Optimal. Leaf size=64 \[ -\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}+\frac {47 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{10 \sqrt {5}} \]
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Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {820, 738, 212}
\begin {gather*} \frac {47 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{10 \sqrt {5}}-\frac {13 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 820
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx &=-\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}+\frac {47}{10} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}-\frac {47}{5} \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}+\frac {47 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{10 \sqrt {5}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 61, normalized size = 0.95 \begin {gather*} -\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}+\frac {47 \tanh ^{-1}\left (\frac {\sqrt {2+5 x+3 x^2}}{\sqrt {5} (1+x)}\right )}{5 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 53, normalized size = 0.83
method | result | size |
default | \(-\frac {47 \sqrt {5}\, \arctanh \left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{50}-\frac {13 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{10 \left (x +\frac {3}{2}\right )}\) | \(53\) |
risch | \(-\frac {13 \sqrt {3 x^{2}+5 x +2}}{5 \left (3+2 x \right )}-\frac {47 \sqrt {5}\, \arctanh \left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{50}\) | \(53\) |
trager | \(-\frac {13 \sqrt {3 x^{2}+5 x +2}}{5 \left (3+2 x \right )}+\frac {47 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-5\right ) x +7 \RootOf \left (\textit {\_Z}^{2}-5\right )+10 \sqrt {3 x^{2}+5 x +2}}{3+2 x}\right )}{50}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 64, normalized size = 1.00 \begin {gather*} -\frac {47}{50} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{5 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.24, size = 80, normalized size = 1.25 \begin {gather*} \frac {47 \, \sqrt {5} {\left (2 \, x + 3\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 260 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{100 \, {\left (2 \, x + 3\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 {x}{4 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 12 x \sqrt {3 x^{2} + 5 x + 2} + 9 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{4 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 12 x \sqrt {3 x^{2} + 5 x + 2} + 9 \sqrt {3 x^{2} + 5 x + 2}}\right )\, 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. 127 vs.
\(2 (50) = 100\).
time = 2.42, size = 127, normalized size = 1.98 \begin {gather*} \frac {1}{50} \, \sqrt {5} {\left (13 \, \sqrt {5} \sqrt {3} + 47 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {47 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{50 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {13 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}}{10 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {x-5}{{\left (2\,x+3\right )}^2\,\sqrt {3\,x^2+5\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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