Optimal. Leaf size=89 \[ -\frac {73 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)}-\frac {13 \sqrt {3 x^2+5 x+2}}{10 (2 x+3)^2}+\frac {389 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{100 \sqrt {5}} \]
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Rubi [A] time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \begin {gather*} -\frac {73 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)}-\frac {13 \sqrt {3 x^2+5 x+2}}{10 (2 x+3)^2}+\frac {389 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{100 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx &=-\frac {13 \sqrt {2+5 x+3 x^2}}{10 (3+2 x)^2}-\frac {1}{10} \int \frac {-\frac {29}{2}+39 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{10 (3+2 x)^2}-\frac {73 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {389}{100} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{10 (3+2 x)^2}-\frac {73 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}-\frac {389}{50} \operatorname {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}}{10 (3+2 x)^2}-\frac {73 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {389 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{100 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 69, normalized size = 0.78 \begin {gather*} \frac {1}{500} \left (-\frac {10 \sqrt {3 x^2+5 x+2} (292 x+503)}{(2 x+3)^2}-389 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 66, normalized size = 0.74 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} (-292 x-503)}{50 (2 x+3)^2}+\frac {389 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{50 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 95, normalized size = 1.07 \begin {gather*} \frac {389 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\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 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (292 \, x + 503\right )}}{1000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 206, normalized size = 2.31 \begin {gather*} \frac {389}{500} \, \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 {778 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 3551 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 13217 \, \sqrt {3} x + 4971 \, \sqrt {3} - 13217 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{50 \, {\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 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.83 \begin {gather*} -\frac {389 \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 )}{500}-\frac {73 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{50 \left (x +\frac {3}{2}\right )}-\frac {13 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{40 \left (x +\frac {3}{2}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 90, normalized size = 1.01 \begin {gather*} -\frac {389}{500} \, \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}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {73 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{25 \, {\left (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 = -0.01 \begin {gather*} -\int \frac {x-5}{{\left (2\,x+3\right )}^3\,\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{8 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 54 x \sqrt {3 x^{2} + 5 x + 2} + 27 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{8 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 54 x \sqrt {3 x^{2} + 5 x + 2} + 27 \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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