Optimal. Leaf size=12 \[ \frac {\sin ^{-1}(5+6 x)}{\sqrt {3}} \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222}
\begin {gather*} \frac {\text {ArcSin}(6 x+5)}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,-5-6 x\right )}{\sqrt {3}}\\ &=\frac {\sin ^{-1}(5+6 x)}{\sqrt {3}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(12)=24\).
time = 0.06, size = 33, normalized size = 2.75 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-2-5 x-3 x^2}}{\sqrt {3} (1+x)}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 12, normalized size = 1.00
method | result | size |
default | \(\frac {\arcsin \left (6 x +5\right ) \sqrt {3}}{3}\) | \(12\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}+3\right ) x +6 \sqrt {-3 x^{2}-5 x -2}-5 \RootOf \left (\textit {\_Z}^{2}+3\right )\right )}{3}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.04, size = 11, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arcsin \left (6 \, x + 5\right ) \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. 40 vs.
\(2 (11) = 22\).
time = 0.38, size = 40, normalized size = 3.33 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} - 5 \, x - 2} {\left (6 \, x + 5\right )}}{6 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}}\right ) \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 {- 3 x^{2} - 5 x - 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (11) = 22\).
time = 0.47, size = 31, normalized size = 2.58 \begin {gather*} \frac {1}{12} \, \sqrt {-3 \, x^{2} - 5 \, x - 2} {\left (6 \, x + 5\right )} + \frac {1}{72} \, \sqrt {3} \arcsin \left (6 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 11, normalized size = 0.92 \begin {gather*} \frac {\sqrt {3}\,\mathrm {asin}\left (6\,x+5\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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