Optimal. Leaf size=10 \[ \frac {\sin ^{-1}(1+b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {55, 633, 222}
\begin {gather*} \frac {\text {ArcSin}(b x+1)}{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 {-b x} \sqrt {2+b x}} \, dx &=\int \frac {1}{\sqrt {-2 b x-b^2 x^2}} \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 b-2 b^2 x\right )}{2 b^2}\\ &=\frac {\sin ^{-1}(1+b x)}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(10)=20\).
time = 0.01, size = 57, normalized size = 5.70 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {2+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{\sqrt {b} \sqrt {-b x (2+b x)}} \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. \(57\) vs.
\(2(10)=20\).
time = 0.15, size = 58, normalized size = 5.80
method | result | size |
meijerg | \(\frac {2 \sqrt {x}\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {b}\, \sqrt {-b x}}\) | \(27\) |
default | \(\frac {\sqrt {-b x \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1}{b}\right )}{\sqrt {-x^{2} b^{2}-2 b x}}\right )}{\sqrt {-b x}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 18, normalized size = 1.80 \begin {gather*} -\frac {\arcsin \left (-\frac {b^{2} x + 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. 26 vs.
\(2 (10) = 20\).
time = 0.76, size = 26, normalized size = 2.60 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b x}}{b x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.57, size = 24, normalized size = 2.40 \begin {gather*} - \frac {2 i \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 17, normalized size = 1.70 \begin {gather*} -\frac {2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {-b x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 34, normalized size = 3.40 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {b\,x+2}\right )}{\sqrt {-b\,x}\,\sqrt {b^2}}\right )}{\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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