Optimal. Leaf size=26 \[ \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{21}} \sqrt {3+5 x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {56, 222}
\begin {gather*} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{21}} \sqrt {5 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-2 x} \sqrt {3+5 x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {21-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{\sqrt {5}}\\ &=\sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{21}} \sqrt {3+5 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 31, normalized size = 1.19 \begin {gather*} -\sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {15}{2}-5 x}}{\sqrt {3+5 x}}\right ) \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 = 2.36, size = 43, normalized size = 1.65 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\left (-\frac {I}{5}\right ) \sqrt {10} \text {ArcCosh}\left [\frac {\sqrt {42} \sqrt {3+5 x}}{21}\right ],\text {Abs}\left [\frac {3}{5}+x\right ]>\frac {21}{10}\right \}\right \},\frac {\sqrt {10} \text {ArcSin}\left [\frac {\sqrt {210} \sqrt {\frac {3}{5}+x}}{21}\right ]}{5}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs.
\(2(18)=36\).
time = 0.17, size = 39, normalized size = 1.50
method | result | size |
default | \(\frac {\sqrt {\left (3-2 x \right ) \left (3+5 x \right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{21}-\frac {3}{7}\right )}{10 \sqrt {3-2 x}\, \sqrt {3+5 x}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 11, normalized size = 0.42 \begin {gather*} -\frac {1}{10} \, \sqrt {10} \arcsin \left (-\frac {20}{21} \, x + \frac {3}{7}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (18) = 36\).
time = 0.30, size = 44, normalized size = 1.69 \begin {gather*} -\frac {1}{5} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 3} - 3 \, \sqrt {5} \sqrt {2}}{10 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.60, size = 56, normalized size = 2.15 \begin {gather*} \begin {cases} - \frac {\sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {210} \sqrt {x + \frac {3}{5}}}{21} \right )}}{5} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {21}{10} \\\frac {\sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {210} \sqrt {x + \frac {3}{5}}}{21} \right )}}{5} & \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 = 35, normalized size = 1.35 \begin {gather*} -\frac {2 \sqrt {5} \arcsin \left (\frac {5 \sqrt {-2 x+3}}{\sqrt {105}}\right )}{\sqrt {2}\cdot 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 40, normalized size = 1.54 \begin {gather*} -\frac {2\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {3}-\sqrt {3-2\,x}\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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