Optimal. Leaf size=40 \[ \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.01, 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 = {50, 63, 208} \begin {gather*} \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{-a+b x} \, dx &=\frac {2 \sqrt {x}}{b}+\frac {a \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{b}\\ &=\frac {2 \sqrt {x}}{b}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 40, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 83, normalized size = 2.08 \begin {gather*} \left [\frac {\sqrt {\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) + 2 \, \sqrt {x}}{b}, \frac {2 \, {\left (\sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + \sqrt {x}\right )}}{b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 33, normalized size = 0.82 \begin {gather*} \frac {2 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} b} + \frac {2 \, \sqrt {x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.80 \begin {gather*} -\frac {2 a \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {2 \sqrt {x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 47, normalized size = 1.18 \begin {gather*} \frac {a \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {2 \, \sqrt {x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 28, normalized size = 0.70 \begin {gather*} \frac {2\,\sqrt {x}}{b}-\frac {2\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 87, normalized size = 2.18 \begin {gather*} \begin {cases} \frac {\sqrt {a} \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{2} \sqrt {\frac {1}{b}}} - \frac {\sqrt {a} \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{2} \sqrt {\frac {1}{b}}} + \frac {2 \sqrt {x}}{b} & \text {for}\: b \neq 0 \\- \frac {2 x^{\frac {3}{2}}}{3 a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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