\(\int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+(2500 x^4-32 x^8+8 x^9) \log (x)+4 x^8 \log ^2(x)} \, dx\) [4250]

Optimal result

Integrand size = 76, antiderivative size = 17 \[ \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx=\frac {5 x}{-4+\frac {625}{2 x^4}+x+\log (x)} \]

[Out]

5*x/(ln(x)+x-4+625/2/x^4)

Rubi [F]

\[ \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx=\int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx \]

[In]

Int[(31250*x^4 - 100*x^8 + 20*x^8*Log[x])/(390625 - 10000*x^4 + 2500*x^5 + 64*x^8 - 32*x^9 + 4*x^10 + (2500*x^
4 - 32*x^8 + 8*x^9)*Log[x] + 4*x^8*Log[x]^2),x]

[Out]

25000*Defer[Int][x^4/(625 - 8*x^4 + 2*x^5 + 2*x^4*Log[x])^2, x] - 20*Defer[Int][x^8/(625 - 8*x^4 + 2*x^5 + 2*x
^4*Log[x])^2, x] - 20*Defer[Int][x^9/(625 - 8*x^4 + 2*x^5 + 2*x^4*Log[x])^2, x] + 10*Defer[Int][x^4/(625 - 8*x
^4 + 2*x^5 + 2*x^4*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {10 x^4 \left (3125-10 x^4+2 x^4 \log (x)\right )}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx \\ & = 10 \int \frac {x^4 \left (3125-10 x^4+2 x^4 \log (x)\right )}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx \\ & = 10 \int \left (-\frac {2 x^4 \left (-1250+x^4+x^5\right )}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2}+\frac {x^4}{625-8 x^4+2 x^5+2 x^4 \log (x)}\right ) \, dx \\ & = 10 \int \frac {x^4}{625-8 x^4+2 x^5+2 x^4 \log (x)} \, dx-20 \int \frac {x^4 \left (-1250+x^4+x^5\right )}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx \\ & = 10 \int \frac {x^4}{625-8 x^4+2 x^5+2 x^4 \log (x)} \, dx-20 \int \left (-\frac {1250 x^4}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2}+\frac {x^8}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2}+\frac {x^9}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2}\right ) \, dx \\ & = 10 \int \frac {x^4}{625-8 x^4+2 x^5+2 x^4 \log (x)} \, dx-20 \int \frac {x^8}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx-20 \int \frac {x^9}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx+25000 \int \frac {x^4}{\left (625-8 x^4+2 x^5+2 x^4 \log (x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx=\frac {10 x^5}{625-8 x^4+2 x^5+2 x^4 \log (x)} \]

[In]

Integrate[(31250*x^4 - 100*x^8 + 20*x^8*Log[x])/(390625 - 10000*x^4 + 2500*x^5 + 64*x^8 - 32*x^9 + 4*x^10 + (2
500*x^4 - 32*x^8 + 8*x^9)*Log[x] + 4*x^8*Log[x]^2),x]

[Out]

(10*x^5)/(625 - 8*x^4 + 2*x^5 + 2*x^4*Log[x])

Maple [A] (verified)

Time = 2.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59

method	result	size
risch	\(\frac {10 x^{5}}{2 x^{4} \ln \left (x \right )+2 x^{5}-8 x^{4}+625}\)	\(27\)
parallelrisch	\(\frac {10 x^{5}}{2 x^{4} \ln \left (x \right )+2 x^{5}-8 x^{4}+625}\)	\(27\)
default	\(-\frac {5 \left (2 x^{4} \ln \left (x \right )-8 x^{4}+625\right )}{2 x^{4} \ln \left (x \right )+2 x^{5}-8 x^{4}+625}\)	\(38\)

[In]

int((20*x^8*ln(x)-100*x^8+31250*x^4)/(4*x^8*ln(x)^2+(8*x^9-32*x^8+2500*x^4)*ln(x)+4*x^10-32*x^9+64*x^8+2500*x^
5-10000*x^4+390625),x,method=_RETURNVERBOSE)

[Out]

10*x^5/(2*x^4*ln(x)+2*x^5-8*x^4+625)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx=\frac {10 \, x^{5}}{2 \, x^{5} + 2 \, x^{4} \log \left (x\right ) - 8 \, x^{4} + 625} \]

[In]

integrate((20*x^8*log(x)-100*x^8+31250*x^4)/(4*x^8*log(x)^2+(8*x^9-32*x^8+2500*x^4)*log(x)+4*x^10-32*x^9+64*x^
8+2500*x^5-10000*x^4+390625),x, algorithm="fricas")

[Out]

10*x^5/(2*x^5 + 2*x^4*log(x) - 8*x^4 + 625)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx=\frac {10 x^{5}}{2 x^{5} + 2 x^{4} \log {\left (x \right )} - 8 x^{4} + 625} \]

[In]

integrate((20*x**8*ln(x)-100*x**8+31250*x**4)/(4*x**8*ln(x)**2+(8*x**9-32*x**8+2500*x**4)*ln(x)+4*x**10-32*x**
9+64*x**8+2500*x**5-10000*x**4+390625),x)

[Out]

10*x**5/(2*x**5 + 2*x**4*log(x) - 8*x**4 + 625)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx=\frac {10 \, x^{5}}{2 \, x^{5} + 2 \, x^{4} \log \left (x\right ) - 8 \, x^{4} + 625} \]

[In]

integrate((20*x^8*log(x)-100*x^8+31250*x^4)/(4*x^8*log(x)^2+(8*x^9-32*x^8+2500*x^4)*log(x)+4*x^10-32*x^9+64*x^
8+2500*x^5-10000*x^4+390625),x, algorithm="maxima")

[Out]

10*x^5/(2*x^5 + 2*x^4*log(x) - 8*x^4 + 625)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx=\frac {10 \, x^{5}}{2 \, x^{5} + 2 \, x^{4} \log \left (x\right ) - 8 \, x^{4} + 625} \]

[In]

integrate((20*x^8*log(x)-100*x^8+31250*x^4)/(4*x^8*log(x)^2+(8*x^9-32*x^8+2500*x^4)*log(x)+4*x^10-32*x^9+64*x^
8+2500*x^5-10000*x^4+390625),x, algorithm="giac")

[Out]

10*x^5/(2*x^5 + 2*x^4*log(x) - 8*x^4 + 625)

Mupad [B] (verification not implemented)

Time = 9.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {31250 x^4-100 x^8+20 x^8 \log (x)}{390625-10000 x^4+2500 x^5+64 x^8-32 x^9+4 x^{10}+\left (2500 x^4-32 x^8+8 x^9\right ) \log (x)+4 x^8 \log ^2(x)} \, dx=\frac {10\,x^5}{2\,x^4\,\ln \left (x\right )-8\,x^4+2\,x^5+625} \]

[In]

int((20*x^8*log(x) + 31250*x^4 - 100*x^8)/(log(x)*(2500*x^4 - 32*x^8 + 8*x^9) + 4*x^8*log(x)^2 - 10000*x^4 + 2
500*x^5 + 64*x^8 - 32*x^9 + 4*x^10 + 390625),x)

[Out]

(10*x^5)/(2*x^4*log(x) - 8*x^4 + 2*x^5 + 625)