Optimal. Leaf size=40 \[ \frac {2 \sqrt {x}}{a}-\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {263, 50, 63, 205} \[ \frac {2 \sqrt {x}}{a}-\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right ) \sqrt {x}} \, dx &=\int \frac {\sqrt {x}}{b+a x} \, dx\\ &=\frac {2 \sqrt {x}}{a}-\frac {b \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{a}\\ &=\frac {2 \sqrt {x}}{a}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {2 \sqrt {x}}{a}-\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {2 \sqrt {x}}{a}-\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 85, normalized size = 2.12 \[ \left [\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {a x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, \sqrt {x}}{a}, -\frac {2 \, {\left (\sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) - \sqrt {x}\right )}}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 31, normalized size = 0.78 \[ -\frac {2 \, b \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {2 \, \sqrt {x}}{a} \]
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 \[ -\frac {2 b \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {2 \sqrt {x}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.65, size = 31, normalized size = 0.78 \[ \frac {2 \, b \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a} + \frac {2 \, \sqrt {x}}{a} \]
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 \[ \frac {2\,\sqrt {x}}{a}-\frac {2\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.35, size = 92, normalized size = 2.30 \[ \begin {cases} \frac {2 \sqrt {x}}{a} + \frac {i \sqrt {b} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{a^{2} \sqrt {\frac {1}{a}}} - \frac {i \sqrt {b} \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{a^{2} \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\\frac {2 x^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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