Optimal. Leaf size=11 \[ \frac {\sin ^{-1}(2 b x+3)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 11, 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 = {53, 619, 216} \begin {gather*} \frac {\sin ^{-1}(2 b x+3)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-b x} \sqrt {2+b x}} \, dx &=\int \frac {1}{\sqrt {-2-3 b x-b^2 x^2}} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{b^2}}} \, dx,x,-3 b-2 b^2 x\right )}{b^2}\\ &=\frac {\sin ^{-1}(3+2 b x)}{b}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 49, normalized size = 4.45 \begin {gather*} \frac {2 \sqrt {b x+1} \sqrt {b x+2} \sinh ^{-1}\left (\sqrt {b x+1}\right )}{b \sqrt {-((b x+1) (b x+2))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.05, size = 26, normalized size = 2.36 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-b x-1}}{\sqrt {b x+2}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 44, normalized size = 4.00 \begin {gather*} -\frac {\arctan \left (\frac {{\left (2 \, b x + 3\right )} \sqrt {b x + 2} \sqrt {-b x - 1}}{2 \, {\left (b^{2} x^{2} + 3 \, b x + 2\right )}}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 13, normalized size = 1.18 \begin {gather*} \frac {2 \, \arcsin \left (\sqrt {b x + 2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 66, normalized size = 6.00 \begin {gather*} \frac {\sqrt {\left (-b x -1\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {3}{2 b}\right )}{\sqrt {-b^{2} x^{2}-3 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 = 3.01, size = 21, normalized size = 1.91 \begin {gather*} -\frac {\arcsin \left (-\frac {2 \, b^{2} x + 3 \, b}{b}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 41, normalized size = 3.73 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {-b\,x-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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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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