Optimal. Leaf size=28 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0218977, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a*x + b*x^2],x]
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Rubi in Sympy [A] time = 1.16208, size = 26, normalized size = 0.93 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a*x)**(1/2),x)
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Mathematica [A] time = 0.0294724, size = 56, normalized size = 2. \[ \frac{2 \sqrt{x} \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b} \sqrt{x (a+b x)}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a*x + b*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 29, normalized size = 1. \[{1\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a*x)^(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^2 + a*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272986, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left ({\left (2 \, b x + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{2} + a x} b\right )}{\sqrt{b}}, \frac{2 \, \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right )}{\sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^2 + a*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x + b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.273711, size = 47, normalized size = 1.68 \[ -\frac{{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^2 + a*x),x, algorithm="giac")
[Out]