Optimal. Leaf size=144 \[ -\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {4632 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}+\frac {3289 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{125 \sqrt {5}} \]
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Rubi [A]
time = 0.06, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {836, 848, 820,
738, 212} \begin {gather*} -\frac {6 (47 x+37)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}-\frac {4632 \sqrt {3 x^2+5 x+2}}{125 (2 x+3)}-\frac {478 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^2}-\frac {2464 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^3}+\frac {3289 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{125 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 820
Rule 836
Rule 848
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2}{5} \int \frac {653+846 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}+\frac {2}{75} \int \frac {-5113-7392 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {1}{375} \int \frac {19035+35850 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {4632 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}+\frac {3289}{125} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {4632 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}-\frac {6578}{125} \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}-\frac {2464 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {478 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac {4632 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}+\frac {3289 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{125 \sqrt {5}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 90, normalized size = 0.62 \begin {gather*} \frac {2 \left (-\frac {5 \sqrt {2+5 x+3 x^2} \left (181559+634312 x+792065 x^2+424938 x^3+83376 x^4\right )}{(1+x) (3+2 x)^3 (2+3 x)}+9867 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )\right )}{1875} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 132, normalized size = 0.92
method | result | size |
risch | \(-\frac {2 \left (83376 x^{4}+424938 x^{3}+792065 x^{2}+634312 x +181559\right )}{375 \left (3+2 x \right )^{3} \sqrt {3 x^{2}+5 x +2}}-\frac {3289 \sqrt {5}\, \arctanh \left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{625}\) | \(73\) |
trager | \(-\frac {2 \left (83376 x^{4}+424938 x^{3}+792065 x^{2}+634312 x +181559\right )}{375 \left (3+2 x \right )^{3} \sqrt {3 x^{2}+5 x +2}}+\frac {3289 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-5\right ) x +7 \RootOf \left (\textit {\_Z}^{2}-5\right )+10 \sqrt {3 x^{2}+5 x +2}}{3+2 x}\right )}{625}\) | \(92\) |
default | \(-\frac {13}{120 \left (x +\frac {3}{2}\right )^{3} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {349}{600 \left (x +\frac {3}{2}\right )^{2} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {271}{75 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}+\frac {3289}{250 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {1158 \left (5+6 x \right )}{125 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {3289 \sqrt {5}\, \arctanh \left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{625}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 225, normalized size = 1.56 \begin {gather*} -\frac {3289}{625} \, \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 {6948 \, x}{125 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {8291}{250 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13}{15 \, {\left (8 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + 36 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {349}{150 \, {\left (4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {542}{75 \, {\left (2 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.37, size = 140, normalized size = 0.97 \begin {gather*} \frac {9867 \, \sqrt {5} {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\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 ) - 20 \, {\left (83376 \, x^{4} + 424938 \, x^{3} + 792065 \, x^{2} + 634312 \, x + 181559\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3750 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\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}{48 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 368 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 1160 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 1920 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1755 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 837 x \sqrt {3 x^{2} + 5 x + 2} + 162 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{48 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 368 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 1160 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 1920 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1755 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 837 x \sqrt {3 x^{2} + 5 x + 2} + 162 \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. 276 vs.
\(2 (118) = 236\).
time = 1.47, size = 276, normalized size = 1.92 \begin {gather*} \frac {3289}{625} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {6 \, {\left (4209 \, x + 2959\right )}}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {2 \, {\left (118356 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 851850 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 6938110 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 8824815 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 15944775 \, \sqrt {3} x + 3678471 \, \sqrt {3} - 15944775 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{1875 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \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 = -0.01 \begin {gather*} -\int \frac {x-5}{{\left (2\,x+3\right )}^4\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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