Optimal. Leaf size=11 \[ \frac {\sin ^{-1}(3+2 b x)}{b} \]
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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 = {55, 633, 222}
\begin {gather*} \frac {\sin ^{-1}(2 b x+3)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 55
Rule 222
Rule 633
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 {\text {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] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(11)=22\).
time = 0.01, size = 59, normalized size = 5.36 \begin {gather*} \frac {2 \sqrt {1+b x} \sqrt {2+b x} \tanh ^{-1}\left (\frac {\sqrt {2+b x}}{\sqrt {1+b x}}\right )}{b \sqrt {-((1+b x) (2+b x))}} \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 in comparison} \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. \(65\) vs.
\(2(11)=22\).
time = 0.17, size = 66, normalized size = 6.00
method | result | size |
default | \(\frac {\sqrt {\left (-b x -1\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {3}{2 b}\right )}{\sqrt {-x^{2} b^{2}-3 b x -2}}\right )}{\sqrt {-b x -1}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (11) = 22\).
time = 0.30, 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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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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Giac [A]
time = 0.00, size = 16, normalized size = 1.45 \begin {gather*} -\frac {2 \arcsin \left (\sqrt {-b x-1}\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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