3.21.27 \(\int \frac {-900+9 x+1209 x \log (x)-400 x^2 \log ^2(x)}{450 x-600 x^2 \log (x)+200 x^3 \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\frac {1}{x^2}\right )-\frac {3}{25 \left (4-\frac {6}{x \log (x)}\right )} \]

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Rubi [F]  time = 0.54, 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 {-900+9 x+1209 x \log (x)-400 x^2 \log ^2(x)}{450 x-600 x^2 \log (x)+200 x^3 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-900 + 9*x + 1209*x*Log[x] - 400*x^2*Log[x]^2)/(450*x - 600*x^2*Log[x] + 200*x^3*Log[x]^2),x]

[Out]

-2*Log[x] + (9*Defer[Int][(-3 + 2*x*Log[x])^(-2), x])/50 + (27*Defer[Int][1/(x*(-3 + 2*x*Log[x])^2), x])/100 +
(9*Defer[Int][1/(x*(-3 + 2*x*Log[x])), x])/100

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-900+9 x+1209 x \log (x)-400 x^2 \log ^2(x)}{50 x (3-2 x \log (x))^2} \, dx\\ &=\frac {1}{50} \int \frac {-900+9 x+1209 x \log (x)-400 x^2 \log ^2(x)}{x (3-2 x \log (x))^2} \, dx\\ &=\frac {1}{50} \int \left (-\frac {100}{x}+\frac {9 (3+2 x)}{2 x (-3+2 x \log (x))^2}+\frac {9}{2 x (-3+2 x \log (x))}\right ) \, dx\\ &=-2 \log (x)+\frac {9}{100} \int \frac {3+2 x}{x (-3+2 x \log (x))^2} \, dx+\frac {9}{100} \int \frac {1}{x (-3+2 x \log (x))} \, dx\\ &=-2 \log (x)+\frac {9}{100} \int \frac {1}{x (-3+2 x \log (x))} \, dx+\frac {9}{100} \int \left (\frac {2}{(-3+2 x \log (x))^2}+\frac {3}{x (-3+2 x \log (x))^2}\right ) \, dx\\ &=-2 \log (x)+\frac {9}{100} \int \frac {1}{x (-3+2 x \log (x))} \, dx+\frac {9}{50} \int \frac {1}{(-3+2 x \log (x))^2} \, dx+\frac {27}{100} \int \frac {1}{x (-3+2 x \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 22, normalized size = 1.00 \begin {gather*} \frac {1}{50} \left (-100 \log (x)+\frac {9}{2 (3-2 x \log (x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-900 + 9*x + 1209*x*Log[x] - 400*x^2*Log[x]^2)/(450*x - 600*x^2*Log[x] + 200*x^3*Log[x]^2),x]

[Out]

(-100*Log[x] + 9/(2*(3 - 2*x*Log[x])))/50

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fricas [A]  time = 0.72, size = 24, normalized size = 1.09 \begin {gather*} -\frac {400 \, x \log \relax (x)^{2} - 600 \, \log \relax (x) + 9}{100 \, {\left (2 \, x \log \relax (x) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-400*x^2*log(x)^2+1209*x*log(x)+9*x-900)/(200*x^3*log(x)^2-600*x^2*log(x)+450*x),x, algorithm="fric
as")

[Out]

-1/100*(400*x*log(x)^2 - 600*log(x) + 9)/(2*x*log(x) - 3)

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giac [A]  time = 0.20, size = 16, normalized size = 0.73 \begin {gather*} -\frac {9}{100 \, {\left (2 \, x \log \relax (x) - 3\right )}} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-400*x^2*log(x)^2+1209*x*log(x)+9*x-900)/(200*x^3*log(x)^2-600*x^2*log(x)+450*x),x, algorithm="giac
")

[Out]

-9/100/(2*x*log(x) - 3) - 2*log(x)

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maple [A]  time = 0.03, size = 17, normalized size = 0.77

method	result	size
norman	\(-\frac {9}{100 \left (2 x \ln \relax (x )-3\right )}-2 \ln \relax (x )\)	\(17\)
risch	\(-\frac {9}{100 \left (2 x \ln \relax (x )-3\right )}-2 \ln \relax (x )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-400*x^2*ln(x)^2+1209*x*ln(x)+9*x-900)/(200*x^3*ln(x)^2-600*x^2*ln(x)+450*x),x,method=_RETURNVERBOSE)

[Out]

-9/100/(2*x*ln(x)-3)-2*ln(x)

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maxima [A]  time = 0.46, size = 16, normalized size = 0.73 \begin {gather*} -\frac {9}{100 \, {\left (2 \, x \log \relax (x) - 3\right )}} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-400*x^2*log(x)^2+1209*x*log(x)+9*x-900)/(200*x^3*log(x)^2-600*x^2*log(x)+450*x),x, algorithm="maxi
ma")

[Out]

-9/100/(2*x*log(x) - 3) - 2*log(x)

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mupad [B]  time = 1.24, size = 16, normalized size = 0.73 \begin {gather*} -2\,\ln \relax (x)-\frac {9}{100\,\left (2\,x\,\ln \relax (x)-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x - 400*x^2*log(x)^2 + 1209*x*log(x) - 900)/(450*x - 600*x^2*log(x) + 200*x^3*log(x)^2),x)

[Out]

- 2*log(x) - 9/(100*(2*x*log(x) - 3))

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sympy [A]  time = 0.11, size = 15, normalized size = 0.68 \begin {gather*} - 2 \log {\relax (x )} - \frac {9}{200 x \log {\relax (x )} - 300} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-400*x**2*ln(x)**2+1209*x*ln(x)+9*x-900)/(200*x**3*ln(x)**2-600*x**2*ln(x)+450*x),x)

[Out]

-2*log(x) - 9/(200*x*log(x) - 300)

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