Optimal. Leaf size=22 \[ -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x-1}}{\sqrt {3}}\right )}{b} \]
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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 = {63, 215} \begin {gather*} -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x-1}}{\sqrt {3}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 215
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+x^2}} \, dx,x,\sqrt {-1-b x}\right )}{b}\\ &=-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-1-b x}}{\sqrt {3}}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.82 \begin {gather*} -\frac {2 \sqrt {-b x-1} \sin ^{-1}\left (\frac {\sqrt {b x+1}}{\sqrt {3}}\right )}{b \sqrt {b x+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 27, normalized size = 1.23 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2-b x}}{\sqrt {-b x-1}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 30, normalized size = 1.36 \begin {gather*} -\frac {\log \left (-2 \, b x + 2 \, \sqrt {-b x + 2} \sqrt {-b x - 1} + 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 25, normalized size = 1.14 \begin {gather*} \frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x - 1}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 70, normalized size = 3.18 \begin {gather*} \frac {\sqrt {\left (-b x -1\right ) \left (-b x +2\right )}\, \ln \left (\frac {b^{2} x -\frac {1}{2} b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}-b x -2}\right )}{\sqrt {-b x -1}\, \sqrt {-b x +2}\, \sqrt {b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 33, normalized size = 1.50 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - b x - 2} b - b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 46, normalized size = 2.09 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {-b\,x-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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- b x - 1} \sqrt {- b x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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