Optimal. Leaf size=62 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
[Out]
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Rubi [A] time = 0.0628911, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x,x]
[Out]
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Rubi in Sympy [A] time = 8.92435, size = 49, normalized size = 0.79 \[ \frac{a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x+a)**2)**(1/2)/x,x)
[Out]
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Mathematica [A] time = 0.0166625, size = 27, normalized size = 0.44 \[ \frac{\sqrt{(a+b x)^2} (a \log (x)+b x)}{a+b x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x,x]
[Out]
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Maple [C] time = 0.039, size = 19, normalized size = 0.3 \[{\it csgn} \left ( bx+a \right ) \left ( bx+a+a\ln \left ( bx \right ) \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x+a)^2)^(1/2)/x,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(sqrt((b*x + a)^2)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230485, size = 11, normalized size = 0.18 \[ b x + a \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.191528, size = 7, normalized size = 0.11 \[ a \log{\left (x \right )} + b x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x+a)**2)**(1/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.208518, size = 28, normalized size = 0.45 \[ b x{\rm sign}\left (b x + a\right ) + a{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/x,x, algorithm="giac")
[Out]