Optimal. Leaf size=40 \[ \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {50, 63, 205} \begin {gather*} \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tan ^{-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 205
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} \tan ^{-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} \tan ^{-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 = 0.02, size = 40, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \tan ^{-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 = 0.98, size = 85, normalized size = 2.12 \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 = 0.99, size = 31, normalized size = 0.78 \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 = 0.01, size = 32, normalized size = 0.80 \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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maxima [A] time = 2.97, size = 31, normalized size = 0.78 \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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mupad [B] time = 0.04, size = 28, normalized size = 0.70 \begin {gather*} \frac {2\,\sqrt {x}}{b}-\frac {2\,\sqrt {a}\,\mathrm {atan}\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 = 0.72, size = 92, normalized size = 2.30 \begin {gather*} \begin {cases} \frac {i \sqrt {a} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{2} \sqrt {\frac {1}{b}}} - \frac {i \sqrt {a} \log {\left (i \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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