Optimal. Leaf size=144 \[ -\frac {6 (47 x+37)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}-\frac {4632 \sqrt {3 x^2+5 x+2}}{125 (2 x+3)}-\frac {478 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^2}-\frac {2464 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^3}+\frac {3289 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{125 \sqrt {5}} \]
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Rubi [A] time = 0.10, 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 = {822, 834, 806, 724, 206} \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 206
Rule 724
Rule 806
Rule 822
Rule 834
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} \operatorname {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 = 0.06, size = 84, normalized size = 0.58 \begin {gather*} \frac {-9867 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {10 \left (83376 x^4+424938 x^3+792065 x^2+634312 x+181559\right )}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}}}{1875} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 93, normalized size = 0.65 \begin {gather*} \frac {6578 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{125 \sqrt {5}}-\frac {2 \sqrt {3 x^2+5 x+2} \left (83376 x^4+424938 x^3+792065 x^2+634312 x+181559\right )}{375 (x+1) (2 x+3)^3 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, 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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giac [B] time = 0.32, 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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maple [A] time = 0.01, size = 132, normalized size = 0.92 \begin {gather*} -\frac {3289 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{625}-\frac {349}{600 \left (x +\frac {3}{2}\right )^{2} \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {271}{75 \left (x +\frac {3}{2}\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {3289}{250 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {1158 \left (6 x +5\right )}{125 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {13}{120 \left (x +\frac {3}{2}\right )^{3} \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, 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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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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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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