\(\int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+(360 x-6000 x^4+15000 x^5-9000 x^6) \log (x)}{(400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}) \log ^2(x)} \, dx\) [2345]

Optimal result

Integrand size = 108, antiderivative size = 32 \[ \int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx=\frac {12}{\left (-4-x+\frac {1}{5} \left (-x+(5-5 x)^2 x^4\right )\right ) \log (x)} \]

[Out]

12/(1/5*(-5*x+5)^2*x^4-6/5*x-4)/ln(x)

Rubi [F]

\[ \int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx=\int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx \]

[In]

Int[(1200 + 360*x - 1500*x^4 + 3000*x^5 - 1500*x^6 + (360*x - 6000*x^4 + 15000*x^5 - 9000*x^6)*Log[x])/((400*x
+ 240*x^2 + 36*x^3 - 1000*x^5 + 1700*x^6 - 400*x^7 - 300*x^8 + 625*x^9 - 2500*x^10 + 3750*x^11 - 2500*x^12 +
625*x^13)*Log[x]^2),x]

[Out]

-60*Defer[Int][1/(x*(-20 - 6*x + 25*x^4 - 50*x^5 + 25*x^6)*Log[x]^2), x] - 120*Defer[Int][(-3 + 50*x^3 - 125*x
^4 + 75*x^5)/((-20 - 6*x + 25*x^4 - 50*x^5 + 25*x^6)^2*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {60 \left (20+6 x-25 x^4+50 x^5-25 x^6-2 x \left (-3+50 x^3-125 x^4+75 x^5\right ) \log (x)\right )}{x \left (20+6 x-25 x^4+50 x^5-25 x^6\right )^2 \log ^2(x)} \, dx \\ & = 60 \int \frac {20+6 x-25 x^4+50 x^5-25 x^6-2 x \left (-3+50 x^3-125 x^4+75 x^5\right ) \log (x)}{x \left (20+6 x-25 x^4+50 x^5-25 x^6\right )^2 \log ^2(x)} \, dx \\ & = 60 \int \left (-\frac {1}{x \left (-20-6 x+25 x^4-50 x^5+25 x^6\right ) \log ^2(x)}-\frac {2 \left (-3+50 x^3-125 x^4+75 x^5\right )}{\left (-20-6 x+25 x^4-50 x^5+25 x^6\right )^2 \log (x)}\right ) \, dx \\ & = -\left (60 \int \frac {1}{x \left (-20-6 x+25 x^4-50 x^5+25 x^6\right ) \log ^2(x)} \, dx\right )-120 \int \frac {-3+50 x^3-125 x^4+75 x^5}{\left (-20-6 x+25 x^4-50 x^5+25 x^6\right )^2 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx=-\frac {60}{\left (20+6 x-25 x^4+50 x^5-25 x^6\right ) \log (x)} \]

[In]

Integrate[(1200 + 360*x - 1500*x^4 + 3000*x^5 - 1500*x^6 + (360*x - 6000*x^4 + 15000*x^5 - 9000*x^6)*Log[x])/(
(400*x + 240*x^2 + 36*x^3 - 1000*x^5 + 1700*x^6 - 400*x^7 - 300*x^8 + 625*x^9 - 2500*x^10 + 3750*x^11 - 2500*x
^12 + 625*x^13)*Log[x]^2),x]

[Out]

-60/((20 + 6*x - 25*x^4 + 50*x^5 - 25*x^6)*Log[x])

Maple [A] (verified)

Time = 18.72 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {60}{\ln \left (x \right ) \left (25 x^{6}-50 x^{5}+25 x^{4}-6 x -20\right )}\)	\(29\)
risch	\(\frac {60}{\ln \left (x \right ) \left (25 x^{6}-50 x^{5}+25 x^{4}-6 x -20\right )}\)	\(29\)
parallelrisch	\(\frac {60}{\ln \left (x \right ) \left (25 x^{6}-50 x^{5}+25 x^{4}-6 x -20\right )}\)	\(29\)

[In]

int(((-9000*x^6+15000*x^5-6000*x^4+360*x)*ln(x)-1500*x^6+3000*x^5-1500*x^4+360*x+1200)/(625*x^13-2500*x^12+375
0*x^11-2500*x^10+625*x^9-300*x^8-400*x^7+1700*x^6-1000*x^5+36*x^3+240*x^2+400*x)/ln(x)^2,x,method=_RETURNVERBO
SE)

[Out]

60/ln(x)/(25*x^6-50*x^5+25*x^4-6*x-20)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx=\frac {60}{{\left (25 \, x^{6} - 50 \, x^{5} + 25 \, x^{4} - 6 \, x - 20\right )} \log \left (x\right )} \]

[In]

integrate(((-9000*x^6+15000*x^5-6000*x^4+360*x)*log(x)-1500*x^6+3000*x^5-1500*x^4+360*x+1200)/(625*x^13-2500*x
^12+3750*x^11-2500*x^10+625*x^9-300*x^8-400*x^7+1700*x^6-1000*x^5+36*x^3+240*x^2+400*x)/log(x)^2,x, algorithm=
"fricas")

[Out]

60/((25*x^6 - 50*x^5 + 25*x^4 - 6*x - 20)*log(x))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx=\frac {60}{\left (25 x^{6} - 50 x^{5} + 25 x^{4} - 6 x - 20\right ) \log {\left (x \right )}} \]

[In]

integrate(((-9000*x**6+15000*x**5-6000*x**4+360*x)*ln(x)-1500*x**6+3000*x**5-1500*x**4+360*x+1200)/(625*x**13-
2500*x**12+3750*x**11-2500*x**10+625*x**9-300*x**8-400*x**7+1700*x**6-1000*x**5+36*x**3+240*x**2+400*x)/ln(x)*
*2,x)

[Out]

60/((25*x**6 - 50*x**5 + 25*x**4 - 6*x - 20)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx=\frac {60}{{\left (25 \, x^{6} - 50 \, x^{5} + 25 \, x^{4} - 6 \, x - 20\right )} \log \left (x\right )} \]

[In]

integrate(((-9000*x^6+15000*x^5-6000*x^4+360*x)*log(x)-1500*x^6+3000*x^5-1500*x^4+360*x+1200)/(625*x^13-2500*x
^12+3750*x^11-2500*x^10+625*x^9-300*x^8-400*x^7+1700*x^6-1000*x^5+36*x^3+240*x^2+400*x)/log(x)^2,x, algorithm=
"maxima")

[Out]

60/((25*x^6 - 50*x^5 + 25*x^4 - 6*x - 20)*log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx=\frac {60}{25 \, x^{6} \log \left (x\right ) - 50 \, x^{5} \log \left (x\right ) + 25 \, x^{4} \log \left (x\right ) - 6 \, x \log \left (x\right ) - 20 \, \log \left (x\right )} \]

[In]

integrate(((-9000*x^6+15000*x^5-6000*x^4+360*x)*log(x)-1500*x^6+3000*x^5-1500*x^4+360*x+1200)/(625*x^13-2500*x
^12+3750*x^11-2500*x^10+625*x^9-300*x^8-400*x^7+1700*x^6-1000*x^5+36*x^3+240*x^2+400*x)/log(x)^2,x, algorithm=
"giac")

[Out]

60/(25*x^6*log(x) - 50*x^5*log(x) + 25*x^4*log(x) - 6*x*log(x) - 20*log(x))

Mupad [B] (verification not implemented)

Time = 9.55 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1200+360 x-1500 x^4+3000 x^5-1500 x^6+\left (360 x-6000 x^4+15000 x^5-9000 x^6\right ) \log (x)}{\left (400 x+240 x^2+36 x^3-1000 x^5+1700 x^6-400 x^7-300 x^8+625 x^9-2500 x^{10}+3750 x^{11}-2500 x^{12}+625 x^{13}\right ) \log ^2(x)} \, dx=-\frac {60}{\ln \left (x\right )\,\left (-25\,x^6+50\,x^5-25\,x^4+6\,x+20\right )} \]

[In]

int((360*x + log(x)*(360*x - 6000*x^4 + 15000*x^5 - 9000*x^6) - 1500*x^4 + 3000*x^5 - 1500*x^6 + 1200)/(log(x)
^2*(400*x + 240*x^2 + 36*x^3 - 1000*x^5 + 1700*x^6 - 400*x^7 - 300*x^8 + 625*x^9 - 2500*x^10 + 3750*x^11 - 250
0*x^12 + 625*x^13)),x)

[Out]

-60/(log(x)*(6*x - 25*x^4 + 50*x^5 - 25*x^6 + 20))