Optimal. Leaf size=26 \[ \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{21}} \sqrt {5 x+3}\right ) \]
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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 = {54, 216} \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 54
Rule 216
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-2 x} \sqrt {3+5 x}} \, dx &=\frac {2 \operatorname {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 = 0.01, size = 45, normalized size = 1.73 \begin {gather*} -\frac {\sqrt {\frac {2}{5}} \sqrt {3-2 x} \sinh ^{-1}\left (\sqrt {\frac {5}{21}} \sqrt {2 x-3}\right )}{\sqrt {2 x-3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 36, normalized size = 1.38 \begin {gather*} -\sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {3-2 x}}{\sqrt {5 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.77, 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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giac [A] time = 0.93, size = 21, normalized size = 0.81 \begin {gather*} \frac {1}{5} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {1}{21} \, \sqrt {42} \sqrt {5 \, x + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 39, normalized size = 1.50 \begin {gather*} \frac {\sqrt {\left (-2 x +3\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{21}-\frac {3}{7}\right )}{10 \sqrt {-2 x +3}\, \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, 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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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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sympy [A] time = 1.07, size = 58, normalized size = 2.23 \begin {gather*} \begin {cases} - \frac {\sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {210} \sqrt {x + \frac {3}{5}}}{21} \right )}}{5} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{21} > 1 \\\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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