Optimal. Leaf size=35 \[ \sqrt {x+1} \sqrt {3 x+2}-\frac {\sinh ^{-1}\left (\sqrt {3 x+2}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {26, 50, 54, 215} \[ \sqrt {x+1} \sqrt {3 x+2}-\frac {\sinh ^{-1}\left (\sqrt {3 x+2}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 26
Rule 50
Rule 54
Rule 215
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x} \sqrt {2+3 x}}{\sqrt {1-x^2}} \, dx &=\int \frac {\sqrt {2+3 x}}{\sqrt {1+x}} \, dx\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {1}{2} \int \frac {1}{\sqrt {1+x} \sqrt {2+3 x}} \, dx\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {2+3 x}\right )}{\sqrt {3}}\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {\sinh ^{-1}\left (\sqrt {2+3 x}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.40 \[ \frac {3 \sqrt {x+1} (3 x+2)-\sqrt {9 x+6} \sinh ^{-1}\left (\sqrt {3 x+2}\right )}{3 \sqrt {3 x+2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 96, normalized size = 2.74 \[ \frac {\sqrt {3} {\left (x - 1\right )} \log \left (-\frac {72 \, x^{3} + 4 \, \sqrt {3} \sqrt {-x^{2} + 1} {\left (6 \, x + 5\right )} \sqrt {3 \, x + 2} \sqrt {-x + 1} + 48 \, x^{2} - 71 \, x - 49}{x - 1}\right ) - 12 \, \sqrt {-x^{2} + 1} \sqrt {3 \, x + 2} \sqrt {-x + 1}}{12 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {3 \, x + 2} \sqrt {-x + 1}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 86, normalized size = 2.46 \[ \frac {\sqrt {-x +1}\, \sqrt {3 x +2}\, \sqrt {-x^{2}+1}\, \left (\sqrt {3}\, \ln \left (\sqrt {3}\, x +\frac {5 \sqrt {3}}{6}+\sqrt {3 x^{2}+5 x +2}\right )-6 \sqrt {3 x^{2}+5 x +2}\right )}{6 \left (x -1\right ) \sqrt {3 x^{2}+5 x +2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {3 \, x + 2} \sqrt {-x + 1}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {3\,x+2}\,\sqrt {1-x}}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - x} \sqrt {3 x + 2}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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