Optimal. Leaf size=44 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}-\frac {2 \sqrt {x}}{b \sqrt {b x+2}} \]
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Rubi [A] time = 0.01, antiderivative size = 44, 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 = {47, 54, 215} \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}-\frac {2 \sqrt {x}}{b \sqrt {b x+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 54
Rule 215
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{(2+b x)^{3/2}} \, dx &=-\frac {2 \sqrt {x}}{b \sqrt {2+b x}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{b}\\ &=-\frac {2 \sqrt {x}}{b \sqrt {2+b x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {2 \sqrt {x}}{b \sqrt {2+b x}}+\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 1.00 \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}-\frac {2 \sqrt {x}}{b \sqrt {b x+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 50, normalized size = 1.14 \begin {gather*} -\frac {2 \log \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{b^{3/2}}-\frac {2 \sqrt {x}}{b \sqrt {b x+2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 117, normalized size = 2.66 \begin {gather*} \left [\frac {{\left (b x + 2\right )} \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) - 2 \, \sqrt {b x + 2} b \sqrt {x}}{b^{3} x + 2 \, b^{2}}, -\frac {2 \, {\left ({\left (b x + 2\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) + \sqrt {b x + 2} b \sqrt {x}\right )}}{b^{3} x + 2 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 10.61, size = 82, normalized size = 1.86 \begin {gather*} -\frac {{\left (\frac {\log \left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{\sqrt {b}} + \frac {8 \, \sqrt {b}}{{\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b}\right )} {\left | b \right |}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 48, normalized size = 1.09 \begin {gather*} \frac {-\frac {\sqrt {\pi }\, \sqrt {2}\, \sqrt {b}\, \sqrt {x}}{\sqrt {\frac {b x}{2}+1}}+2 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {2}\, \sqrt {b}\, \sqrt {x}}{2}\right )}{\sqrt {\pi }\, b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 57, normalized size = 1.30 \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 )}{b^{\frac {3}{2}}} - \frac {2 \, \sqrt {x}}{\sqrt {b x + 2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x}}{{\left (b\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.56, size = 41, normalized size = 0.93 \begin {gather*} - \frac {2 \sqrt {x}}{b \sqrt {b x + 2}} + \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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