Optimal. Leaf size=16 \[ -\frac {\sin ^{-1}\left (\frac {1}{5} (1-2 b x)\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, 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 {\text {ArcSin}\left (\frac {1}{5} (1-2 b x)\right )}{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 {3-b x} \sqrt {2+b x}} \, dx &=\int \frac {1}{\sqrt {6+b x-b^2 x^2}} \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{25 b^2}}} \, dx,x,b-2 b^2 x\right )}{5 b^2}\\ &=-\frac {\sin ^{-1}\left (\frac {1}{5} (1-2 b x)\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 26, normalized size = 1.62 \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. \(64\) vs.
\(2(11)=22\).
time = 0.16, size = 65, normalized size = 4.06
method | result | size |
default | \(\frac {\sqrt {\left (-b x +3\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x -\frac {1}{2 b}\right )}{\sqrt {-x^{2} b^{2}+b x +6}}\right )}{\sqrt {-b x +3}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 21, normalized size = 1.31 \begin {gather*} -\frac {\arcsin \left (-\frac {2 \, b^{2} x - b}{5 \, 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 = 1.00, size = 44, normalized size = 2.75 \begin {gather*} -\frac {\arctan \left (\frac {{\left (2 \, b x - 1\right )} \sqrt {b x + 2} \sqrt {-b x + 3}}{2 \, {\left (b^{2} x^{2} - 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 [A]
time = 1.26, size = 18, normalized size = 1.12 \begin {gather*} \frac {2 \, \arcsin \left (\frac {1}{5} \, \sqrt {5} \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 = 0.08, size = 44, normalized size = 2.75 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {3}-\sqrt {3-b\,x}\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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