Optimal. Leaf size=40 \[ \sqrt {x} \sqrt {2+b x}+\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.00, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 56, 221}
\begin {gather*} \sqrt {x} \sqrt {b x+2}+\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {\sqrt {2+b x}}{\sqrt {x}} \, dx &=\sqrt {x} \sqrt {2+b x}+\int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx\\ &=\sqrt {x} \sqrt {2+b x}+2 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x} \sqrt {2+b x}+\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 46, normalized size = 1.15 \begin {gather*} \sqrt {x} \sqrt {2+b x}-\frac {2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 58, normalized size = 1.45
method | result | size |
meijerg | \(-\frac {-\sqrt {\pi }\, \sqrt {b}\, \sqrt {x}\, \sqrt {2}\, \sqrt {\frac {b x}{2}+1}-2 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {b}\, \sqrt {\pi }}\) | \(49\) |
default | \(\sqrt {x}\, \sqrt {b x +2}+\frac {\sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{\sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(58\) |
risch | \(\sqrt {x}\, \sqrt {b x +2}+\frac {\sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{\sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (29) = 58\).
time = 0.53, size = 68, normalized size = 1.70 \begin {gather*} -\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}} - \frac {2 \, \sqrt {b x + 2}}{{\left (b - \frac {b x + 2}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 86, normalized size = 2.15 \begin {gather*} \left [\frac {\sqrt {b x + 2} b \sqrt {x} + \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{b}, \frac {\sqrt {b x + 2} b \sqrt {x} - 2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.88, size = 37, normalized size = 0.92 \begin {gather*} \sqrt {x} \sqrt {b x + 2} + \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 40, normalized size = 1.00 \begin {gather*} \sqrt {x}\,\sqrt {b\,x+2}-\frac {4\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {2}-\sqrt {b\,x+2}}\right )}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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