Optimal. Leaf size=24 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {54, 215} \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 215
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 55, normalized size = 2.29 \[ \left [\frac {\log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{\sqrt {b}}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 46, normalized size = 1.92 \[ \frac {\sqrt {\left (b x +2\right ) x}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{\sqrt {b x +2}\, \sqrt {b}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.96, size = 41, normalized size = 1.71 \[ -\frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 30, normalized size = 1.25 \[ \frac {4\,\mathrm {atan}\left (\frac {\sqrt {2}-\sqrt {b\,x+2}}{\sqrt {-b}\,\sqrt {x}}\right )}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.02, size = 24, normalized size = 1.00 \[ \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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