3.74.16 \(\int \frac {4 x-8 x \log (x)+84 \log ^2(x)}{5 x^4+(10 x^2-210 x^3) \log (x)+(5-210 x+2205 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {4 x}{5 \left (x+x \left (-21 x+\frac {x^2}{\log (x)}\right )\right )} \]

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Rubi [F]  time = 0.99, 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 {4 x-8 x \log (x)+84 \log ^2(x)}{5 x^4+\left (10 x^2-210 x^3\right ) \log (x)+\left (5-210 x+2205 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x - 8*x*Log[x] + 84*Log[x]^2)/(5*x^4 + (10*x^2 - 210*x^3)*Log[x] + (5 - 210*x + 2205*x^2)*Log[x]^2),x]

[Out]

4/(5*(1 - 21*x)) + (4*Defer[Int][(x^2 + Log[x] - 21*x*Log[x])^(-2), x])/46305 + (4*Defer[Int][x/(x^2 + Log[x]
- 21*x*Log[x])^2, x])/5 - (4*Defer[Int][x^2/(x^2 + Log[x] - 21*x*Log[x])^2, x])/105 + (4*Defer[Int][1/((-1 + 2
1*x)^2*(x^2 + Log[x] - 21*x*Log[x])^2), x])/46305 + (8*Defer[Int][1/((-1 + 21*x)*(x^2 + Log[x] - 21*x*Log[x])^
2), x])/46305 - (8*Defer[Int][1/((-1 + 21*x)^2*(x^2 + Log[x] - 21*x*Log[x])), x])/105 - (8*Defer[Int][1/((-1 +
21*x)*(x^2 + Log[x] - 21*x*Log[x])), x])/105

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (x-2 x \log (x)+21 \log ^2(x)\right )}{5 \left (x^2+\log (x)-21 x \log (x)\right )^2} \, dx\\ &=\frac {4}{5} \int \frac {x-2 x \log (x)+21 \log ^2(x)}{\left (x^2+\log (x)-21 x \log (x)\right )^2} \, dx\\ &=\frac {4}{5} \int \left (\frac {21}{(-1+21 x)^2}+\frac {x \left (1-42 x+443 x^2-21 x^3\right )}{(1-21 x)^2 \left (x^2+\log (x)-21 x \log (x)\right )^2}-\frac {2 x}{(-1+21 x)^2 \left (x^2+\log (x)-21 x \log (x)\right )}\right ) \, dx\\ &=\frac {4}{5 (1-21 x)}+\frac {4}{5} \int \frac {x \left (1-42 x+443 x^2-21 x^3\right )}{(1-21 x)^2 \left (x^2+\log (x)-21 x \log (x)\right )^2} \, dx-\frac {8}{5} \int \frac {x}{(-1+21 x)^2 \left (x^2+\log (x)-21 x \log (x)\right )} \, dx\\ &=\frac {4}{5 (1-21 x)}+\frac {4}{5} \int \left (\frac {1}{9261 \left (x^2+\log (x)-21 x \log (x)\right )^2}+\frac {x}{\left (x^2+\log (x)-21 x \log (x)\right )^2}-\frac {x^2}{21 \left (x^2+\log (x)-21 x \log (x)\right )^2}+\frac {1}{9261 (-1+21 x)^2 \left (x^2+\log (x)-21 x \log (x)\right )^2}+\frac {2}{9261 (-1+21 x) \left (x^2+\log (x)-21 x \log (x)\right )^2}\right ) \, dx-\frac {8}{5} \int \left (\frac {1}{21 (-1+21 x)^2 \left (x^2+\log (x)-21 x \log (x)\right )}+\frac {1}{21 (-1+21 x) \left (x^2+\log (x)-21 x \log (x)\right )}\right ) \, dx\\ &=\frac {4}{5 (1-21 x)}+\frac {4 \int \frac {1}{\left (x^2+\log (x)-21 x \log (x)\right )^2} \, dx}{46305}+\frac {4 \int \frac {1}{(-1+21 x)^2 \left (x^2+\log (x)-21 x \log (x)\right )^2} \, dx}{46305}+\frac {8 \int \frac {1}{(-1+21 x) \left (x^2+\log (x)-21 x \log (x)\right )^2} \, dx}{46305}-\frac {4}{105} \int \frac {x^2}{\left (x^2+\log (x)-21 x \log (x)\right )^2} \, dx-\frac {8}{105} \int \frac {1}{(-1+21 x)^2 \left (x^2+\log (x)-21 x \log (x)\right )} \, dx-\frac {8}{105} \int \frac {1}{(-1+21 x) \left (x^2+\log (x)-21 x \log (x)\right )} \, dx+\frac {4}{5} \int \frac {x}{\left (x^2+\log (x)-21 x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 19, normalized size = 0.83 \begin {gather*} \frac {4 \log (x)}{5 \left (x^2+\log (x)-21 x \log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x - 8*x*Log[x] + 84*Log[x]^2)/(5*x^4 + (10*x^2 - 210*x^3)*Log[x] + (5 - 210*x + 2205*x^2)*Log[x]^
2),x]

[Out]

(4*Log[x])/(5*(x^2 + Log[x] - 21*x*Log[x]))

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fricas [A]  time = 0.54, size = 19, normalized size = 0.83 \begin {gather*} \frac {4 \, \log \relax (x)}{5 \, {\left (x^{2} - {\left (21 \, x - 1\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((84*log(x)^2-8*x*log(x)+4*x)/((2205*x^2-210*x+5)*log(x)^2+(-210*x^3+10*x^2)*log(x)+5*x^4),x, algorit
hm="fricas")

[Out]

4/5*log(x)/(x^2 - (21*x - 1)*log(x))

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giac [A]  time = 0.15, size = 44, normalized size = 1.91 \begin {gather*} \frac {4 \, x^{2}}{5 \, {\left (21 \, x^{3} - 441 \, x^{2} \log \relax (x) - x^{2} + 42 \, x \log \relax (x) - \log \relax (x)\right )}} - \frac {4}{5 \, {\left (21 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((84*log(x)^2-8*x*log(x)+4*x)/((2205*x^2-210*x+5)*log(x)^2+(-210*x^3+10*x^2)*log(x)+5*x^4),x, algorit
hm="giac")

[Out]

4/5*x^2/(21*x^3 - 441*x^2*log(x) - x^2 + 42*x*log(x) - log(x)) - 4/5/(21*x - 1)

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maple [A]  time = 0.04, size = 18, normalized size = 0.78

method	result	size
norman	\(\frac {4 \ln \relax (x )}{5 \left (x^{2}-21 x \ln \relax (x )+\ln \relax (x )\right )}\)	\(18\)
risch	\(-\frac {4}{5 \left (21 x -1\right )}+\frac {4 x^{2}}{5 \left (21 x -1\right ) \left (x^{2}-21 x \ln \relax (x )+\ln \relax (x )\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((84*ln(x)^2-8*x*ln(x)+4*x)/((2205*x^2-210*x+5)*ln(x)^2+(-210*x^3+10*x^2)*ln(x)+5*x^4),x,method=_RETURNVERB
OSE)

[Out]

4/5*ln(x)/(x^2-21*x*ln(x)+ln(x))

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maxima [A]  time = 0.38, size = 19, normalized size = 0.83 \begin {gather*} \frac {4 \, \log \relax (x)}{5 \, {\left (x^{2} - {\left (21 \, x - 1\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((84*log(x)^2-8*x*log(x)+4*x)/((2205*x^2-210*x+5)*log(x)^2+(-210*x^3+10*x^2)*log(x)+5*x^4),x, algorit
hm="maxima")

[Out]

4/5*log(x)/(x^2 - (21*x - 1)*log(x))

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mupad [B]  time = 4.59, size = 21, normalized size = 0.91 \begin {gather*} \frac {4\,\ln \relax (x)}{5\,\left (\ln \relax (x)-21\,x\,\ln \relax (x)+x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + 84*log(x)^2 - 8*x*log(x))/(log(x)*(10*x^2 - 210*x^3) + log(x)^2*(2205*x^2 - 210*x + 5) + 5*x^4),x)

[Out]

(4*log(x))/(5*(log(x) - 21*x*log(x) + x^2))

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sympy [A]  time = 0.23, size = 36, normalized size = 1.57 \begin {gather*} - \frac {4 x^{2}}{- 105 x^{3} + 5 x^{2} + \left (2205 x^{2} - 210 x + 5\right ) \log {\relax (x )}} - \frac {84}{2205 x - 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((84*ln(x)**2-8*x*ln(x)+4*x)/((2205*x**2-210*x+5)*ln(x)**2+(-210*x**3+10*x**2)*ln(x)+5*x**4),x)

[Out]

-4*x**2/(-105*x**3 + 5*x**2 + (2205*x**2 - 210*x + 5)*log(x)) - 84/(2205*x - 105)

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