\(\int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+(1920 x^3-2048 x^6+8192 x^7) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx\) [1298]

Optimal result

Integrand size = 83, antiderivative size = 24 \[ \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx=\frac {x}{-15-16 x^2 (-x+4 x (x+\log (5 x)))} \]

[Out]

x/(-15-16*(4*(ln(5*x)+x)*x-x)*x^2)

Rubi [F]

\[ \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx=\int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx \]

[In]

Int[(-15 + 32*x^3 + 192*x^4 + 128*x^3*Log[5*x])/(225 - 480*x^3 + 1920*x^4 + 256*x^6 - 2048*x^7 + 4096*x^8 + (1
920*x^3 - 2048*x^6 + 8192*x^7)*Log[5*x] + 4096*x^6*Log[5*x]^2),x]

[Out]

-45*Defer[Int][(15 - 16*x^3 + 64*x^4 + 64*x^3*Log[5*x])^(-2), x] + 64*Defer[Int][x^3/(15 - 16*x^3 + 64*x^4 + 6
4*x^3*Log[5*x])^2, x] + 64*Defer[Int][x^4/(15 - 16*x^3 + 64*x^4 + 64*x^3*Log[5*x])^2, x] + 2*Defer[Int][(15 -
16*x^3 + 64*x^4 + 64*x^3*Log[5*x])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2} \, dx \\ & = \int \left (\frac {-45+64 x^3+64 x^4}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2}+\frac {2}{15-16 x^3+64 x^4+64 x^3 \log (5 x)}\right ) \, dx \\ & = 2 \int \frac {1}{15-16 x^3+64 x^4+64 x^3 \log (5 x)} \, dx+\int \frac {-45+64 x^3+64 x^4}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2} \, dx \\ & = 2 \int \frac {1}{15-16 x^3+64 x^4+64 x^3 \log (5 x)} \, dx+\int \left (-\frac {45}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2}+\frac {64 x^3}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2}+\frac {64 x^4}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2}\right ) \, dx \\ & = 2 \int \frac {1}{15-16 x^3+64 x^4+64 x^3 \log (5 x)} \, dx-45 \int \frac {1}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2} \, dx+64 \int \frac {x^3}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2} \, dx+64 \int \frac {x^4}{\left (15-16 x^3+64 x^4+64 x^3 \log (5 x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx=-\frac {x}{15-16 x^3+64 x^4+64 x^3 \log (5 x)} \]

[In]

Integrate[(-15 + 32*x^3 + 192*x^4 + 128*x^3*Log[5*x])/(225 - 480*x^3 + 1920*x^4 + 256*x^6 - 2048*x^7 + 4096*x^
8 + (1920*x^3 - 2048*x^6 + 8192*x^7)*Log[5*x] + 4096*x^6*Log[5*x]^2),x]

[Out]

-(x/(15 - 16*x^3 + 64*x^4 + 64*x^3*Log[5*x]))

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(-\frac {625 x}{40000 x^{4}+40000 x^{3} \ln \left (5 x \right )-10000 x^{3}+9375}\)	\(27\)
default	\(-\frac {625 x}{40000 x^{4}+40000 x^{3} \ln \left (5 x \right )-10000 x^{3}+9375}\)	\(27\)
risch	\(-\frac {x}{64 x^{3} \ln \left (5 x \right )+64 x^{4}-16 x^{3}+15}\)	\(27\)
parallelrisch	\(-\frac {x}{64 x^{3} \ln \left (5 x \right )+64 x^{4}-16 x^{3}+15}\)	\(27\)

[In]

int((128*x^3*ln(5*x)+192*x^4+32*x^3-15)/(4096*x^6*ln(5*x)^2+(8192*x^7-2048*x^6+1920*x^3)*ln(5*x)+4096*x^8-2048
*x^7+256*x^6+1920*x^4-480*x^3+225),x,method=_RETURNVERBOSE)

[Out]

-625*x/(40000*x^4+40000*x^3*ln(5*x)-10000*x^3+9375)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx=-\frac {x}{64 \, x^{4} + 64 \, x^{3} \log \left (5 \, x\right ) - 16 \, x^{3} + 15} \]

[In]

integrate((128*x^3*log(5*x)+192*x^4+32*x^3-15)/(4096*x^6*log(5*x)^2+(8192*x^7-2048*x^6+1920*x^3)*log(5*x)+4096
*x^8-2048*x^7+256*x^6+1920*x^4-480*x^3+225),x, algorithm="fricas")

[Out]

-x/(64*x^4 + 64*x^3*log(5*x) - 16*x^3 + 15)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx=- \frac {x}{64 x^{4} + 64 x^{3} \log {\left (5 x \right )} - 16 x^{3} + 15} \]

[In]

integrate((128*x**3*ln(5*x)+192*x**4+32*x**3-15)/(4096*x**6*ln(5*x)**2+(8192*x**7-2048*x**6+1920*x**3)*ln(5*x)
+4096*x**8-2048*x**7+256*x**6+1920*x**4-480*x**3+225),x)

[Out]

-x/(64*x**4 + 64*x**3*log(5*x) - 16*x**3 + 15)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx=-\frac {x}{64 \, x^{4} + 16 \, x^{3} {\left (4 \, \log \left (5\right ) - 1\right )} + 64 \, x^{3} \log \left (x\right ) + 15} \]

[In]

integrate((128*x^3*log(5*x)+192*x^4+32*x^3-15)/(4096*x^6*log(5*x)^2+(8192*x^7-2048*x^6+1920*x^3)*log(5*x)+4096
*x^8-2048*x^7+256*x^6+1920*x^4-480*x^3+225),x, algorithm="maxima")

[Out]

-x/(64*x^4 + 16*x^3*(4*log(5) - 1) + 64*x^3*log(x) + 15)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx=-\frac {x}{64 \, x^{4} + 64 \, x^{3} \log \left (5 \, x\right ) - 16 \, x^{3} + 15} \]

[In]

integrate((128*x^3*log(5*x)+192*x^4+32*x^3-15)/(4096*x^6*log(5*x)^2+(8192*x^7-2048*x^6+1920*x^3)*log(5*x)+4096
*x^8-2048*x^7+256*x^6+1920*x^4-480*x^3+225),x, algorithm="giac")

[Out]

-x/(64*x^4 + 64*x^3*log(5*x) - 16*x^3 + 15)

Mupad [B] (verification not implemented)

Time = 8.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-15+32 x^3+192 x^4+128 x^3 \log (5 x)}{225-480 x^3+1920 x^4+256 x^6-2048 x^7+4096 x^8+\left (1920 x^3-2048 x^6+8192 x^7\right ) \log (5 x)+4096 x^6 \log ^2(5 x)} \, dx=-\frac {x}{64\,x^3\,\ln \left (5\,x\right )-16\,x^3+64\,x^4+15} \]

[In]

int((128*x^3*log(5*x) + 32*x^3 + 192*x^4 - 15)/(log(5*x)*(1920*x^3 - 2048*x^6 + 8192*x^7) - 480*x^3 + 1920*x^4
+ 256*x^6 - 2048*x^7 + 4096*x^8 + 4096*x^6*log(5*x)^2 + 225),x)

[Out]

-x/(64*x^3*log(5*x) - 16*x^3 + 64*x^4 + 15)