3.2413 \(\int \frac{1}{x \sqrt{a^2-2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{\log (x) (a-b x)}{a \sqrt{a^2-2 a b x+b^2 x^2}}-\frac{(a-b x) \log (a-b x)}{a \sqrt{a^2-2 a b x+b^2 x^2}} \]

[Out]

((a - b*x)*Log[x])/(a*Sqrt[a^2 - 2*a*b*x + b^2*x^2]) - ((a - b*x)*Log[a - b*x])/
(a*Sqrt[a^2 - 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.0806786, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\log (x) (a-b x)}{a \sqrt{a^2-2 a b x+b^2 x^2}}-\frac{(a-b x) \log (a-b x)}{a \sqrt{a^2-2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a^2 - 2*a*b*x + b^2*x^2]),x]

[Out]

((a - b*x)*Log[x])/(a*Sqrt[a^2 - 2*a*b*x + b^2*x^2]) - ((a - b*x)*Log[a - b*x])/
(a*Sqrt[a^2 - 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 15.6807, size = 63, normalized size = 0.89 \[ \frac{\sqrt{a^{2} - 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a \left (a - b x\right )} - \frac{\sqrt{a^{2} - 2 a b x + b^{2} x^{2}} \log{\left (a - b x \right )}}{a \left (a - b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/((b*x-a)**2)**(1/2),x)

[Out]

sqrt(a**2 - 2*a*b*x + b**2*x**2)*log(x)/(a*(a - b*x)) - sqrt(a**2 - 2*a*b*x + b*
*2*x**2)*log(a - b*x)/(a*(a - b*x))

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Mathematica [A]  time = 0.0342609, size = 34, normalized size = 0.48 \[ \frac{(a-b x) (\log (x)-\log (a-b x))}{a \sqrt{(a-b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[a^2 - 2*a*b*x + b^2*x^2]),x]

[Out]

((a - b*x)*(Log[x] - Log[a - b*x]))/(a*Sqrt[(a - b*x)^2])

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Maple [A]  time = 0.019, size = 36, normalized size = 0.5 \[{\frac{ \left ( bx-a \right ) \left ( \ln \left ( bx-a \right ) -\ln \left ( x \right ) \right ) }{a}{\frac{1}{\sqrt{ \left ( bx-a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/((b*x-a)^2)^(1/2),x)

[Out]

(b*x-a)*(ln(b*x-a)-ln(x))/((b*x-a)^2)^(1/2)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x - a)^2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221619, size = 23, normalized size = 0.32 \[ \frac{\log \left (b x - a\right ) - \log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x - a)^2)*x),x, algorithm="fricas")

[Out]

(log(b*x - a) - log(x))/a

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Sympy [A]  time = 0.389651, size = 10, normalized size = 0.14 \[ \frac{- \log{\left (x \right )} + \log{\left (- \frac{a}{b} + x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/((b*x-a)**2)**(1/2),x)

[Out]

(-log(x) + log(-a/b + x))/a

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GIAC/XCAS [A]  time = 0.20665, size = 42, normalized size = 0.59 \[{\left (\frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{a} - \frac{{\rm ln}\left ({\left | x \right |}\right )}{a}\right )}{\rm sign}\left (b x - a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x - a)^2)*x),x, algorithm="giac")

[Out]

(ln(abs(b*x - a))/a - ln(abs(x))/a)*sign(b*x - a)