3.47.99 \(\int \frac {-80+184 x+41 x^2+(-18+38 x+4 x^2) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+(4000+2440 x+488 x^2+32 x^3) \log (x)+(600+242 x+24 x^2) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx\)

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

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Rubi [F]  time = 1.89, 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 {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-80 + 184*x + 41*x^2 + (-18 + 38*x + 4*x^2)*Log[x] + (-1 + 2*x)*Log[x]^2)/(10000 + 8200*x + 2481*x^2 + 32
8*x^3 + 16*x^4 + (4000 + 2440*x + 488*x^2 + 32*x^3)*Log[x] + (600 + 242*x + 24*x^2)*Log[x]^2 + (40 + 8*x)*Log[
x]^3 + Log[x]^4),x]

[Out]

20*Defer[Int][(100 + 41*x + 4*x^2 + 20*Log[x] + 4*x*Log[x] + Log[x]^2)^(-2), x] + 25*Defer[Int][x/(100 + 41*x
+ 4*x^2 + 20*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] - 37*Defer[Int][x^2/(100 + 41*x + 4*x^2 + 20*Log[x] + 4*x*L
og[x] + Log[x]^2)^2, x] - 8*Defer[Int][x^3/(100 + 41*x + 4*x^2 + 20*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] + 2*
Defer[Int][Log[x]/(100 + 41*x + 4*x^2 + 20*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] + 2*Defer[Int][(x*Log[x])/(10
0 + 41*x + 4*x^2 + 20*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] - 4*Defer[Int][(x^2*Log[x])/(100 + 41*x + 4*x^2 +
20*Log[x] + 4*x*Log[x] + Log[x]^2)^2, x] - Defer[Int][(100 + 41*x + 4*x^2 + 20*Log[x] + 4*x*Log[x] + Log[x]^2)
^(-1), x] + 2*Defer[Int][x/(100 + 41*x + 4*x^2 + 20*Log[x] + 4*x*Log[x] + Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-80+184 x+41 x^2+2 \left (-9+19 x+2 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{\left (100+41 x+4 x^2+4 (5+x) \log (x)+\log ^2(x)\right )^2} \, dx\\ &=\int \left (-\frac {(-1+x) \left (20+45 x+8 x^2+2 \log (x)+4 x \log (x)\right )}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {-1+2 x}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)}\right ) \, dx\\ &=-\int \frac {(-1+x) \left (20+45 x+8 x^2+2 \log (x)+4 x \log (x)\right )}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {-1+2 x}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)} \, dx\\ &=-\int \left (\frac {-20-45 x-8 x^2-2 \log (x)-4 x \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {x \left (20+45 x+8 x^2+2 \log (x)+4 x \log (x)\right )}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}\right ) \, dx+\int \left (-\frac {1}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)}+\frac {2 x}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)}\right ) \, dx\\ &=2 \int \frac {x}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)} \, dx-\int \frac {-20-45 x-8 x^2-2 \log (x)-4 x \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x \left (20+45 x+8 x^2+2 \log (x)+4 x \log (x)\right )}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {1}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)} \, dx\\ &=2 \int \frac {x}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)} \, dx-\int \frac {1}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)} \, dx-\int \left (-\frac {20}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}-\frac {45 x}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}-\frac {8 x^2}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}-\frac {2 \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}-\frac {4 x \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}\right ) \, dx-\int \left (\frac {20 x}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {45 x^2}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {8 x^3}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {2 x \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}+\frac {4 x^2 \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2}\right ) \, dx\\ &=2 \int \frac {\log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-2 \int \frac {x \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx+2 \int \frac {x}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)} \, dx+4 \int \frac {x \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-4 \int \frac {x^2 \log (x)}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx+8 \int \frac {x^2}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-8 \int \frac {x^3}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx+20 \int \frac {1}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-20 \int \frac {x}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx+45 \int \frac {x}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-45 \int \frac {x^2}{\left (100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {1}{100+41 x+4 x^2+20 \log (x)+4 x \log (x)+\log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 28, normalized size = 1.12 \begin {gather*} \frac {(-1+x) x}{100+41 x+4 x^2+4 (5+x) \log (x)+\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-80 + 184*x + 41*x^2 + (-18 + 38*x + 4*x^2)*Log[x] + (-1 + 2*x)*Log[x]^2)/(10000 + 8200*x + 2481*x^
2 + 328*x^3 + 16*x^4 + (4000 + 2440*x + 488*x^2 + 32*x^3)*Log[x] + (600 + 242*x + 24*x^2)*Log[x]^2 + (40 + 8*x
)*Log[x]^3 + Log[x]^4),x]

[Out]

((-1 + x)*x)/(100 + 41*x + 4*x^2 + 4*(5 + x)*Log[x] + Log[x]^2)

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fricas [A]  time = 0.52, size = 31, normalized size = 1.24 \begin {gather*} \frac {x^{2} - x}{4 \, x^{2} + 4 \, {\left (x + 5\right )} \log \relax (x) + \log \relax (x)^{2} + 41 \, x + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log(x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+
600)*log(x)^2+(32*x^3+488*x^2+2440*x+4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm="fricas")

[Out]

(x^2 - x)/(4*x^2 + 4*(x + 5)*log(x) + log(x)^2 + 41*x + 100)

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giac [A]  time = 0.25, size = 33, normalized size = 1.32 \begin {gather*} \frac {x^{2} - x}{4 \, x^{2} + 4 \, x \log \relax (x) + \log \relax (x)^{2} + 41 \, x + 20 \, \log \relax (x) + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log(x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+
600)*log(x)^2+(32*x^3+488*x^2+2440*x+4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm="giac")

[Out]

(x^2 - x)/(4*x^2 + 4*x*log(x) + log(x)^2 + 41*x + 20*log(x) + 100)

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maple [A]  time = 0.06, size = 31, normalized size = 1.24

method	result	size
risch	\(\frac {x \left (x -1\right )}{\ln \relax (x )^{2}+4 x \ln \relax (x )+4 x^{2}+20 \ln \relax (x )+41 x +100}\)	\(31\)
norman	\(\frac {x^{2}-x}{\ln \relax (x )^{2}+4 x \ln \relax (x )+4 x^{2}+20 \ln \relax (x )+41 x +100}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-1)*ln(x)^2+(4*x^2+38*x-18)*ln(x)+41*x^2+184*x-80)/(ln(x)^4+(8*x+40)*ln(x)^3+(24*x^2+242*x+600)*ln(x)
^2+(32*x^3+488*x^2+2440*x+4000)*ln(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x,method=_RETURNVERBOSE)

[Out]

x*(x-1)/(ln(x)^2+4*x*ln(x)+4*x^2+20*ln(x)+41*x+100)

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maxima [A]  time = 0.42, size = 31, normalized size = 1.24 \begin {gather*} \frac {x^{2} - x}{4 \, x^{2} + 4 \, {\left (x + 5\right )} \log \relax (x) + \log \relax (x)^{2} + 41 \, x + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log(x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+
600)*log(x)^2+(32*x^3+488*x^2+2440*x+4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm="maxima")

[Out]

(x^2 - x)/(4*x^2 + 4*(x + 5)*log(x) + log(x)^2 + 41*x + 100)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {184\,x+\ln \relax (x)\,\left (4\,x^2+38\,x-18\right )+41\,x^2+{\ln \relax (x)}^2\,\left (2\,x-1\right )-80}{8200\,x+{\ln \relax (x)}^2\,\left (24\,x^2+242\,x+600\right )+{\ln \relax (x)}^4+2481\,x^2+328\,x^3+16\,x^4+{\ln \relax (x)}^3\,\left (8\,x+40\right )+\ln \relax (x)\,\left (32\,x^3+488\,x^2+2440\,x+4000\right )+10000} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((184*x + log(x)*(38*x + 4*x^2 - 18) + 41*x^2 + log(x)^2*(2*x - 1) - 80)/(8200*x + log(x)^2*(242*x + 24*x^2
+ 600) + log(x)^4 + 2481*x^2 + 328*x^3 + 16*x^4 + log(x)^3*(8*x + 40) + log(x)*(2440*x + 488*x^2 + 32*x^3 + 4
000) + 10000),x)

[Out]

int((184*x + log(x)*(38*x + 4*x^2 - 18) + 41*x^2 + log(x)^2*(2*x - 1) - 80)/(8200*x + log(x)^2*(242*x + 24*x^2
+ 600) + log(x)^4 + 2481*x^2 + 328*x^3 + 16*x^4 + log(x)^3*(8*x + 40) + log(x)*(2440*x + 488*x^2 + 32*x^3 + 4
000) + 10000), x)

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sympy [A]  time = 0.17, size = 27, normalized size = 1.08 \begin {gather*} \frac {x^{2} - x}{4 x^{2} + 41 x + \left (4 x + 20\right ) \log {\relax (x )} + \log {\relax (x )}^{2} + 100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*ln(x)**2+(4*x**2+38*x-18)*ln(x)+41*x**2+184*x-80)/(ln(x)**4+(8*x+40)*ln(x)**3+(24*x**2+242*
x+600)*ln(x)**2+(32*x**3+488*x**2+2440*x+4000)*ln(x)+16*x**4+328*x**3+2481*x**2+8200*x+10000),x)

[Out]

(x**2 - x)/(4*x**2 + 41*x + (4*x + 20)*log(x) + log(x)**2 + 100)

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