3.52.33 \(\int \frac {x^2+5 x^4+(-x^2-15 x^4) \log (x)+(-5-10 x^2) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+(320 x^4+1600 x^6) \log (x)+(160 x^2+2400 x^4) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx\)

Optimal. Leaf size=28 \[ \frac {\log (x)}{4 x \left (4+5 \left (2 x+\frac {2 \log (x)}{x}\right )^2\right )} \]

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Rubi [F]  time = 1.80, 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 {x^2+5 x^4+\left (-x^2-15 x^4\right ) \log (x)+\left (-5-10 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 x^4+160 x^6+400 x^8+\left (320 x^4+1600 x^6\right ) \log (x)+\left (160 x^2+2400 x^4\right ) \log ^2(x)+1600 x^2 \log ^3(x)+400 \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2 + 5*x^4 + (-x^2 - 15*x^4)*Log[x] + (-5 - 10*x^2)*Log[x]^2 + 5*Log[x]^3)/(16*x^4 + 160*x^6 + 400*x^8 +
(320*x^4 + 1600*x^6)*Log[x] + (160*x^2 + 2400*x^4)*Log[x]^2 + 1600*x^2*Log[x]^3 + 400*Log[x]^4),x]

[Out]

Defer[Int][(-x^2 - 5*x^4 - 10*x^2*Log[x] - 5*Log[x]^2)^(-1), x]/16 + Defer[Int][x^2/(x^2 + 5*x^4 + 10*x^2*Log[
x] + 5*Log[x]^2)^2, x]/8 + (7*Defer[Int][x^4/(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2)^2, x])/8 + (5*Defer[In
t][x^6/(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2)^2, x])/4 + Defer[Int][(x^2*Log[x])/(x^2 + 5*x^4 + 10*x^2*Log
[x] + 5*Log[x]^2)^2, x]/2 + (5*Defer[Int][(x^4*Log[x])/(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2)^2, x])/4 - D
efer[Int][x^2/(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2), x]/4 + Defer[Int][Log[x]/(x^2 + 5*x^4 + 10*x^2*Log[x
] + 5*Log[x]^2), x]/16

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2+5 x^4-\left (x^2+15 x^4\right ) \log (x)-5 \left (1+2 x^2\right ) \log ^2(x)+5 \log ^3(x)}{16 \left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx\\ &=\frac {1}{16} \int \frac {x^2+5 x^4-\left (x^2+15 x^4\right ) \log (x)-5 \left (1+2 x^2\right ) \log ^2(x)+5 \log ^3(x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx\\ &=\frac {1}{16} \int \left (\frac {2 x^2 \left (1+7 x^2+10 x^4+4 \log (x)+10 x^2 \log (x)\right )}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {-1-4 x^2+\log (x)}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)}\right ) \, dx\\ &=\frac {1}{16} \int \frac {-1-4 x^2+\log (x)}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)} \, dx+\frac {1}{8} \int \frac {x^2 \left (1+7 x^2+10 x^4+4 \log (x)+10 x^2 \log (x)\right )}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx\\ &=\frac {1}{16} \int \left (\frac {1}{-x^2-5 x^4-10 x^2 \log (x)-5 \log ^2(x)}-\frac {4 x^2}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)}+\frac {\log (x)}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)}\right ) \, dx+\frac {1}{8} \int \left (\frac {x^2}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {7 x^4}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {10 x^6}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {4 x^2 \log (x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}+\frac {10 x^4 \log (x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2}\right ) \, dx\\ &=\frac {1}{16} \int \frac {1}{-x^2-5 x^4-10 x^2 \log (x)-5 \log ^2(x)} \, dx+\frac {1}{16} \int \frac {\log (x)}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)} \, dx+\frac {1}{8} \int \frac {x^2}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx-\frac {1}{4} \int \frac {x^2}{x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)} \, dx+\frac {1}{2} \int \frac {x^2 \log (x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx+\frac {7}{8} \int \frac {x^4}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx+\frac {5}{4} \int \frac {x^6}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx+\frac {5}{4} \int \frac {x^4 \log (x)}{\left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 31, normalized size = 1.11 \begin {gather*} \frac {x \log (x)}{16 \left (x^2+5 x^4+10 x^2 \log (x)+5 \log ^2(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + 5*x^4 + (-x^2 - 15*x^4)*Log[x] + (-5 - 10*x^2)*Log[x]^2 + 5*Log[x]^3)/(16*x^4 + 160*x^6 + 400
*x^8 + (320*x^4 + 1600*x^6)*Log[x] + (160*x^2 + 2400*x^4)*Log[x]^2 + 1600*x^2*Log[x]^3 + 400*Log[x]^4),x]

[Out]

(x*Log[x])/(16*(x^2 + 5*x^4 + 10*x^2*Log[x] + 5*Log[x]^2))

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fricas [A]  time = 0.49, size = 29, normalized size = 1.04 \begin {gather*} \frac {x \log \relax (x)}{16 \, {\left (5 \, x^{4} + 10 \, x^{2} \log \relax (x) + x^{2} + 5 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)^3+(-10*x^2-5)*log(x)^2+(-15*x^4-x^2)*log(x)+5*x^4+x^2)/(400*log(x)^4+1600*x^2*log(x)^3+(24
00*x^4+160*x^2)*log(x)^2+(1600*x^6+320*x^4)*log(x)+400*x^8+160*x^6+16*x^4),x, algorithm="fricas")

[Out]

1/16*x*log(x)/(5*x^4 + 10*x^2*log(x) + x^2 + 5*log(x)^2)

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giac [A]  time = 0.19, size = 29, normalized size = 1.04 \begin {gather*} \frac {x \log \relax (x)}{16 \, {\left (5 \, x^{4} + 10 \, x^{2} \log \relax (x) + x^{2} + 5 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)^3+(-10*x^2-5)*log(x)^2+(-15*x^4-x^2)*log(x)+5*x^4+x^2)/(400*log(x)^4+1600*x^2*log(x)^3+(24
00*x^4+160*x^2)*log(x)^2+(1600*x^6+320*x^4)*log(x)+400*x^8+160*x^6+16*x^4),x, algorithm="giac")

[Out]

1/16*x*log(x)/(5*x^4 + 10*x^2*log(x) + x^2 + 5*log(x)^2)

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maple [A]  time = 0.03, size = 30, normalized size = 1.07

method	result	size
risch	\(\frac {x \ln \relax (x )}{80 x^{4}+160 x^{2} \ln \relax (x )+80 \ln \relax (x )^{2}+16 x^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*ln(x)^3+(-10*x^2-5)*ln(x)^2+(-15*x^4-x^2)*ln(x)+5*x^4+x^2)/(400*ln(x)^4+1600*x^2*ln(x)^3+(2400*x^4+160*
x^2)*ln(x)^2+(1600*x^6+320*x^4)*ln(x)+400*x^8+160*x^6+16*x^4),x,method=_RETURNVERBOSE)

[Out]

1/16*x*ln(x)/(5*x^4+10*x^2*ln(x)+5*ln(x)^2+x^2)

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maxima [A]  time = 0.39, size = 29, normalized size = 1.04 \begin {gather*} \frac {x \log \relax (x)}{16 \, {\left (5 \, x^{4} + 10 \, x^{2} \log \relax (x) + x^{2} + 5 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)^3+(-10*x^2-5)*log(x)^2+(-15*x^4-x^2)*log(x)+5*x^4+x^2)/(400*log(x)^4+1600*x^2*log(x)^3+(24
00*x^4+160*x^2)*log(x)^2+(1600*x^6+320*x^4)*log(x)+400*x^8+160*x^6+16*x^4),x, algorithm="maxima")

[Out]

1/16*x*log(x)/(5*x^4 + 10*x^2*log(x) + x^2 + 5*log(x)^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {5\,{\ln \relax (x)}^3-{\ln \relax (x)}^2\,\left (10\,x^2+5\right )-\ln \relax (x)\,\left (15\,x^4+x^2\right )+x^2+5\,x^4}{\ln \relax (x)\,\left (1600\,x^6+320\,x^4\right )+400\,{\ln \relax (x)}^4+{\ln \relax (x)}^2\,\left (2400\,x^4+160\,x^2\right )+1600\,x^2\,{\ln \relax (x)}^3+16\,x^4+160\,x^6+400\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*log(x)^3 - log(x)^2*(10*x^2 + 5) - log(x)*(x^2 + 15*x^4) + x^2 + 5*x^4)/(log(x)*(320*x^4 + 1600*x^6) +
400*log(x)^4 + log(x)^2*(160*x^2 + 2400*x^4) + 1600*x^2*log(x)^3 + 16*x^4 + 160*x^6 + 400*x^8),x)

[Out]

int((5*log(x)^3 - log(x)^2*(10*x^2 + 5) - log(x)*(x^2 + 15*x^4) + x^2 + 5*x^4)/(log(x)*(320*x^4 + 1600*x^6) +
400*log(x)^4 + log(x)^2*(160*x^2 + 2400*x^4) + 1600*x^2*log(x)^3 + 16*x^4 + 160*x^6 + 400*x^8), x)

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sympy [A]  time = 0.16, size = 29, normalized size = 1.04 \begin {gather*} \frac {x \log {\relax (x )}}{80 x^{4} + 160 x^{2} \log {\relax (x )} + 16 x^{2} + 80 \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*ln(x)**3+(-10*x**2-5)*ln(x)**2+(-15*x**4-x**2)*ln(x)+5*x**4+x**2)/(400*ln(x)**4+1600*x**2*ln(x)**
3+(2400*x**4+160*x**2)*ln(x)**2+(1600*x**6+320*x**4)*ln(x)+400*x**8+160*x**6+16*x**4),x)

[Out]

x*log(x)/(80*x**4 + 160*x**2*log(x) + 16*x**2 + 80*log(x)**2)

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