\(\int \frac {36+49 x+(-33 x-52 x^2) \log (9 x)+(6 x^2+13 x^3) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx\) [6827]

Optimal result

Integrand size = 65, antiderivative size = 29 \[ \int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx=\frac {-13+\frac {-3+\frac {3}{x \left (-\frac {2}{x}+\log (9 x)\right )}}{x}}{x} \]

[Out]

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

Rubi [F]

\[ \int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx=\int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx \]

[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{align*} \text {integral}& = \int \frac {36+49 x-x (33+52 x) \log (9 x)+x^2 (6+13 x) \log ^2(9 x)}{x^3 (2-x \log (9 x))^2} \, dx \\ & = \int \left (\frac {6+13 x}{x^3}-\frac {3 (2+x)}{x^3 (-2+x \log (9 x))^2}-\frac {9}{x^3 (-2+x \log (9 x))}\right ) \, dx \\ & = -\left (3 \int \frac {2+x}{x^3 (-2+x \log (9 x))^2} \, dx\right )-9 \int \frac {1}{x^3 (-2+x \log (9 x))} \, dx+\int \frac {6+13 x}{x^3} \, dx \\ & = -\frac {(6+13 x)^2}{12 x^2}-3 \int \left (\frac {2}{x^3 (-2+x \log (9 x))^2}+\frac {1}{x^2 (-2+x \log (9 x))^2}\right ) \, dx-9 \int \frac {1}{x^3 (-2+x \log (9 x))} \, dx \\ & = -\frac {(6+13 x)^2}{12 x^2}-3 \int \frac {1}{x^2 (-2+x \log (9 x))^2} \, dx-6 \int \frac {1}{x^3 (-2+x \log (9 x))^2} \, dx-9 \int \frac {1}{x^3 (-2+x \log (9 x))} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx=\frac {-3-13 x+\frac {3}{-2+x \log (9 x)}}{x^2} \]

[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

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {13 x +3}{x^{2}}+\frac {3}{x^{2} \left (x \ln \left (9 x \right )-2\right )}\)	\(27\)
norman	\(\frac {9-13 x^{2} \ln \left (9 x \right )+26 x -3 x \ln \left (9 x \right )}{x^{2} \left (x \ln \left (9 x \right )-2\right )}\)	\(36\)
parallelrisch	\(\frac {9-13 x^{2} \ln \left (9 x \right )+26 x -3 x \ln \left (9 x \right )}{x^{2} \left (x \ln \left (9 x \right )-2\right )}\)	\(36\)
derivativedivides	\(\frac {729-243 x \ln \left (9 x \right )-1053 x^{2} \ln \left (9 x \right )+2106 x}{9 x^{2} \left (9 x \ln \left (9 x \right )-18\right )}\)	\(38\)
default	\(\frac {729-243 x \ln \left (9 x \right )-1053 x^{2} \ln \left (9 x \right )+2106 x}{9 x^{2} \left (9 x \ln \left (9 x \right )-18\right )}\)	\(38\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx=-\frac {{\left (13 \, x^{2} + 3 \, x\right )} \log \left (9 \, x\right ) - 26 \, x - 9}{x^{3} \log \left (9 \, x\right ) - 2 \, x^{2}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx=\frac {3}{x^{3} \log {\left (9 x \right )} - 2 x^{2}} + \frac {- 13 x - 3}{x^{2}} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx=-\frac {26 \, x^{2} \log \left (3\right ) + 2 \, x {\left (3 \, \log \left (3\right ) - 13\right )} + {\left (13 \, x^{2} + 3 \, x\right )} \log \left (x\right ) - 9}{2 \, x^{3} \log \left (3\right ) + x^{3} \log \left (x\right ) - 2 \, x^{2}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx=\frac {3}{x^{3} \log \left (9 \, x\right ) - 2 \, x^{2}} - \frac {13 \, x + 3}{x^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 11.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {36+49 x+\left (-33 x-52 x^2\right ) \log (9 x)+\left (6 x^2+13 x^3\right ) \log ^2(9 x)}{4 x^3-4 x^4 \log (9 x)+x^5 \log ^2(9 x)} \, dx=\frac {26\,x-3\,x\,\ln \left (9\,x\right )-13\,x^2\,\ln \left (9\,x\right )+9}{x^2\,\left (x\,\ln \left (9\,x\right )-2\right )} \]

[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))