\(\int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx\) [5609]

Optimal result

Integrand size = 67, antiderivative size = 26 \[ \int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx=\frac {x (5+25 x)}{5 \left (\frac {3}{5}+x \log (3)-4 \log ^2(x)\right )} \]

[Out]

1/5*(25*x+5)*x/(3/5+x*ln(3)-4*ln(x)^2)

Rubi [F]

\[ \int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx=\int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx \]

[In]

Int[(15 + 150*x + 125*x^2*Log[3] + (200 + 1000*x)*Log[x] + (-100 - 1000*x)*Log[x]^2)/(9 + 30*x*Log[3] + 25*x^2
*Log[3]^2 + (-120 - 200*x*Log[3])*Log[x]^2 + 400*Log[x]^4),x]

[Out]

-25*Log[3]*Defer[Int][x/(3 + 5*x*Log[3] - 20*Log[x]^2)^2, x] - 125*Log[3]*Defer[Int][x^2/(3 + 5*x*Log[3] - 20*
Log[x]^2)^2, x] + 5*Defer[Int][(3 + 5*x*Log[3] - 20*Log[x]^2)^(-1), x] + 50*Defer[Int][x/(3 + 5*x*Log[3] - 20*
Log[x]^2), x] + 200*Defer[Int][Log[x]/(-3 - 5*x*Log[3] + 20*Log[x]^2)^2, x] + 1000*Defer[Int][(x*Log[x])/(-3 -
5*x*Log[3] + 20*Log[x]^2)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (3+30 x+25 x^2 \log (3)+40 (1+5 x) \log (x)-20 (1+10 x) \log ^2(x)\right )}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2} \, dx \\ & = 5 \int \frac {3+30 x+25 x^2 \log (3)+40 (1+5 x) \log (x)-20 (1+10 x) \log ^2(x)}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2} \, dx \\ & = 5 \int \left (-\frac {5 (1+5 x) (x \log (3)-8 \log (x))}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2}+\frac {1+10 x}{3+5 x \log (3)-20 \log ^2(x)}\right ) \, dx \\ & = 5 \int \frac {1+10 x}{3+5 x \log (3)-20 \log ^2(x)} \, dx-25 \int \frac {(1+5 x) (x \log (3)-8 \log (x))}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2} \, dx \\ & = 5 \int \left (\frac {1}{3+5 x \log (3)-20 \log ^2(x)}+\frac {10 x}{3+5 x \log (3)-20 \log ^2(x)}\right ) \, dx-25 \int \left (\frac {x \log (3)-8 \log (x)}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2}+\frac {5 x (x \log (3)-8 \log (x))}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2}\right ) \, dx \\ & = 5 \int \frac {1}{3+5 x \log (3)-20 \log ^2(x)} \, dx-25 \int \frac {x \log (3)-8 \log (x)}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2} \, dx+50 \int \frac {x}{3+5 x \log (3)-20 \log ^2(x)} \, dx-125 \int \frac {x (x \log (3)-8 \log (x))}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2} \, dx \\ & = 5 \int \frac {1}{3+5 x \log (3)-20 \log ^2(x)} \, dx-25 \int \left (\frac {x \log (3)}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2}-\frac {8 \log (x)}{\left (-3-5 x \log (3)+20 \log ^2(x)\right )^2}\right ) \, dx+50 \int \frac {x}{3+5 x \log (3)-20 \log ^2(x)} \, dx-125 \int \left (\frac {x^2 \log (3)}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2}-\frac {8 x \log (x)}{\left (-3-5 x \log (3)+20 \log ^2(x)\right )^2}\right ) \, dx \\ & = 5 \int \frac {1}{3+5 x \log (3)-20 \log ^2(x)} \, dx+50 \int \frac {x}{3+5 x \log (3)-20 \log ^2(x)} \, dx+200 \int \frac {\log (x)}{\left (-3-5 x \log (3)+20 \log ^2(x)\right )^2} \, dx+1000 \int \frac {x \log (x)}{\left (-3-5 x \log (3)+20 \log ^2(x)\right )^2} \, dx-(25 \log (3)) \int \frac {x}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2} \, dx-(125 \log (3)) \int \frac {x^2}{\left (3+5 x \log (3)-20 \log ^2(x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx=\frac {5 x (1+5 x)}{3+5 x \log (3)-20 \log ^2(x)} \]

[In]

Integrate[(15 + 150*x + 125*x^2*Log[3] + (200 + 1000*x)*Log[x] + (-100 - 1000*x)*Log[x]^2)/(9 + 30*x*Log[3] +
25*x^2*Log[3]^2 + (-120 - 200*x*Log[3])*Log[x]^2 + 400*Log[x]^4),x]

[Out]

(5*x*(1 + 5*x))/(3 + 5*x*Log[3] - 20*Log[x]^2)

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
risch	\(\frac {5 \left (1+5 x \right ) x}{5 x \ln \left (3\right )-20 \ln \left (x \right )^{2}+3}\)	\(24\)
default	\(\frac {25 x^{2}+5 x}{5 x \ln \left (3\right )-20 \ln \left (x \right )^{2}+3}\)	\(25\)
norman	\(\frac {25 x^{2}+5 x}{5 x \ln \left (3\right )-20 \ln \left (x \right )^{2}+3}\)	\(26\)
parallelrisch	\(\frac {75 x^{2}+15 x}{15 x \ln \left (3\right )-60 \ln \left (x \right )^{2}+9}\)	\(27\)

[In]

int(((-1000*x-100)*ln(x)^2+(1000*x+200)*ln(x)+125*x^2*ln(3)+150*x+15)/(400*ln(x)^4+(-200*x*ln(3)-120)*ln(x)^2+
25*x^2*ln(3)^2+30*x*ln(3)+9),x,method=_RETURNVERBOSE)

[Out]

5*(1+5*x)*x/(5*x*ln(3)-20*ln(x)^2+3)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx=\frac {5 \, {\left (5 \, x^{2} + x\right )}}{5 \, x \log \left (3\right ) - 20 \, \log \left (x\right )^{2} + 3} \]

[In]

integrate(((-1000*x-100)*log(x)^2+(1000*x+200)*log(x)+125*x^2*log(3)+150*x+15)/(400*log(x)^4+(-200*x*log(3)-12
0)*log(x)^2+25*x^2*log(3)^2+30*x*log(3)+9),x, algorithm="fricas")

[Out]

5*(5*x^2 + x)/(5*x*log(3) - 20*log(x)^2 + 3)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx=\frac {- 25 x^{2} - 5 x}{- 5 x \log {\left (3 \right )} + 20 \log {\left (x \right )}^{2} - 3} \]

[In]

integrate(((-1000*x-100)*ln(x)**2+(1000*x+200)*ln(x)+125*x**2*ln(3)+150*x+15)/(400*ln(x)**4+(-200*x*ln(3)-120)
*ln(x)**2+25*x**2*ln(3)**2+30*x*ln(3)+9),x)

[Out]

(-25*x**2 - 5*x)/(-5*x*log(3) + 20*log(x)**2 - 3)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx=\frac {5 \, {\left (5 \, x^{2} + x\right )}}{5 \, x \log \left (3\right ) - 20 \, \log \left (x\right )^{2} + 3} \]

[In]

integrate(((-1000*x-100)*log(x)^2+(1000*x+200)*log(x)+125*x^2*log(3)+150*x+15)/(400*log(x)^4+(-200*x*log(3)-12
0)*log(x)^2+25*x^2*log(3)^2+30*x*log(3)+9),x, algorithm="maxima")

[Out]

5*(5*x^2 + x)/(5*x*log(3) - 20*log(x)^2 + 3)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx=\frac {5 \, {\left (5 \, x^{2} + x\right )}}{5 \, x \log \left (3\right ) - 20 \, \log \left (x\right )^{2} + 3} \]

[In]

integrate(((-1000*x-100)*log(x)^2+(1000*x+200)*log(x)+125*x^2*log(3)+150*x+15)/(400*log(x)^4+(-200*x*log(3)-12
0)*log(x)^2+25*x^2*log(3)^2+30*x*log(3)+9),x, algorithm="giac")

[Out]

5*(5*x^2 + x)/(5*x*log(3) - 20*log(x)^2 + 3)

Mupad [F(-1)]

Timed out. \[ \int \frac {15+150 x+125 x^2 \log (3)+(200+1000 x) \log (x)+(-100-1000 x) \log ^2(x)}{9+30 x \log (3)+25 x^2 \log ^2(3)+(-120-200 x \log (3)) \log ^2(x)+400 \log ^4(x)} \, dx=\int \frac {150\,x+\ln \left (x\right )\,\left (1000\,x+200\right )+125\,x^2\,\ln \left (3\right )-{\ln \left (x\right )}^2\,\left (1000\,x+100\right )+15}{25\,x^2\,{\ln \left (3\right )}^2+30\,x\,\ln \left (3\right )+400\,{\ln \left (x\right )}^4-{\ln \left (x\right )}^2\,\left (200\,x\,\ln \left (3\right )+120\right )+9} \,d x \]

[In]

int((150*x + log(x)*(1000*x + 200) + 125*x^2*log(3) - log(x)^2*(1000*x + 100) + 15)/(25*x^2*log(3)^2 + 30*x*lo
g(3) + 400*log(x)^4 - log(x)^2*(200*x*log(3) + 120) + 9),x)

[Out]

int((150*x + log(x)*(1000*x + 200) + 125*x^2*log(3) - log(x)^2*(1000*x + 100) + 15)/(25*x^2*log(3)^2 + 30*x*lo
g(3) + 400*log(x)^4 - log(x)^2*(200*x*log(3) + 120) + 9), x)