3.103.81 \(\int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+(2 x-500 x^2+100 x^3+62500 x^6) \log (x)+(-10+3750 x^4) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+(-500 x^2+100 x^3+62500 x^6) \log (x)+(-10+2 x+3750 x^4) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {x}{1+\frac {x}{-5+\left (25 x^2+\log (x)\right )^2}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.55, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {25+50 x^3-6250 x^4+2500 x^5+390625 x^8+\left (2 x-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{25-10 x+x^2-6250 x^4+1250 x^5+390625 x^8+\left (-500 x^2+100 x^3+62500 x^6\right ) \log (x)+\left (-10+2 x+3750 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25 + 50*x^3 - 6250*x^4 + 2500*x^5 + 390625*x^8 + (2*x - 500*x^2 + 100*x^3 + 62500*x^6)*Log[x] + (-10 + 37
50*x^4)*Log[x]^2 + 100*x^2*Log[x]^3 + Log[x]^4)/(25 - 10*x + x^2 - 6250*x^4 + 1250*x^5 + 390625*x^8 + (-500*x^
2 + 100*x^3 + 62500*x^6)*Log[x] + (-10 + 2*x + 3750*x^4)*Log[x]^2 + 100*x^2*Log[x]^3 + Log[x]^4),x]

[Out]

x + Defer[Int][x^2/(-5 + x + 625*x^4 + 50*x^2*Log[x] + Log[x]^2)^2, x] + 50*Defer[Int][x^3/(-5 + x + 625*x^4 +
50*x^2*Log[x] + Log[x]^2)^2, x] + 2500*Defer[Int][x^5/(-5 + x + 625*x^4 + 50*x^2*Log[x] + Log[x]^2)^2, x] + 2
*Defer[Int][(x*Log[x])/(-5 + x + 625*x^4 + 50*x^2*Log[x] + Log[x]^2)^2, x] + 100*Defer[Int][(x^3*Log[x])/(-5 +
x + 625*x^4 + 50*x^2*Log[x] + Log[x]^2)^2, x] - 2*Defer[Int][x/(-5 + x + 625*x^4 + 50*x^2*Log[x] + Log[x]^2),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \left (1+2 x^3-250 x^4+100 x^5+15625 x^8\right )+2 x \left (1-250 x+50 x^2+31250 x^5\right ) \log (x)+10 \left (-1+375 x^4\right ) \log ^2(x)+100 x^2 \log ^3(x)+\log ^4(x)}{\left (5-x-625 x^4-50 x^2 \log (x)-\log ^2(x)\right )^2} \, dx\\ &=\int \left (1+\frac {x \left (x+50 x^2+2500 x^4+2 \log (x)+100 x^2 \log (x)\right )}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}-\frac {2 x}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)}\right ) \, dx\\ &=x-2 \int \frac {x}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \, dx+\int \frac {x \left (x+50 x^2+2500 x^4+2 \log (x)+100 x^2 \log (x)\right )}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx\\ &=x-2 \int \frac {x}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \, dx+\int \left (\frac {x^2}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}+\frac {50 x^3}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}+\frac {2500 x^5}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}+\frac {2 x \log (x)}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}+\frac {100 x^3 \log (x)}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2}\right ) \, dx\\ &=x+2 \int \frac {x \log (x)}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx-2 \int \frac {x}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \, dx+50 \int \frac {x^3}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx+100 \int \frac {x^3 \log (x)}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx+2500 \int \frac {x^5}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x^2}{\left (-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 28, normalized size = 1.27 \begin {gather*} x-\frac {x^2}{-5+x+625 x^4+50 x^2 \log (x)+\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + 50*x^3 - 6250*x^4 + 2500*x^5 + 390625*x^8 + (2*x - 500*x^2 + 100*x^3 + 62500*x^6)*Log[x] + (-1
0 + 3750*x^4)*Log[x]^2 + 100*x^2*Log[x]^3 + Log[x]^4)/(25 - 10*x + x^2 - 6250*x^4 + 1250*x^5 + 390625*x^8 + (-
500*x^2 + 100*x^3 + 62500*x^6)*Log[x] + (-10 + 2*x + 3750*x^4)*Log[x]^2 + 100*x^2*Log[x]^3 + Log[x]^4),x]

[Out]

x - x^2/(-5 + x + 625*x^4 + 50*x^2*Log[x] + Log[x]^2)

________________________________________________________________________________________

fricas [A]  time = 0.95, size = 44, normalized size = 2.00 \begin {gather*} \frac {625 \, x^{5} + 50 \, x^{3} \log \relax (x) + x \log \relax (x)^{2} - 5 \, x}{625 \, x^{4} + 50 \, x^{2} \log \relax (x) + \log \relax (x)^{2} + x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^4+100*x^2*log(x)^3+(3750*x^4-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2+2*x)*log(x)+390625*x^8+
2500*x^5-6250*x^4+50*x^3+25)/(log(x)^4+100*x^2*log(x)^3+(3750*x^4+2*x-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2)
*log(x)+390625*x^8+1250*x^5-6250*x^4+x^2-10*x+25),x, algorithm="fricas")

[Out]

(625*x^5 + 50*x^3*log(x) + x*log(x)^2 - 5*x)/(625*x^4 + 50*x^2*log(x) + log(x)^2 + x - 5)

________________________________________________________________________________________

giac [A]  time = 0.27, size = 28, normalized size = 1.27 \begin {gather*} x - \frac {x^{2}}{625 \, x^{4} + 50 \, x^{2} \log \relax (x) + \log \relax (x)^{2} + x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^4+100*x^2*log(x)^3+(3750*x^4-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2+2*x)*log(x)+390625*x^8+
2500*x^5-6250*x^4+50*x^3+25)/(log(x)^4+100*x^2*log(x)^3+(3750*x^4+2*x-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2)
*log(x)+390625*x^8+1250*x^5-6250*x^4+x^2-10*x+25),x, algorithm="giac")

[Out]

x - x^2/(625*x^4 + 50*x^2*log(x) + log(x)^2 + x - 5)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 29, normalized size = 1.32

method	result	size
risch	\(x -\frac {x^{2}}{625 x^{4}+50 x^{2} \ln \relax (x )+\ln \relax (x )^{2}+x -5}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)^4+100*x^2*ln(x)^3+(3750*x^4-10)*ln(x)^2+(62500*x^6+100*x^3-500*x^2+2*x)*ln(x)+390625*x^8+2500*x^5-6
250*x^4+50*x^3+25)/(ln(x)^4+100*x^2*ln(x)^3+(3750*x^4+2*x-10)*ln(x)^2+(62500*x^6+100*x^3-500*x^2)*ln(x)+390625
*x^8+1250*x^5-6250*x^4+x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

x-x^2/(625*x^4+50*x^2*ln(x)+ln(x)^2+x-5)

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 44, normalized size = 2.00 \begin {gather*} \frac {625 \, x^{5} + 50 \, x^{3} \log \relax (x) + x \log \relax (x)^{2} - 5 \, x}{625 \, x^{4} + 50 \, x^{2} \log \relax (x) + \log \relax (x)^{2} + x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^4+100*x^2*log(x)^3+(3750*x^4-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2+2*x)*log(x)+390625*x^8+
2500*x^5-6250*x^4+50*x^3+25)/(log(x)^4+100*x^2*log(x)^3+(3750*x^4+2*x-10)*log(x)^2+(62500*x^6+100*x^3-500*x^2)
*log(x)+390625*x^8+1250*x^5-6250*x^4+x^2-10*x+25),x, algorithm="maxima")

[Out]

(625*x^5 + 50*x^3*log(x) + x*log(x)^2 - 5*x)/(625*x^4 + 50*x^2*log(x) + log(x)^2 + x - 5)

________________________________________________________________________________________

mupad [B]  time = 6.98, size = 28, normalized size = 1.27 \begin {gather*} x-\frac {x^2}{625\,x^4+50\,x^2\,\ln \relax (x)+x+{\ln \relax (x)}^2-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(2*x - 500*x^2 + 100*x^3 + 62500*x^6) + log(x)^2*(3750*x^4 - 10) + log(x)^4 + 100*x^2*log(x)^3 + 5
0*x^3 - 6250*x^4 + 2500*x^5 + 390625*x^8 + 25)/(log(x)^2*(2*x + 3750*x^4 - 10) - 10*x + log(x)^4 + log(x)*(100
*x^3 - 500*x^2 + 62500*x^6) + 100*x^2*log(x)^3 + x^2 - 6250*x^4 + 1250*x^5 + 390625*x^8 + 25),x)

[Out]

x - x^2/(x + 50*x^2*log(x) + log(x)^2 + 625*x^4 - 5)

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 26, normalized size = 1.18 \begin {gather*} - \frac {x^{2}}{625 x^{4} + 50 x^{2} \log {\relax (x )} + x + \log {\relax (x )}^{2} - 5} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)**4+100*x**2*ln(x)**3+(3750*x**4-10)*ln(x)**2+(62500*x**6+100*x**3-500*x**2+2*x)*ln(x)+390625*
x**8+2500*x**5-6250*x**4+50*x**3+25)/(ln(x)**4+100*x**2*ln(x)**3+(3750*x**4+2*x-10)*ln(x)**2+(62500*x**6+100*x
**3-500*x**2)*ln(x)+390625*x**8+1250*x**5-6250*x**4+x**2-10*x+25),x)

[Out]

-x**2/(625*x**4 + 50*x**2*log(x) + x + log(x)**2 - 5) + x

________________________________________________________________________________________