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 = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1979, 620, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 1979
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x (a+b x)}} \, dx &=\int \frac {1}{\sqrt {a x+b x^2}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [B] time = 0.00, size = 57, normalized size = 2.04 \[ \frac {2 \sqrt {a} \sqrt {x} \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {x (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 62, normalized size = 2.21 \[ \left [\frac {\log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{\sqrt {b}}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 35, normalized size = 1.25 \[ -\frac {\log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 1.04 \[ \frac {\ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 27, normalized size = 0.96 \[ \frac {\log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 28, normalized size = 1.00 \[ \frac {\ln \left (\frac {\frac {a}{2}+b\,x}{\sqrt {b}}+\sqrt {b\,x^2+a\,x}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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