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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]