3.2412 \(\int \frac{1}{x \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{\log (x) (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}} \]

[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 = 0.0789686, antiderivative size = 68, 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 = 14.803, size = 63, normalized size = 0.93 \[ \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a \left (a + b x\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a \left (a + b x\right )} \]

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 = 0.0302518, size = 31, normalized size = 0.46 \[ \frac{(a+b x) (\log (x)-\log (a+b x))}{a \sqrt{(a+b x)^2}} \]

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 = 0.006, size = 30, normalized size = 0.4 \[{\frac{ \left ( bx+a \right ) \left ( \ln \left ( x \right ) -\ln \left ( bx+a \right ) \right ) }{a}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

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(x)-ln(b*x+a))/((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 = 0.219408, size = 22, 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 = 0.374341, size = 10, normalized size = 0.15 \[ \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a} \]

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 = 0.205319, size = 38, normalized size = 0.56 \[ -{\left (\frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{a} - \frac{{\rm ln}\left ({\left | x \right |}\right )}{a}\right )}{\rm sign}\left (b x + a\right ) \]

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)