3.69.27 $\int \frac{36+49x+(-33x-52x^2)\log (9x)+(6x^2+13x^3){\log }^2(9x)}{4x^3-4x^4\log (9x)+x^5{\log }^2(9x)}dx$

Optimal. Leaf size=29 $$\frac{-13+\frac{-3+\frac{3}{x(-\frac{2}{x}+\log (9x))}}{x}}{x}$$

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Rubi [F]  time = 0.70, 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 = {} $$\begin{array}{c}\int \frac{36+49x+(-33x-52x^2)\log (9x)+(6x^2+13x^3){\log }^2(9x)}{4x^3-4x^4\log (9x)+x^5{\log }^2(9x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(36 + 49*x + (-33*x - 52*x^2)*Log[9*x] + (6*x^2 + 13*x^3)*Log[9*x]^2)/(4*x^3 - 4*x^4*Log[9*x] + x^5*Log[9*
x]^2),x]

[Out]

-1/12*(6 + 13*x)^2/x^2 - 6*Defer[Int][1/(x^3*(-2 + x*Log[9*x])^2), x] - 3*Defer[Int][1/(x^2*(-2 + x*Log[9*x])^
2), x] - 9*Defer[Int][1/(x^3*(-2 + x*Log[9*x])), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{36+49x-x(33+52x)\log (9x)+x^2(6+13x){\log }^2(9x)}{x^3(2-x\log (9x){)}^2}dx\\ & =\int (\frac{6+13x}{x^3}-\frac{3(2+x)}{x^3(-2+x\log (9x){)}^2}-\frac{9}{x^3(-2+x\log (9x))})dx\\ & =-(3\int \frac{2+x}{x^3(-2+x\log (9x){)}^2}dx)-9\int \frac{1}{x^3(-2+x\log (9x))}dx+\int \frac{6+13x}{x^3}dx\\ & =-\frac{(6+13x{)}^2}{12x^2}-3\int (\frac{2}{x^3(-2+x\log (9x){)}^2}+\frac{1}{x^2(-2+x\log (9x){)}^2})dx-9\int \frac{1}{x^3(-2+x\log (9x))}dx\\ & =-\frac{(6+13x{)}^2}{12x^2}-3\int \frac{1}{x^2(-2+x\log (9x){)}^2}dx-6\int \frac{1}{x^3(-2+x\log (9x){)}^2}dx-9\int \frac{1}{x^3(-2+x\log (9x))}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.17, size = 21, normalized size = 0.72 $$\begin{array}{c}\frac{-3-13x+\frac{3}{-2+x\log (9x)}}{x^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(36 + 49*x + (-33*x - 52*x^2)*Log[9*x] + (6*x^2 + 13*x^3)*Log[9*x]^2)/(4*x^3 - 4*x^4*Log[9*x] + x^5*
Log[9*x]^2),x]

[Out]

(-3 - 13*x + 3/(-2 + x*Log[9*x]))/x^2

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fricas [A]  time = 0.58, size = 37, normalized size = 1.28 $$\begin{array}{c}-\frac{(13x^2+3x)\log \left(9x\right)-26x-9}{x^3\log \left(9x\right)-2x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13*x^3+6*x^2)*log(9*x)^2+(-52*x^2-33*x)*log(9*x)+49*x+36)/(x^5*log(9*x)^2-4*x^4*log(9*x)+4*x^3),x,
algorithm="fricas")

[Out]

-((13*x^2 + 3*x)*log(9*x) - 26*x - 9)/(x^3*log(9*x) - 2*x^2)

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giac [A]  time = 0.18, size = 29, normalized size = 1.00 $$\begin{array}{c}\frac{3}{x^3\log \left(9x\right)-2x^2}-\frac{13x+3}{x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13*x^3+6*x^2)*log(9*x)^2+(-52*x^2-33*x)*log(9*x)+49*x+36)/(x^5*log(9*x)^2-4*x^4*log(9*x)+4*x^3),x,
algorithm="giac")

[Out]

3/(x^3*log(9*x) - 2*x^2) - (13*x + 3)/x^2

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maple [A]  time = 0.05, size = 27, normalized size = 0.93

method	result	size
risch	$-\frac{13x+3}{x^2}+\frac{3}{x^2(x\ln \left(9x\right)-2)}$	$27$
norman	$\frac{9-13x^2\ln \left(9x\right)+26x-3x\ln \left(9x\right)}{x^2(x\ln \left(9x\right)-2)}$	$36$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((13*x^3+6*x^2)*ln(9*x)^2+(-52*x^2-33*x)*ln(9*x)+49*x+36)/(x^5*ln(9*x)^2-4*x^4*ln(9*x)+4*x^3),x,method=_RE
TURNVERBOSE)

[Out]

-(13*x+3)/x^2+3/x^2/(x*ln(9*x)-2)

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maxima [A]  time = 0.50, size = 53, normalized size = 1.83 $$\begin{array}{c}-\frac{26x^2\log (3)+2x(3\log (3)-13)+(13x^2+3x)\log (x)-9}{2x^3\log (3)+x^3\log (x)-2x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13*x^3+6*x^2)*log(9*x)^2+(-52*x^2-33*x)*log(9*x)+49*x+36)/(x^5*log(9*x)^2-4*x^4*log(9*x)+4*x^3),x,
algorithm="maxima")

[Out]

-(26*x^2*log(3) + 2*x*(3*log(3) - 13) + (13*x^2 + 3*x)*log(x) - 9)/(2*x^3*log(3) + x^3*log(x) - 2*x^2)

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mupad [B]  time = 4.24, size = 35, normalized size = 1.21 $$\begin{array}{c}\frac{26x-3x\ln \left(9x\right)-13x^2\ln \left(9x\right)+9}{x^2(x\ln \left(9x\right)-2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((49*x - log(9*x)*(33*x + 52*x^2) + log(9*x)^2*(6*x^2 + 13*x^3) + 36)/(4*x^3 - 4*x^4*log(9*x) + x^5*log(9*x
)^2),x)

[Out]

(26*x - 3*x*log(9*x) - 13*x^2*log(9*x) + 9)/(x^2*(x*log(9*x) - 2))

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sympy [A]  time = 0.17, size = 24, normalized size = 0.83 $$\begin{array}{c}\frac{3}{x^3\log \left(9x\right)-2x^2}+\frac{-13x-3}{x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((13*x**3+6*x**2)*ln(9*x)**2+(-52*x**2-33*x)*ln(9*x)+49*x+36)/(x**5*ln(9*x)**2-4*x**4*ln(9*x)+4*x**3
),x)

[Out]

3/(x**3*log(9*x) - 2*x**2) + (-13*x - 3)/x**2

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