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

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

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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 = {} $$\begin{array}{c}\int \frac{8x^4-4x^5+(-20x^4+8x^5)\log (2x)+(72-72x+18x^2){\log }^3(2x)}{(36-36x+9x^2){\log }^3(2x)}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{8x^4-4x^5+(-20x^4+8x^5)\log (2x)+(72-72x+18x^2){\log }^3(2x)}{9(-2+x{)}^2{\log }^3(2x)}dx\\ & =\frac{1}{9}\int \frac{8x^4-4x^5+(-20x^4+8x^5)\log (2x)+(72-72x+18x^2){\log }^3(2x)}{(-2+x{)}^2{\log }^3(2x)}dx\\ & =\frac{1}{9}\int (18-\frac{4x^4}{(-2+x){\log }^3(2x)}+\frac{4x^4(-5+2x)}{(-2+x{)}^2{\log }^2(2x)})dx\\ & =2x-\frac{4}{9}\int \frac{x^4}{(-2+x){\log }^3(2x)}dx+\frac{4}{9}\int \frac{x^4(-5+2x)}{(-2+x{)}^2{\log }^2(2x)}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{2(x^5+9(x^2-2x)\log {\left(2x\right)}^2)}{9(x-2)\log {\left(2x\right)}^2}\end{array}$$

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 $$\begin{array}{c}\frac{2x^5}{9(x\log {\left(2x\right)}^2-2\log {\left(2x\right)}^2)}+2x\end{array}$$

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

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 $$\begin{array}{c}\frac{2(x^5+9x^2\log (2{)}^2-18x\log (2{)}^2+9(x^2-2x)\log (x{)}^2+18(x^2\log (2)-2x\log (2))\log (x))}{9(x\log (2{)}^2+(x-2)\log (x{)}^2-2\log (2{)}^2+2(x\log (2)-2\log (2))\log (x))}\end{array}$$

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 $$\begin{array}{c}\frac{2x^5}{9{\ln \left(2x\right)}^2(x-2)}+\frac{2x(9x-18)}{9(x-2)}\end{array}$$

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 $$\begin{array}{c}\frac{2x^5}{(9x-18)\log {\left(2x\right)}^2}+2x\end{array}$$

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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