Optimal. Leaf size=22 \[ -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-3-b x}}{\sqrt {5}}\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {65, 221}
\begin {gather*} -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x-3}}{\sqrt {5}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 221
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 {5+x^2}} \, dx,x,\sqrt {-3-b x}\right )}{b}\\ &=-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-3-b x}}{\sqrt {5}}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 27, normalized size = 1.23 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2-b x}}{\sqrt {-3-b x}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \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. \(68\) vs.
\(2(19)=38\).
time = 0.17, size = 69, normalized size = 3.14
method | result | size |
default | \(\frac {\sqrt {\left (-b x -3\right ) \left (-b x +2\right )}\, \ln \left (\frac {\frac {1}{2} b +b^{2} x}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+b x -6}\right )}{\sqrt {-b x -3}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 30, normalized size = 1.36 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + b x - 6} b + b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 30, normalized size = 1.36 \begin {gather*} -\frac {\log \left (-2 \, b x + 2 \, \sqrt {-b x + 2} \sqrt {-b x - 3} - 1\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 [A]
time = 0.00, size = 27, normalized size = 1.23 \begin {gather*} \frac {2 \ln \left (\sqrt {-b x+2}-\sqrt {-b x-3}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 52, normalized size = 2.36 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {-b\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {2-b\,x}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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