0.88 \begin {gather*} 2 x+\frac {2 x^5}{9 (-2+x) \log ^2(2 x)} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.62, size = 32, normalized size = 1.28 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.24, size = 28, normalized size = 1.12 \begin {gather*} \frac {2 \, x^{5}}{9 \, {\left (x \log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right )^{2}\right )}} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.10, size = 21, normalized size = 0.84

method	result	size
risch	\(2 x +\frac {2 x^{5}}{9 \left (x -2\right ) \ln \left (2 x \right )^{2}}\)	\(21\)
norman	\(\frac {-8 \ln \left (2 x \right )^{2}+\frac {2 x^{5}}{9}+2 x^{2} \ln \left (2 x \right )^{2}}{\left (x -2\right ) \ln \left (2 x \right )^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[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/(x-2)/ln(2*x)^2

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maxima [B]  time = 0.51, size = 87, normalized size = 3.48 \begin {gather*} \frac {2 \, {\left (x^{5} + 9 \, x^{2} \log \relax (2)^{2} - 18 \, x \log \relax (2)^{2} + 9 \, {\left (x^{2} - 2 \, x\right )} \log \relax (x)^{2} + 18 \, {\left (x^{2} \log \relax (2) - 2 \, x \log \relax (2)\right )} \log \relax (x)\right )}}{9 \, {\left (x \log \relax (2)^{2} + {\left (x - 2\right )} \log \relax (x)^{2} - 2 \, \log \relax (2)^{2} + 2 \, {\left (x \log \relax (2) - 2 \, \log \relax (2)\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 3.13, size = 20, normalized size = 0.80 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.16, size = 19, normalized size = 0.76 \begin {gather*} \frac {2 x^{5}}{\left (9 x - 18\right ) \log {\left (2 x \right )}^{2}} + 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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