\(\int \frac {30-75 x+10 x^2-25 x^3+(-55 x-70 x^2+25 x^3) \log (x)+(-10 x+25 x^2+10 x \log (x)) \log (5 x^2)}{(4 x-20 x^2+25 x^3) \log ^2(x)} \, dx\) [9609]

Optimal result

Integrand size = 74, antiderivative size = 27 \[ \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx=\frac {5 \left (-3+x \left (-x+\log \left (5 x^2\right )\right )\right )}{(2-5 x) \log (x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx=\int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx \]

[In]

Int[(30 - 75*x + 10*x^2 - 25*x^3 + (-55*x - 70*x^2 + 25*x^3)*Log[x] + (-10*x + 25*x^2 + 10*x*Log[x])*Log[5*x^2
])/((4*x - 20*x^2 + 25*x^3)*Log[x]^2),x]

[Out]

-75*Defer[Int][1/((-2 + 5*x)^2*Log[x]^2), x] + 30*Defer[Int][1/(x*(-2 + 5*x)^2*Log[x]^2), x] + 10*Defer[Int][x
/((-2 + 5*x)^2*Log[x]^2), x] - 25*Defer[Int][x^2/((-2 + 5*x)^2*Log[x]^2), x] + 5*Defer[Int][(-11 - 14*x + 5*x^
2)/((-2 + 5*x)^2*Log[x]), x] + 5*Defer[Int][Log[5*x^2]/((-2 + 5*x)*Log[x]^2), x] + 10*Defer[Int][Log[5*x^2]/((
-2 + 5*x)^2*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{x \left (4-20 x+25 x^2\right ) \log ^2(x)} \, dx \\ & = \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{x (-2+5 x)^2 \log ^2(x)} \, dx \\ & = \int \left (-\frac {75}{(-2+5 x)^2 \log ^2(x)}+\frac {30}{x (-2+5 x)^2 \log ^2(x)}+\frac {10 x}{(-2+5 x)^2 \log ^2(x)}-\frac {25 x^2}{(-2+5 x)^2 \log ^2(x)}+\frac {5 \left (-11-14 x+5 x^2\right )}{(-2+5 x)^2 \log (x)}+\frac {5 (-2+5 x+2 \log (x)) \log \left (5 x^2\right )}{(-2+5 x)^2 \log ^2(x)}\right ) \, dx \\ & = 5 \int \frac {-11-14 x+5 x^2}{(-2+5 x)^2 \log (x)} \, dx+5 \int \frac {(-2+5 x+2 \log (x)) \log \left (5 x^2\right )}{(-2+5 x)^2 \log ^2(x)} \, dx+10 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx-25 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx+30 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-75 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx \\ & = 5 \int \frac {-11-14 x+5 x^2}{(-2+5 x)^2 \log (x)} \, dx+5 \int \left (-\frac {2 \log \left (5 x^2\right )}{(-2+5 x)^2 \log ^2(x)}+\frac {5 x \log \left (5 x^2\right )}{(-2+5 x)^2 \log ^2(x)}+\frac {2 \log \left (5 x^2\right )}{(-2+5 x)^2 \log (x)}\right ) \, dx+10 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx-25 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx+30 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-75 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx \\ & = 5 \int \frac {-11-14 x+5 x^2}{(-2+5 x)^2 \log (x)} \, dx+10 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx-10 \int \frac {\log \left (5 x^2\right )}{(-2+5 x)^2 \log ^2(x)} \, dx+10 \int \frac {\log \left (5 x^2\right )}{(-2+5 x)^2 \log (x)} \, dx-25 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx+25 \int \frac {x \log \left (5 x^2\right )}{(-2+5 x)^2 \log ^2(x)} \, dx+30 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-75 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx \\ & = 5 \int \frac {-11-14 x+5 x^2}{(-2+5 x)^2 \log (x)} \, dx+10 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx-10 \int \frac {\log \left (5 x^2\right )}{(-2+5 x)^2 \log ^2(x)} \, dx+10 \int \frac {\log \left (5 x^2\right )}{(-2+5 x)^2 \log (x)} \, dx-25 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx+25 \int \left (\frac {2 \log \left (5 x^2\right )}{5 (-2+5 x)^2 \log ^2(x)}+\frac {\log \left (5 x^2\right )}{5 (-2+5 x) \log ^2(x)}\right ) \, dx+30 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-75 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx \\ & = 5 \int \frac {-11-14 x+5 x^2}{(-2+5 x)^2 \log (x)} \, dx+5 \int \frac {\log \left (5 x^2\right )}{(-2+5 x) \log ^2(x)} \, dx+10 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx+10 \int \frac {\log \left (5 x^2\right )}{(-2+5 x)^2 \log (x)} \, dx-25 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx+30 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-75 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx=2+\frac {5 \left (3+x^2-x \log \left (5 x^2\right )\right )}{(-2+5 x) \log (x)} \]

[In]

Integrate[(30 - 75*x + 10*x^2 - 25*x^3 + (-55*x - 70*x^2 + 25*x^3)*Log[x] + (-10*x + 25*x^2 + 10*x*Log[x])*Log
[5*x^2])/((4*x - 20*x^2 + 25*x^3)*Log[x]^2),x]

[Out]

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

Maple [A] (verified)

Time = 32.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11

method	result	size
parallelrisch	\(\frac {1800+600 x^{2}-600 x \ln \left (5 x^{2}\right )}{120 \ln \left (x \right ) \left (5 x -2\right )}\)	\(30\)
risch	\(-\frac {4}{5 x -2}+\frac {15+\frac {5 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}-5 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\frac {5 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}-5 x \ln \left (5\right )+5 x^{2}}{\ln \left (x \right ) \left (5 x -2\right )}\)	\(88\)

[In]

int(((10*x*ln(x)+25*x^2-10*x)*ln(5*x^2)+(25*x^3-70*x^2-55*x)*ln(x)-25*x^3+10*x^2-75*x+30)/(25*x^3-20*x^2+4*x)/
ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/120*(1800+600*x^2-600*x*ln(5*x^2))/ln(x)/(5*x-2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx=\frac {5 \, x^{2} - 5 \, x \log \left (5\right ) - 4 \, \log \left (x\right ) + 15}{{\left (5 \, x - 2\right )} \log \left (x\right )} \]

[In]

integrate(((10*x*log(x)+25*x^2-10*x)*log(5*x^2)+(25*x^3-70*x^2-55*x)*log(x)-25*x^3+10*x^2-75*x+30)/(25*x^3-20*
x^2+4*x)/log(x)^2,x, algorithm="fricas")

[Out]

(5*x^2 - 5*x*log(5) - 4*log(x) + 15)/((5*x - 2)*log(x))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx=- \frac {20}{25 x - 10} + \frac {5 x^{2} - 5 x \log {\left (5 \right )} + 15}{\left (5 x - 2\right ) \log {\left (x \right )}} \]

[In]

integrate(((10*x*ln(x)+25*x**2-10*x)*ln(5*x**2)+(25*x**3-70*x**2-55*x)*ln(x)-25*x**3+10*x**2-75*x+30)/(25*x**3
-20*x**2+4*x)/ln(x)**2,x)

[Out]

-20/(25*x - 10) + (5*x**2 - 5*x*log(5) + 15)/((5*x - 2)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx=\frac {5 \, x^{2} - 5 \, x \log \left (5\right ) - 4 \, \log \left (x\right ) + 15}{{\left (5 \, x - 2\right )} \log \left (x\right )} \]

[In]

integrate(((10*x*log(x)+25*x^2-10*x)*log(5*x^2)+(25*x^3-70*x^2-55*x)*log(x)-25*x^3+10*x^2-75*x+30)/(25*x^3-20*
x^2+4*x)/log(x)^2,x, algorithm="maxima")

[Out]

(5*x^2 - 5*x*log(5) - 4*log(x) + 15)/((5*x - 2)*log(x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx=\frac {5 \, {\left (x^{2} - x \log \left (5\right ) + 3\right )}}{5 \, x \log \left (x\right ) - 2 \, \log \left (x\right )} - \frac {4}{5 \, x - 2} \]

[In]

integrate(((10*x*log(x)+25*x^2-10*x)*log(5*x^2)+(25*x^3-70*x^2-55*x)*log(x)-25*x^3+10*x^2-75*x+30)/(25*x^3-20*
x^2+4*x)/log(x)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 15.73 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {30-75 x+10 x^2-25 x^3+\left (-55 x-70 x^2+25 x^3\right ) \log (x)+\left (-10 x+25 x^2+10 x \log (x)\right ) \log \left (5 x^2\right )}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx=\frac {5\,\left (x^2-x\,\ln \left (5\,x^2\right )+3\right )}{\ln \left (x\right )\,\left (5\,x-2\right )} \]

[In]

int(-(75*x - log(5*x^2)*(10*x*log(x) - 10*x + 25*x^2) - 10*x^2 + 25*x^3 + log(x)*(55*x + 70*x^2 - 25*x^3) - 30
)/(log(x)^2*(4*x - 20*x^2 + 25*x^3)),x)

[Out]

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