\(\int \frac {750 \log (x)+375 \log ^2(x)+(-6075 x^4-180 x^5) \log ^4(x)}{625+(750 x+6750 x^4+150 x^5) \log ^2(x)+(225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}) \log ^4(x)} \, dx\) [8281]

Optimal result

Integrand size = 87, antiderivative size = 28 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\frac {x}{x+x^4 \left (9+\frac {x}{5}+\frac {5}{3 x^4 \log ^2(x)}\right )} \]

[Out]

x/(x^4*(9+1/5*x+5/3/x^4/ln(x)^2)+x)

Rubi [F(-1)]

Timed out. \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\text {\$Aborted} \]

[In]

Int[(750*Log[x] + 375*Log[x]^2 + (-6075*x^4 - 180*x^5)*Log[x]^4)/(625 + (750*x + 6750*x^4 + 150*x^5)*Log[x]^2
+ (225*x^2 + 4050*x^5 + 90*x^6 + 18225*x^8 + 810*x^9 + 9*x^10)*Log[x]^4),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\frac {15 x \log ^2(x)}{25+3 x \left (5+45 x^3+x^4\right ) \log ^2(x)} \]

[In]

Integrate[(750*Log[x] + 375*Log[x]^2 + (-6075*x^4 - 180*x^5)*Log[x]^4)/(625 + (750*x + 6750*x^4 + 150*x^5)*Log
[x]^2 + (225*x^2 + 4050*x^5 + 90*x^6 + 18225*x^8 + 810*x^9 + 9*x^10)*Log[x]^4),x]

[Out]

(15*x*Log[x]^2)/(25 + 3*x*(5 + 45*x^3 + x^4)*Log[x]^2)

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {15 x \ln \left (x \right )^{2}}{3 x^{5} \ln \left (x \right )^{2}+135 x^{4} \ln \left (x \right )^{2}+15 x \ln \left (x \right )^{2}+25}\)	\(37\)
parallelrisch	\(\frac {15 x \ln \left (x \right )^{2}}{3 x^{5} \ln \left (x \right )^{2}+135 x^{4} \ln \left (x \right )^{2}+15 x \ln \left (x \right )^{2}+25}\)	\(37\)
risch	\(\frac {5}{x^{4}+45 x^{3}+5}-\frac {125}{\left (x^{4}+45 x^{3}+5\right ) \left (3 x^{5} \ln \left (x \right )^{2}+135 x^{4} \ln \left (x \right )^{2}+15 x \ln \left (x \right )^{2}+25\right )}\)	\(59\)

[In]

int(((-180*x^5-6075*x^4)*ln(x)^4+375*ln(x)^2+750*ln(x))/((9*x^10+810*x^9+18225*x^8+90*x^6+4050*x^5+225*x^2)*ln
(x)^4+(150*x^5+6750*x^4+750*x)*ln(x)^2+625),x,method=_RETURNVERBOSE)

[Out]

15*x*ln(x)^2/(3*x^5*ln(x)^2+135*x^4*ln(x)^2+15*x*ln(x)^2+25)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\frac {15 \, x \log \left (x\right )^{2}}{3 \, {\left (x^{5} + 45 \, x^{4} + 5 \, x\right )} \log \left (x\right )^{2} + 25} \]

[In]

integrate(((-180*x^5-6075*x^4)*log(x)^4+375*log(x)^2+750*log(x))/((9*x^10+810*x^9+18225*x^8+90*x^6+4050*x^5+22
5*x^2)*log(x)^4+(150*x^5+6750*x^4+750*x)*log(x)^2+625),x, algorithm="fricas")

[Out]

15*x*log(x)^2/(3*(x^5 + 45*x^4 + 5*x)*log(x)^2 + 25)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).

Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=- \frac {125}{25 x^{4} + 1125 x^{3} + \left (3 x^{9} + 270 x^{8} + 6075 x^{7} + 30 x^{5} + 1350 x^{4} + 75 x\right ) \log {\left (x \right )}^{2} + 125} + \frac {5}{x^{4} + 45 x^{3} + 5} \]

[In]

integrate(((-180*x**5-6075*x**4)*ln(x)**4+375*ln(x)**2+750*ln(x))/((9*x**10+810*x**9+18225*x**8+90*x**6+4050*x
**5+225*x**2)*ln(x)**4+(150*x**5+6750*x**4+750*x)*ln(x)**2+625),x)

[Out]

-125/(25*x**4 + 1125*x**3 + (3*x**9 + 270*x**8 + 6075*x**7 + 30*x**5 + 1350*x**4 + 75*x)*log(x)**2 + 125) + 5/
(x**4 + 45*x**3 + 5)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\frac {15 \, x \log \left (x\right )^{2}}{3 \, {\left (x^{5} + 45 \, x^{4} + 5 \, x\right )} \log \left (x\right )^{2} + 25} \]

[In]

integrate(((-180*x^5-6075*x^4)*log(x)^4+375*log(x)^2+750*log(x))/((9*x^10+810*x^9+18225*x^8+90*x^6+4050*x^5+22
5*x^2)*log(x)^4+(150*x^5+6750*x^4+750*x)*log(x)^2+625),x, algorithm="maxima")

[Out]

15*x*log(x)^2/(3*(x^5 + 45*x^4 + 5*x)*log(x)^2 + 25)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).

Time = 0.42 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.96 \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=-\frac {125}{3 \, x^{9} \log \left (x\right )^{2} + 270 \, x^{8} \log \left (x\right )^{2} + 6075 \, x^{7} \log \left (x\right )^{2} + 30 \, x^{5} \log \left (x\right )^{2} + 1350 \, x^{4} \log \left (x\right )^{2} + 25 \, x^{4} + 1125 \, x^{3} + 75 \, x \log \left (x\right )^{2} + 125} + \frac {5}{x^{4} + 45 \, x^{3} + 5} \]

[In]

integrate(((-180*x^5-6075*x^4)*log(x)^4+375*log(x)^2+750*log(x))/((9*x^10+810*x^9+18225*x^8+90*x^6+4050*x^5+22
5*x^2)*log(x)^4+(150*x^5+6750*x^4+750*x)*log(x)^2+625),x, algorithm="giac")

[Out]

-125/(3*x^9*log(x)^2 + 270*x^8*log(x)^2 + 6075*x^7*log(x)^2 + 30*x^5*log(x)^2 + 1350*x^4*log(x)^2 + 25*x^4 + 1
125*x^3 + 75*x*log(x)^2 + 125) + 5/(x^4 + 45*x^3 + 5)

Mupad [F(-1)]

Timed out. \[ \int \frac {750 \log (x)+375 \log ^2(x)+\left (-6075 x^4-180 x^5\right ) \log ^4(x)}{625+\left (750 x+6750 x^4+150 x^5\right ) \log ^2(x)+\left (225 x^2+4050 x^5+90 x^6+18225 x^8+810 x^9+9 x^{10}\right ) \log ^4(x)} \, dx=\int \frac {\left (-180\,x^5-6075\,x^4\right )\,{\ln \left (x\right )}^4+375\,{\ln \left (x\right )}^2+750\,\ln \left (x\right )}{\left (9\,x^{10}+810\,x^9+18225\,x^8+90\,x^6+4050\,x^5+225\,x^2\right )\,{\ln \left (x\right )}^4+\left (150\,x^5+6750\,x^4+750\,x\right )\,{\ln \left (x\right )}^2+625} \,d x \]

[In]

int((750*log(x) + 375*log(x)^2 - log(x)^4*(6075*x^4 + 180*x^5))/(log(x)^2*(750*x + 6750*x^4 + 150*x^5) + log(x
)^4*(225*x^2 + 4050*x^5 + 90*x^6 + 18225*x^8 + 810*x^9 + 9*x^10) + 625),x)

[Out]

int((750*log(x) + 375*log(x)^2 - log(x)^4*(6075*x^4 + 180*x^5))/(log(x)^2*(750*x + 6750*x^4 + 150*x^5) + log(x
)^4*(225*x^2 + 4050*x^5 + 90*x^6 + 18225*x^8 + 810*x^9 + 9*x^10) + 625), x)