Optimal. Leaf size=164 \[ -\frac {15891 \sqrt {3 x^2+5 x+2}}{6250 (2 x+3)}-\frac {1007 \sqrt {3 x^2+5 x+2}}{600 (2 x+3)^2}-\frac {2321 \sqrt {3 x^2+5 x+2}}{1875 (2 x+3)^3}-\frac {443 \sqrt {3 x^2+5 x+2}}{500 (2 x+3)^4}-\frac {13 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^5}+\frac {128381 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{50000 \sqrt {5}} \]
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Rubi [A] time = 0.14, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \[ -\frac {15891 \sqrt {3 x^2+5 x+2}}{6250 (2 x+3)}-\frac {1007 \sqrt {3 x^2+5 x+2}}{600 (2 x+3)^2}-\frac {2321 \sqrt {3 x^2+5 x+2}}{1875 (2 x+3)^3}-\frac {443 \sqrt {3 x^2+5 x+2}}{500 (2 x+3)^4}-\frac {13 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^5}+\frac {128381 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{50000 \sqrt {5}} \]
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)^6 \sqrt {2+5 x+3 x^2}} \, dx &=-\frac {13 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac {1}{25} \int \frac {\frac {25}{2}+156 x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac {443 \sqrt {2+5 x+3 x^2}}{500 (3+2 x)^4}+\frac {1}{500} \int \frac {-\frac {2677}{2}-3987 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac {443 \sqrt {2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac {2321 \sqrt {2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac {\int \frac {\frac {41237}{2}+55704 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx}{7500}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac {443 \sqrt {2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac {2321 \sqrt {2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac {1007 \sqrt {2+5 x+3 x^2}}{600 (3+2 x)^2}+\frac {\int \frac {-\frac {179415}{2}-377625 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx}{75000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac {443 \sqrt {2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac {2321 \sqrt {2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac {1007 \sqrt {2+5 x+3 x^2}}{600 (3+2 x)^2}-\frac {15891 \sqrt {2+5 x+3 x^2}}{6250 (3+2 x)}+\frac {128381 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{50000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac {443 \sqrt {2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac {2321 \sqrt {2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac {1007 \sqrt {2+5 x+3 x^2}}{600 (3+2 x)^2}-\frac {15891 \sqrt {2+5 x+3 x^2}}{6250 (3+2 x)}-\frac {128381 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{25000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac {443 \sqrt {2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac {2321 \sqrt {2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac {1007 \sqrt {2+5 x+3 x^2}}{600 (3+2 x)^2}-\frac {15891 \sqrt {2+5 x+3 x^2}}{6250 (3+2 x)}+\frac {128381 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{50000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 84, normalized size = 0.51 \[ \frac {-385143 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {10 \sqrt {3 x^2+5 x+2} \left (3051072 x^4+19313432 x^3+46092332 x^2+49233702 x+19918587\right )}{(2 x+3)^5}}{750000} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 140, normalized size = 0.85 \[ \frac {385143 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (3051072 \, x^{4} + 19313432 \, x^{3} + 46092332 \, x^{2} + 49233702 \, x + 19918587\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{1500000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 359, normalized size = 2.19 \[ \frac {128381}{250000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {6162288 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 83190888 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 1461489304 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 4863585804 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 30365807072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 40931011758 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 107175203674 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 58461317289 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 54344360217 \, \sqrt {3} x + 7303159752 \, \sqrt {3} - 54344360217 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{75000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 137, normalized size = 0.84 \[ -\frac {128381 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{250000}-\frac {443 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{8000 \left (x +\frac {3}{2}\right )^{4}}-\frac {2321 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{15000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1007 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{2400 \left (x +\frac {3}{2}\right )^{2}}-\frac {15891 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{12500 \left (x +\frac {3}{2}\right )}-\frac {13 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{800 \left (x +\frac {3}{2}\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 198, normalized size = 1.21 \[ -\frac {128381}{250000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {443 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {2321 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{1875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1007 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{600 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {15891 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{6250 \, {\left (2 \, x + 3\right )}} \]
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 \[ -\int \frac {x-5}{{\left (2\,x+3\right )}^6\,\sqrt {3\,x^2+5\,x+2}} \,d x \]
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 \[ - \int \frac {x}{64 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 576 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 2160 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 4320 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 4860 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2916 x \sqrt {3 x^{2} + 5 x + 2} + 729 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{64 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 576 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 2160 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 4320 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 4860 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2916 x \sqrt {3 x^{2} + 5 x + 2} + 729 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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