Optimal. Leaf size=29 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {65, 211}
\begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (b+a x)} \, dx &=2 \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.13, size = 92, normalized size = 3.17 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\sqrt {x}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {2 \sqrt {x}}{b},a\text {==}0\right \},\left \{\frac {-2}{a \sqrt {x}},b\text {==}0\right \}\right \},-\frac {\text {Log}\left [\sqrt {x}+\sqrt {-\frac {b}{a}}\right ]}{a \sqrt {-\frac {b}{a}}}+\frac {\text {Log}\left [\sqrt {x}-\sqrt {-\frac {b}{a}}\right ]}{a \sqrt {-\frac {b}{a}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.05, size = 19, normalized size = 0.66
method | result | size |
derivativedivides | \(\frac {2 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(19\) |
default | \(\frac {2 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 18, normalized size = 0.62 \begin {gather*} \frac {2 \, \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 68, normalized size = 2.34 \begin {gather*} \left [-\frac {\sqrt {-a b} \log \left (\frac {a x - b - 2 \, \sqrt {-a b} \sqrt {x}}{a x + b}\right )}{a b}, -\frac {2 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a \sqrt {x}}\right )}{a b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 73, normalized size = 2.52 \begin {gather*} \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{b} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: b = 0 \\\frac {\log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{a \sqrt {- \frac {b}{a}}} - \frac {\log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{a \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 26, normalized size = 0.90 \begin {gather*} \frac {2 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 19, normalized size = 0.66 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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