3.54.18 \(\int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+(-16 x-72 x^2-96 x^3-32 x^4) \log (x)+(-24-60 x-20 x^2+16 x^3+16 x^4) \log ^2(x)+(12 x+24 x^2) \log ^3(x)+9 \log ^4(x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {x}{-2-x+\log (x) \left (x+\frac {6 \log (x)}{4+8 x}\right )} \]

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Rubi [F]  time = 2.80, 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 {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8 - 36*x - 48*x^2 - 16*x^3 + (-12 - 24*x)*Log[x] + (6 + 24*x)*Log[x]^2)/(16 + 80*x + 132*x^2 + 80*x^3 +
16*x^4 + (-16*x - 72*x^2 - 96*x^3 - 32*x^4)*Log[x] + (-24 - 60*x - 20*x^2 + 16*x^3 + 16*x^4)*Log[x]^2 + (12*x
+ 24*x^2)*Log[x]^3 + 9*Log[x]^4),x]

[Out]

16*Defer[Int][x/(-4 - 10*x - 4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3*Log[x]^2)^2, x] + 40*Defer[Int][x^2/(-4 - 1
0*x - 4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3*Log[x]^2)^2, x] + 16*Defer[Int][x^3/(-4 - 10*x - 4*x^2 + 2*x*Log[x
] + 4*x^2*Log[x] + 3*Log[x]^2)^2, x] - 12*Defer[Int][Log[x]/(-4 - 10*x - 4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3
*Log[x]^2)^2, x] - 28*Defer[Int][(x*Log[x])/(-4 - 10*x - 4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3*Log[x]^2)^2, x]
- 24*Defer[Int][(x^2*Log[x])/(-4 - 10*x - 4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3*Log[x]^2)^2, x] - 32*Defer[In
t][(x^3*Log[x])/(-4 - 10*x - 4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3*Log[x]^2)^2, x] + 2*Defer[Int][(-4 - 10*x -
4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3*Log[x]^2)^(-1), x] + 8*Defer[Int][x/(-4 - 10*x - 4*x^2 + 2*x*Log[x] + 4
*x^2*Log[x] + 3*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 (2+x) (1+2 x)^2-12 (1+2 x) \log (x)+6 (1+4 x) \log ^2(x)}{\left (2 \left (2+5 x+2 x^2\right )-2 x (1+2 x) \log (x)-3 \log ^2(x)\right )^2} \, dx\\ &=\int \left (-\frac {4 (1+2 x) \left (-4 x-2 x^2+3 \log (x)+x \log (x)+4 x^2 \log (x)\right )}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}+\frac {2 (1+4 x)}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)}\right ) \, dx\\ &=2 \int \frac {1+4 x}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \, dx-4 \int \frac {(1+2 x) \left (-4 x-2 x^2+3 \log (x)+x \log (x)+4 x^2 \log (x)\right )}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx\\ &=2 \int \left (\frac {1}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)}+\frac {4 x}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)}\right ) \, dx-4 \int \left (\frac {-4 x-2 x^2+3 \log (x)+x \log (x)+4 x^2 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}+\frac {2 x \left (-4 x-2 x^2+3 \log (x)+x \log (x)+4 x^2 \log (x)\right )}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}\right ) \, dx\\ &=2 \int \frac {1}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \, dx-4 \int \frac {-4 x-2 x^2+3 \log (x)+x \log (x)+4 x^2 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx-8 \int \frac {x \left (-4 x-2 x^2+3 \log (x)+x \log (x)+4 x^2 \log (x)\right )}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx+8 \int \frac {x}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \, dx\\ &=2 \int \frac {1}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \, dx-4 \int \left (-\frac {4 x}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}-\frac {2 x^2}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}+\frac {3 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}+\frac {x \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}+\frac {4 x^2 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}\right ) \, dx+8 \int \frac {x}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \, dx-8 \int \left (-\frac {4 x^2}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}-\frac {2 x^3}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}+\frac {3 x \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}+\frac {x^2 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}+\frac {4 x^3 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2}\right ) \, dx\\ &=2 \int \frac {1}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \, dx-4 \int \frac {x \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx+8 \int \frac {x^2}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx-8 \int \frac {x^2 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx+8 \int \frac {x}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \, dx-12 \int \frac {\log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx+16 \int \frac {x}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx+16 \int \frac {x^3}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx-16 \int \frac {x^2 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx-24 \int \frac {x \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx+32 \int \frac {x^2}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx-32 \int \frac {x^3 \log (x)}{\left (-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.74, size = 39, normalized size = 1.56 \begin {gather*} \frac {2 \left (x+2 x^2\right )}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 - 36*x - 48*x^2 - 16*x^3 + (-12 - 24*x)*Log[x] + (6 + 24*x)*Log[x]^2)/(16 + 80*x + 132*x^2 + 80*
x^3 + 16*x^4 + (-16*x - 72*x^2 - 96*x^3 - 32*x^4)*Log[x] + (-24 - 60*x - 20*x^2 + 16*x^3 + 16*x^4)*Log[x]^2 +
(12*x + 24*x^2)*Log[x]^3 + 9*Log[x]^4),x]

[Out]

(2*(x + 2*x^2))/(-4 - 10*x - 4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3*Log[x]^2)

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fricas [A]  time = 0.72, size = 38, normalized size = 1.52 \begin {gather*} -\frac {2 \, {\left (2 \, x^{2} + x\right )}}{4 \, x^{2} - 2 \, {\left (2 \, x^{2} + x\right )} \log \relax (x) - 3 \, \log \relax (x)^{2} + 10 \, x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+6)*log(x)^2+(-24*x-12)*log(x)-16*x^3-48*x^2-36*x-8)/(9*log(x)^4+(24*x^2+12*x)*log(x)^3+(16*x^
4+16*x^3-20*x^2-60*x-24)*log(x)^2+(-32*x^4-96*x^3-72*x^2-16*x)*log(x)+16*x^4+80*x^3+132*x^2+80*x+16),x, algori
thm="fricas")

[Out]

-2*(2*x^2 + x)/(4*x^2 - 2*(2*x^2 + x)*log(x) - 3*log(x)^2 + 10*x + 4)

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giac [A]  time = 1.31, size = 39, normalized size = 1.56 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} + x\right )}}{4 \, x^{2} \log \relax (x) - 4 \, x^{2} + 2 \, x \log \relax (x) + 3 \, \log \relax (x)^{2} - 10 \, x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+6)*log(x)^2+(-24*x-12)*log(x)-16*x^3-48*x^2-36*x-8)/(9*log(x)^4+(24*x^2+12*x)*log(x)^3+(16*x^
4+16*x^3-20*x^2-60*x-24)*log(x)^2+(-32*x^4-96*x^3-72*x^2-16*x)*log(x)+16*x^4+80*x^3+132*x^2+80*x+16),x, algori
thm="giac")

[Out]

2*(2*x^2 + x)/(4*x^2*log(x) - 4*x^2 + 2*x*log(x) + 3*log(x)^2 - 10*x - 4)

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maple [A]  time = 0.08, size = 39, normalized size = 1.56

method	result	size
risch	\(\frac {2 \left (2 x +1\right ) x}{4 x^{2} \ln \relax (x )+3 \ln \relax (x )^{2}+2 x \ln \relax (x )-4 x^{2}-10 x -4}\)	\(39\)
norman	\(\frac {4 x^{2}+2 x}{4 x^{2} \ln \relax (x )+3 \ln \relax (x )^{2}+2 x \ln \relax (x )-4 x^{2}-10 x -4}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x+6)*ln(x)^2+(-24*x-12)*ln(x)-16*x^3-48*x^2-36*x-8)/(9*ln(x)^4+(24*x^2+12*x)*ln(x)^3+(16*x^4+16*x^3-2
0*x^2-60*x-24)*ln(x)^2+(-32*x^4-96*x^3-72*x^2-16*x)*ln(x)+16*x^4+80*x^3+132*x^2+80*x+16),x,method=_RETURNVERBO
SE)

[Out]

2*(2*x+1)*x/(4*x^2*ln(x)+3*ln(x)^2+2*x*ln(x)-4*x^2-10*x-4)

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maxima [A]  time = 0.41, size = 38, normalized size = 1.52 \begin {gather*} -\frac {2 \, {\left (2 \, x^{2} + x\right )}}{4 \, x^{2} - 2 \, {\left (2 \, x^{2} + x\right )} \log \relax (x) - 3 \, \log \relax (x)^{2} + 10 \, x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+6)*log(x)^2+(-24*x-12)*log(x)-16*x^3-48*x^2-36*x-8)/(9*log(x)^4+(24*x^2+12*x)*log(x)^3+(16*x^
4+16*x^3-20*x^2-60*x-24)*log(x)^2+(-32*x^4-96*x^3-72*x^2-16*x)*log(x)+16*x^4+80*x^3+132*x^2+80*x+16),x, algori
thm="maxima")

[Out]

-2*(2*x^2 + x)/(4*x^2 - 2*(2*x^2 + x)*log(x) - 3*log(x)^2 + 10*x + 4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {36\,x+\ln \relax (x)\,\left (24\,x+12\right )+48\,x^2+16\,x^3-{\ln \relax (x)}^2\,\left (24\,x+6\right )+8}{80\,x+{\ln \relax (x)}^3\,\left (24\,x^2+12\,x\right )-\ln \relax (x)\,\left (32\,x^4+96\,x^3+72\,x^2+16\,x\right )+9\,{\ln \relax (x)}^4-{\ln \relax (x)}^2\,\left (-16\,x^4-16\,x^3+20\,x^2+60\,x+24\right )+132\,x^2+80\,x^3+16\,x^4+16} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(36*x + log(x)*(24*x + 12) + 48*x^2 + 16*x^3 - log(x)^2*(24*x + 6) + 8)/(80*x + log(x)^3*(12*x + 24*x^2)
- log(x)*(16*x + 72*x^2 + 96*x^3 + 32*x^4) + 9*log(x)^4 - log(x)^2*(60*x + 20*x^2 - 16*x^3 - 16*x^4 + 24) + 13
2*x^2 + 80*x^3 + 16*x^4 + 16),x)

[Out]

int(-(36*x + log(x)*(24*x + 12) + 48*x^2 + 16*x^3 - log(x)^2*(24*x + 6) + 8)/(80*x + log(x)^3*(12*x + 24*x^2)
- log(x)*(16*x + 72*x^2 + 96*x^3 + 32*x^4) + 9*log(x)^4 - log(x)^2*(60*x + 20*x^2 - 16*x^3 - 16*x^4 + 24) + 13
2*x^2 + 80*x^3 + 16*x^4 + 16), x)

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sympy [A]  time = 0.21, size = 36, normalized size = 1.44 \begin {gather*} \frac {4 x^{2} + 2 x}{- 4 x^{2} - 10 x + \left (4 x^{2} + 2 x\right ) \log {\relax (x )} + 3 \log {\relax (x )}^{2} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x+6)*ln(x)**2+(-24*x-12)*ln(x)-16*x**3-48*x**2-36*x-8)/(9*ln(x)**4+(24*x**2+12*x)*ln(x)**3+(16*
x**4+16*x**3-20*x**2-60*x-24)*ln(x)**2+(-32*x**4-96*x**3-72*x**2-16*x)*ln(x)+16*x**4+80*x**3+132*x**2+80*x+16)
,x)

[Out]

(4*x**2 + 2*x)/(-4*x**2 - 10*x + (4*x**2 + 2*x)*log(x) + 3*log(x)**2 - 4)

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