Optimal. Leaf size=26 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-3-b x}}{\sqrt {2+b x}}\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {65, 223, 209}
\begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {\sqrt {-b x-3}}{\sqrt {b x+2}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 223
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3-b x} \sqrt {2+b x}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2}} \, dx,x,\sqrt {-3-b x}\right )}{b}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {-3-b x}}{\sqrt {2+b x}}\right )}{b}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {-3-b x}}{\sqrt {2+b x}}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-3-b x}}{\sqrt {2+b x}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs.
\(2(22)=44\).
time = 0.16, size = 66, normalized size = 2.54
method | result | size |
default | \(\frac {\sqrt {\left (-b x -3\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {5}{2 b}\right )}{\sqrt {-x^{2} b^{2}-5 b x -6}}\right )}{\sqrt {-b x -3}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 21, normalized size = 0.81 \begin {gather*} -\frac {\arcsin \left (-\frac {2 \, b^{2} x + 5 \, b}{b}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.99, size = 44, normalized size = 1.69 \begin {gather*} -\frac {\arctan \left (\frac {{\left (2 \, b x + 5\right )} \sqrt {b x + 2} \sqrt {-b x - 3}}{2 \, {\left (b^{2} x^{2} + 5 \, b x + 6\right )}}\right )}{b} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {- b x - 3} \sqrt {b x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 2.16, size = 23, normalized size = 0.88 \begin {gather*} \frac {2 i \, \log \left (\sqrt {b x + 3} - \sqrt {b x + 2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 47, normalized size = 1.81 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {-b\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {b\,x+2}\right )\,\sqrt {b^2}}\right )}{\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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