3.70.42 \(\int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+(-8 x^2+120 x^3+108 x^4+24 x^5) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+(-4 x^2+40 x^3+36 x^4+8 x^5) \log (x)+4 x^4 \log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ x \left (5-\frac {2 x}{x+\frac {2}{2 x-4 x \left (5+\frac {5}{x}+x+\log (x)\right )}}\right ) \]

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Rubi [F]  time = 1.07, 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 {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+\left (-8 x^2+120 x^3+108 x^4+24 x^5\right ) \log (x)+12 x^4 \log ^2(x)}{1-20 x+82 x^2+176 x^3+121 x^4+36 x^5+4 x^6+\left (-4 x^2+40 x^3+36 x^4+8 x^5\right ) \log (x)+4 x^4 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5 - 60*x + 268*x^2 + 536*x^3 + 363*x^4 + 108*x^5 + 12*x^6 + (-8*x^2 + 120*x^3 + 108*x^4 + 24*x^5)*Log[x]
+ 12*x^4*Log[x]^2)/(1 - 20*x + 82*x^2 + 176*x^3 + 121*x^4 + 36*x^5 + 4*x^6 + (-4*x^2 + 40*x^3 + 36*x^4 + 8*x^5
)*Log[x] + 4*x^4*Log[x]^2),x]

[Out]

3*x + 4*Defer[Int][(-1 + 10*x + 9*x^2 + 2*x^3 + 2*x^2*Log[x])^(-2), x] - 20*Defer[Int][x/(-1 + 10*x + 9*x^2 +
2*x^3 + 2*x^2*Log[x])^2, x] + 4*Defer[Int][x^2/(-1 + 10*x + 9*x^2 + 2*x^3 + 2*x^2*Log[x])^2, x] + 4*Defer[Int]
[x^3/(-1 + 10*x + 9*x^2 + 2*x^3 + 2*x^2*Log[x])^2, x] + 2*Defer[Int][(-1 + 10*x + 9*x^2 + 2*x^3 + 2*x^2*Log[x]
)^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5-60 x+268 x^2+536 x^3+363 x^4+108 x^5+12 x^6+4 x^2 \left (-2+30 x+27 x^2+6 x^3\right ) \log (x)+12 x^4 \log ^2(x)}{\left (1-10 x-9 x^2-2 x^3-2 x^2 \log (x)\right )^2} \, dx\\ &=\int \left (3+\frac {4 \left (1-5 x+x^2+x^3\right )}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2}+\frac {2}{-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)}\right ) \, dx\\ &=3 x+2 \int \frac {1}{-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)} \, dx+4 \int \frac {1-5 x+x^2+x^3}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2} \, dx\\ &=3 x+2 \int \frac {1}{-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)} \, dx+4 \int \left (\frac {1}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2}-\frac {5 x}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2}+\frac {x^2}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2}+\frac {x^3}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2}\right ) \, dx\\ &=3 x+2 \int \frac {1}{-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)} \, dx+4 \int \frac {1}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2} \, dx+4 \int \frac {x^2}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2} \, dx+4 \int \frac {x^3}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2} \, dx-20 \int \frac {x}{\left (-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 31, normalized size = 0.97 \begin {gather*} 3 x-\frac {2 x}{-1+10 x+9 x^2+2 x^3+2 x^2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - 60*x + 268*x^2 + 536*x^3 + 363*x^4 + 108*x^5 + 12*x^6 + (-8*x^2 + 120*x^3 + 108*x^4 + 24*x^5)*L
og[x] + 12*x^4*Log[x]^2)/(1 - 20*x + 82*x^2 + 176*x^3 + 121*x^4 + 36*x^5 + 4*x^6 + (-4*x^2 + 40*x^3 + 36*x^4 +
8*x^5)*Log[x] + 4*x^4*Log[x]^2),x]

[Out]

3*x - (2*x)/(-1 + 10*x + 9*x^2 + 2*x^3 + 2*x^2*Log[x])

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fricas [A]  time = 1.12, size = 51, normalized size = 1.59 \begin {gather*} \frac {6 \, x^{4} + 6 \, x^{3} \log \relax (x) + 27 \, x^{3} + 30 \, x^{2} - 5 \, x}{2 \, x^{3} + 2 \, x^{2} \log \relax (x) + 9 \, x^{2} + 10 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^4*log(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*log(x)+12*x^6+108*x^5+363*x^4+536*x^3+268*x^2-60*x+5
)/(4*x^4*log(x)^2+(8*x^5+36*x^4+40*x^3-4*x^2)*log(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x, algorithm=
"fricas")

[Out]

(6*x^4 + 6*x^3*log(x) + 27*x^3 + 30*x^2 - 5*x)/(2*x^3 + 2*x^2*log(x) + 9*x^2 + 10*x - 1)

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giac [A]  time = 0.20, size = 31, normalized size = 0.97 \begin {gather*} 3 \, x - \frac {2 \, x}{2 \, x^{3} + 2 \, x^{2} \log \relax (x) + 9 \, x^{2} + 10 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^4*log(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*log(x)+12*x^6+108*x^5+363*x^4+536*x^3+268*x^2-60*x+5
)/(4*x^4*log(x)^2+(8*x^5+36*x^4+40*x^3-4*x^2)*log(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x, algorithm=
"giac")

[Out]

3*x - 2*x/(2*x^3 + 2*x^2*log(x) + 9*x^2 + 10*x - 1)

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maple [A]  time = 0.05, size = 32, normalized size = 1.00

method	result	size
risch	\(3 x -\frac {2 x}{2 x^{2} \ln \relax (x )+2 x^{3}+9 x^{2}+10 x -1}\)	\(32\)
norman	\(\frac {-5 x +27 x^{3}+30 x^{2}+6 x^{4}+6 x^{3} \ln \relax (x )}{2 x^{2} \ln \relax (x )+2 x^{3}+9 x^{2}+10 x -1}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^4*ln(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*ln(x)+12*x^6+108*x^5+363*x^4+536*x^3+268*x^2-60*x+5)/(4*x^4
*ln(x)^2+(8*x^5+36*x^4+40*x^3-4*x^2)*ln(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x,method=_RETURNVERBOSE
)

[Out]

3*x-2*x/(2*x^2*ln(x)+2*x^3+9*x^2+10*x-1)

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maxima [A]  time = 0.40, size = 51, normalized size = 1.59 \begin {gather*} \frac {6 \, x^{4} + 6 \, x^{3} \log \relax (x) + 27 \, x^{3} + 30 \, x^{2} - 5 \, x}{2 \, x^{3} + 2 \, x^{2} \log \relax (x) + 9 \, x^{2} + 10 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^4*log(x)^2+(24*x^5+108*x^4+120*x^3-8*x^2)*log(x)+12*x^6+108*x^5+363*x^4+536*x^3+268*x^2-60*x+5
)/(4*x^4*log(x)^2+(8*x^5+36*x^4+40*x^3-4*x^2)*log(x)+4*x^6+36*x^5+121*x^4+176*x^3+82*x^2-20*x+1),x, algorithm=
"maxima")

[Out]

(6*x^4 + 6*x^3*log(x) + 27*x^3 + 30*x^2 - 5*x)/(2*x^3 + 2*x^2*log(x) + 9*x^2 + 10*x - 1)

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mupad [B]  time = 4.25, size = 31, normalized size = 0.97 \begin {gather*} 3\,x-\frac {2\,x}{10\,x+2\,x^2\,\ln \relax (x)+9\,x^2+2\,x^3-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^4*log(x)^2 - 60*x + log(x)*(120*x^3 - 8*x^2 + 108*x^4 + 24*x^5) + 268*x^2 + 536*x^3 + 363*x^4 + 108*
x^5 + 12*x^6 + 5)/(4*x^4*log(x)^2 - 20*x + log(x)*(40*x^3 - 4*x^2 + 36*x^4 + 8*x^5) + 82*x^2 + 176*x^3 + 121*x
^4 + 36*x^5 + 4*x^6 + 1),x)

[Out]

3*x - (2*x)/(10*x + 2*x^2*log(x) + 9*x^2 + 2*x^3 - 1)

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sympy [A]  time = 0.18, size = 29, normalized size = 0.91 \begin {gather*} 3 x - \frac {2 x}{2 x^{3} + 2 x^{2} \log {\relax (x )} + 9 x^{2} + 10 x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**4*ln(x)**2+(24*x**5+108*x**4+120*x**3-8*x**2)*ln(x)+12*x**6+108*x**5+363*x**4+536*x**3+268*x*
*2-60*x+5)/(4*x**4*ln(x)**2+(8*x**5+36*x**4+40*x**3-4*x**2)*ln(x)+4*x**6+36*x**5+121*x**4+176*x**3+82*x**2-20*
x+1),x)

[Out]

3*x - 2*x/(2*x**3 + 2*x**2*log(x) + 9*x**2 + 10*x - 1)

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