3.162 \(\int \frac {1}{(5-6 x)^2 x^2} \, dx\)

Optimal. Leaf size=35 \[ \frac {6}{25 (5-6 x)}-\frac {1}{25 x}-\frac {12}{125} \log (5-6 x)+\frac {12 \log (x)}{125} \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {44} \[ \frac {6}{25 (5-6 x)}-\frac {1}{25 x}-\frac {12}{125} \log (5-6 x)+\frac {12 \log (x)}{125} \]

Antiderivative was successfully verified.

[In]

Int[1/((5 - 6*x)^2*x^2),x]

[Out]

6/(25*(5 - 6*x)) - 1/(25*x) - (12*Log[5 - 6*x])/125 + (12*Log[x])/125

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {1}{(5-6 x)^2 x^2} \, dx &=\int \left (\frac {1}{25 x^2}+\frac {12}{125 x}+\frac {36}{25 (-5+6 x)^2}-\frac {72}{125 (-5+6 x)}\right ) \, dx\\ &=\frac {6}{25 (5-6 x)}-\frac {1}{25 x}-\frac {12}{125} \log (5-6 x)+\frac {12 \log (x)}{125}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 0.89 \[ \frac {1}{125} \left (\frac {30}{5-6 x}-\frac {5}{x}-12 \log (5-6 x)+12 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((5 - 6*x)^2*x^2),x]

[Out]

(30/(5 - 6*x) - 5/x - 12*Log[5 - 6*x] + 12*Log[x])/125

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IntegrateAlgebraic [A]  time = 0.02, size = 36, normalized size = 1.03 \[ \frac {5-12 x}{25 x (6 x-5)}+\frac {12 \log (x)}{125}-\frac {12}{125} \log (6 x-5) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((5 - 6*x)^2*x^2),x]

[Out]

(5 - 12*x)/(25*x*(-5 + 6*x)) + (12*Log[x])/125 - (12*Log[-5 + 6*x])/125

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fricas [A]  time = 0.74, size = 48, normalized size = 1.37 \[ -\frac {12 \, {\left (6 \, x^{2} - 5 \, x\right )} \log \left (6 \, x - 5\right ) - 12 \, {\left (6 \, x^{2} - 5 \, x\right )} \log \relax (x) + 60 \, x - 25}{125 \, {\left (6 \, x^{2} - 5 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5-6*x)^2/x^2,x, algorithm="fricas")

[Out]

-1/125*(12*(6*x^2 - 5*x)*log(6*x - 5) - 12*(6*x^2 - 5*x)*log(x) + 60*x - 25)/(6*x^2 - 5*x)

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giac [A]  time = 0.98, size = 40, normalized size = 1.14 \[ -\frac {6}{25 \, {\left (6 \, x - 5\right )}} + \frac {6}{125 \, {\left (\frac {5}{6 \, x - 5} + 1\right )}} + \frac {12}{125} \, \log \left ({\left | -\frac {5}{6 \, x - 5} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5-6*x)^2/x^2,x, algorithm="giac")

[Out]

-6/25/(6*x - 5) + 6/125/(5/(6*x - 5) + 1) + 12/125*log(abs(-5/(6*x - 5) - 1))

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maple [A]  time = 0.31, size = 28, normalized size = 0.80

method	result	size
default	\(-\frac {1}{25 x}+\frac {12 \ln \relax (x )}{125}-\frac {6}{25 \left (-5+6 x \right )}-\frac {12 \ln \left (-5+6 x \right )}{125}\)	\(28\)
risch	\(\frac {-\frac {12 x}{25}+\frac {1}{5}}{x \left (-5+6 x \right )}+\frac {12 \ln \relax (x )}{125}-\frac {12 \ln \left (-5+6 x \right )}{125}\)	\(31\)
norman	\(\frac {\frac {1}{5}-\frac {72 x^{2}}{125}}{x \left (-5+6 x \right )}+\frac {12 \ln \relax (x )}{125}-\frac {12 \ln \left (-5+6 x \right )}{125}\)	\(32\)
meijerg	\(\frac {108 x}{625 \left (3-\frac {18 x}{5}\right )}-\frac {12 \ln \left (1-\frac {6 x}{5}\right )}{125}+\frac {6}{125}+\frac {12 \ln \relax (x )}{125}+\frac {12 \ln \relax (2)}{125}+\frac {12 \ln \relax (3)}{125}-\frac {12 \ln \relax (5)}{125}+\frac {12 i \pi }{125}-\frac {1}{25 x}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5-6*x)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

-1/25/x+12/125*ln(x)-6/25/(-5+6*x)-12/125*ln(-5+6*x)

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maxima [A]  time = 0.43, size = 31, normalized size = 0.89 \[ -\frac {12 \, x - 5}{25 \, {\left (6 \, x^{2} - 5 \, x\right )}} - \frac {12}{125} \, \log \left (6 \, x - 5\right ) + \frac {12}{125} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5-6*x)^2/x^2,x, algorithm="maxima")

[Out]

-1/25*(12*x - 5)/(6*x^2 - 5*x) - 12/125*log(6*x - 5) + 12/125*log(x)

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mupad [B]  time = 0.22, size = 34, normalized size = 0.97 \[ \frac {1}{5\,x\,\left (6\,x-5\right )}-\frac {12}{25\,\left (6\,x-5\right )}-\frac {12\,\ln \left (\frac {6\,x-5}{x}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(6*x - 5)^2),x)

[Out]

1/(5*x*(6*x - 5)) - 12/(25*(6*x - 5)) - (12*log((6*x - 5)/x))/125

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sympy [A]  time = 0.13, size = 29, normalized size = 0.83 \[ \frac {5 - 12 x}{150 x^{2} - 125 x} + \frac {12 \log {\relax (x )}}{125} - \frac {12 \log {\left (x - \frac {5}{6} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5-6*x)**2/x**2,x)

[Out]

(5 - 12*x)/(150*x**2 - 125*x) + 12*log(x)/125 - 12*log(x - 5/6)/125

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