\(\int \frac {-3264 x+864 x^2-76 x^3+(-32 x+16 x^2-2 x^3) \log (5 x^2)}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+(64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6) \log (5 x^2)+(16 x^4-8 x^5+x^6) \log ^2(5 x^2)} \, dx\) [7535]

Optimal result

Integrand size = 124, antiderivative size = 30 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {1}{2+x^2 \left (1+\frac {(-20+x)^2}{4-x}+x+\log \left (5 x^2\right )\right )} \]

[Out]

1/(x^2*(ln(5*x^2)+1+(x-20)^2/(-x+4)+x)+2)

Rubi [F]

\[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx \]

[In]

Int[(-3264*x + 864*x^2 - 76*x^3 + (-32*x + 16*x^2 - 2*x^3)*Log[5*x^2])/(64 - 32*x + 6468*x^2 - 2208*x^3 + 1633
64*x^4 - 29896*x^5 + 1369*x^6 + (64*x^2 - 32*x^3 + 3236*x^4 - 1104*x^5 + 74*x^6)*Log[5*x^2] + (16*x^4 - 8*x^5
+ x^6)*Log[5*x^2]^2),x]

[Out]

-32*Defer[Int][(-8 + 2*x - 404*x^2 + 37*x^3 - 4*x^2*Log[5*x^2] + x^3*Log[5*x^2])^(-2), x] + 64*Defer[Int][1/(x
*(-8 + 2*x - 404*x^2 + 37*x^3 - 4*x^2*Log[5*x^2] + x^3*Log[5*x^2])^2), x] - 28*Defer[Int][x/(-8 + 2*x - 404*x^
2 + 37*x^3 - 4*x^2*Log[5*x^2] + x^3*Log[5*x^2])^2, x] - 240*Defer[Int][x^2/(-8 + 2*x - 404*x^2 + 37*x^3 - 4*x^
2*Log[5*x^2] + x^3*Log[5*x^2])^2, x] - 2*Defer[Int][x^3/(-8 + 2*x - 404*x^2 + 37*x^3 - 4*x^2*Log[5*x^2] + x^3*
Log[5*x^2])^2, x] - 2*Defer[Int][(-8 + 2*x - 404*x^2 + 37*x^3 - 4*x^2*Log[5*x^2] + x^3*Log[5*x^2])^(-1), x] +
8*Defer[Int][1/(x*(-8 + 2*x - 404*x^2 + 37*x^3 - 4*x^2*Log[5*x^2] + x^3*Log[5*x^2])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \left (-1632+432 x-38 x^2-(-4+x)^2 \log \left (5 x^2\right )\right )}{\left (8-2 x+404 x^2-37 x^3-(-4+x) x^2 \log \left (5 x^2\right )\right )^2} \, dx \\ & = 2 \int \frac {x \left (-1632+432 x-38 x^2-(-4+x)^2 \log \left (5 x^2\right )\right )}{\left (8-2 x+404 x^2-37 x^3-(-4+x) x^2 \log \left (5 x^2\right )\right )^2} \, dx \\ & = 2 \int \left (\frac {32-16 x-14 x^2-120 x^3-x^4}{x \left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2}+\frac {4-x}{x \left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )}\right ) \, dx \\ & = 2 \int \frac {32-16 x-14 x^2-120 x^3-x^4}{x \left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2} \, dx+2 \int \frac {4-x}{x \left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )} \, dx \\ & = 2 \int \left (-\frac {16}{\left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2}+\frac {32}{x \left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2}-\frac {14 x}{\left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2}-\frac {120 x^2}{\left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2}-\frac {x^3}{\left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2}\right ) \, dx+2 \int \left (-\frac {1}{-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )}+\frac {4}{x \left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )}\right ) \, dx \\ & = -\left (2 \int \frac {x^3}{\left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2} \, dx\right )-2 \int \frac {1}{-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )} \, dx+8 \int \frac {1}{x \left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )} \, dx-28 \int \frac {x}{\left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2} \, dx-32 \int \frac {1}{\left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2} \, dx+64 \int \frac {1}{x \left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2} \, dx-240 \int \frac {x^2}{\left (-8+2 x-404 x^2+37 x^3-4 x^2 \log \left (5 x^2\right )+x^3 \log \left (5 x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {-4+x}{-8+2 x-404 x^2+37 x^3+(-4+x) x^2 \log \left (5 x^2\right )} \]

[In]

Integrate[(-3264*x + 864*x^2 - 76*x^3 + (-32*x + 16*x^2 - 2*x^3)*Log[5*x^2])/(64 - 32*x + 6468*x^2 - 2208*x^3
+ 163364*x^4 - 29896*x^5 + 1369*x^6 + (64*x^2 - 32*x^3 + 3236*x^4 - 1104*x^5 + 74*x^6)*Log[5*x^2] + (16*x^4 -
8*x^5 + x^6)*Log[5*x^2]^2),x]

[Out]

(-4 + x)/(-8 + 2*x - 404*x^2 + 37*x^3 + (-4 + x)*x^2*Log[5*x^2])

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43

method	result	size
norman	\(\frac {x -4}{x^{3} \ln \left (5 x^{2}\right )-4 \ln \left (5 x^{2}\right ) x^{2}+37 x^{3}-404 x^{2}+2 x -8}\)	\(43\)
risch	\(\frac {x -4}{x^{3} \ln \left (5 x^{2}\right )-4 \ln \left (5 x^{2}\right ) x^{2}+37 x^{3}-404 x^{2}+2 x -8}\)	\(43\)
parallelrisch	\(\frac {2 x -8}{2 x^{3} \ln \left (5 x^{2}\right )-8 \ln \left (5 x^{2}\right ) x^{2}+74 x^{3}-808 x^{2}+4 x -16}\)	\(46\)

[In]

int(((-2*x^3+16*x^2-32*x)*ln(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8*x^5+16*x^4)*ln(5*x^2)^2+(74*x^6-1104*x^5+32
36*x^4-32*x^3+64*x^2)*ln(5*x^2)+1369*x^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x,method=_RETURNVERBO
SE)

[Out]

(x-4)/(x^3*ln(5*x^2)-4*ln(5*x^2)*x^2+37*x^3-404*x^2+2*x-8)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{37 \, x^{3} - 404 \, x^{2} + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (5 \, x^{2}\right ) + 2 \, x - 8} \]

[In]

integrate(((-2*x^3+16*x^2-32*x)*log(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8*x^5+16*x^4)*log(5*x^2)^2+(74*x^6-110
4*x^5+3236*x^4-32*x^3+64*x^2)*log(5*x^2)+1369*x^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x, algorithm
="fricas")

[Out]

(x - 4)/(37*x^3 - 404*x^2 + (x^3 - 4*x^2)*log(5*x^2) + 2*x - 8)

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{37 x^{3} - 404 x^{2} + 2 x + \left (x^{3} - 4 x^{2}\right ) \log {\left (5 x^{2} \right )} - 8} \]

[In]

integrate(((-2*x**3+16*x**2-32*x)*ln(5*x**2)-76*x**3+864*x**2-3264*x)/((x**6-8*x**5+16*x**4)*ln(5*x**2)**2+(74
*x**6-1104*x**5+3236*x**4-32*x**3+64*x**2)*ln(5*x**2)+1369*x**6-29896*x**5+163364*x**4-2208*x**3+6468*x**2-32*
x+64),x)

[Out]

(x - 4)/(37*x**3 - 404*x**2 + 2*x + (x**3 - 4*x**2)*log(5*x**2) - 8)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{x^{3} {\left (\log \left (5\right ) + 37\right )} - 4 \, x^{2} {\left (\log \left (5\right ) + 101\right )} + 2 \, {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x\right ) + 2 \, x - 8} \]

[In]

integrate(((-2*x^3+16*x^2-32*x)*log(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8*x^5+16*x^4)*log(5*x^2)^2+(74*x^6-110
4*x^5+3236*x^4-32*x^3+64*x^2)*log(5*x^2)+1369*x^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x, algorithm
="maxima")

[Out]

(x - 4)/(x^3*(log(5) + 37) - 4*x^2*(log(5) + 101) + 2*(x^3 - 4*x^2)*log(x) + 2*x - 8)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{x^{3} \log \left (5 \, x^{2}\right ) + 37 \, x^{3} - 4 \, x^{2} \log \left (5 \, x^{2}\right ) - 404 \, x^{2} + 2 \, x - 8} \]

[In]

integrate(((-2*x^3+16*x^2-32*x)*log(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8*x^5+16*x^4)*log(5*x^2)^2+(74*x^6-110
4*x^5+3236*x^4-32*x^3+64*x^2)*log(5*x^2)+1369*x^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x, algorithm
="giac")

[Out]

(x - 4)/(x^3*log(5*x^2) + 37*x^3 - 4*x^2*log(5*x^2) - 404*x^2 + 2*x - 8)

Mupad [F(-1)]

Timed out. \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\int -\frac {3264\,x+\ln \left (5\,x^2\right )\,\left (2\,x^3-16\,x^2+32\,x\right )-864\,x^2+76\,x^3}{{\ln \left (5\,x^2\right )}^2\,\left (x^6-8\,x^5+16\,x^4\right )-32\,x+\ln \left (5\,x^2\right )\,\left (74\,x^6-1104\,x^5+3236\,x^4-32\,x^3+64\,x^2\right )+6468\,x^2-2208\,x^3+163364\,x^4-29896\,x^5+1369\,x^6+64} \,d x \]

[In]

int(-(3264*x + log(5*x^2)*(32*x - 16*x^2 + 2*x^3) - 864*x^2 + 76*x^3)/(log(5*x^2)^2*(16*x^4 - 8*x^5 + x^6) - 3
2*x + log(5*x^2)*(64*x^2 - 32*x^3 + 3236*x^4 - 1104*x^5 + 74*x^6) + 6468*x^2 - 2208*x^3 + 163364*x^4 - 29896*x
^5 + 1369*x^6 + 64),x)

[Out]

int(-(3264*x + log(5*x^2)*(32*x - 16*x^2 + 2*x^3) - 864*x^2 + 76*x^3)/(log(5*x^2)^2*(16*x^4 - 8*x^5 + x^6) - 3
2*x + log(5*x^2)*(64*x^2 - 32*x^3 + 3236*x^4 - 1104*x^5 + 74*x^6) + 6468*x^2 - 2208*x^3 + 163364*x^4 - 29896*x
^5 + 1369*x^6 + 64), x)