Optimal. Leaf size=197 \[ -\frac{(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}+\frac{7 (548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^4}+\frac{7 (33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{7680 (2 x+3)^3}-\frac{21 (21974 x+47145) \sqrt{3 x^2+5 x+2}}{10240 (2 x+3)}+\frac{30275 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}-\frac{2345091 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{20480 \sqrt{5}} \]
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Rubi [A] time = 0.126487, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {812, 810, 843, 621, 206, 724} \[ -\frac{(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}+\frac{7 (548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^4}+\frac{7 (33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{7680 (2 x+3)^3}-\frac{21 (21974 x+47145) \sqrt{3 x^2+5 x+2}}{10240 (2 x+3)}+\frac{30275 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}-\frac{2345091 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{20480 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 812
Rule 810
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx &=-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}-\frac{7}{120} \int \frac{(-230-274 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx\\ &=\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{7 \int \frac{(-11292-13112 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{1536}\\ &=\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}-\frac{7 \int \frac{(1351944+1582128 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{122880}\\ &=-\frac{21 (47145+21974 x) \sqrt{2+5 x+3 x^2}}{10240 (3+2 x)}+\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{7 \int \frac{21287376+24912000 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{983040}\\ &=-\frac{21 (47145+21974 x) \sqrt{2+5 x+3 x^2}}{10240 (3+2 x)}+\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{90825 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{1024}-\frac{2345091 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{20480}\\ &=-\frac{21 (47145+21974 x) \sqrt{2+5 x+3 x^2}}{10240 (3+2 x)}+\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{90825}{512} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )+\frac{2345091 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{10240}\\ &=-\frac{21 (47145+21974 x) \sqrt{2+5 x+3 x^2}}{10240 (3+2 x)}+\frac{7 (42733+33142 x) \left (2+5 x+3 x^2\right )^{3/2}}{7680 (3+2 x)^3}+\frac{7 (1003+548 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^4}-\frac{(27+5 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^5}+\frac{30275 \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{1024}-\frac{2345091 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{20480 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.167127, size = 130, normalized size = 0.66 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (46080 x^7-257280 x^6+483840 x^5+27897856 x^4+127665096 x^3+242016116 x^2+213122626 x+72189541\right )}{(2 x+3)^5}+2345091 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )+3027500 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{102400} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 316, normalized size = 1.6 \begin{align*} -{\frac{13}{800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{27}{8000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{251}{5000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}+{\frac{10023}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{95295+114354\,x}{25000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}+{\frac{614355+737226\,x}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{19059}{12500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{185185+222222\,x}{16000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{186165+223398\,x}{6400}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{30275\,\sqrt{3}}{1024}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{2345091\,\sqrt{5}}{102400}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }-{\frac{335013}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{2345091}{400000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{781697}{64000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2345091}{102400}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.21978, size = 440, normalized size = 2.23 \begin{align*} -\frac{30069}{100000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{251 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{625 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{10023 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{25000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{368613}{50000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{112329}{400000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{19059 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{5000 \,{\left (2 \, x + 3\right )}} + \frac{111111}{8000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{40957}{64000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{111699}{3200} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{30275}{1024} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{2345091}{102400} \, \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{855771}{51200} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65764, size = 683, normalized size = 3.47 \begin{align*} \frac{3027500 \, \sqrt{3}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 2345091 \, \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 (46080 \, x^{7} - 257280 \, x^{6} + 483840 \, x^{5} + 27897856 \, x^{4} + 127665096 \, x^{3} + 242016116 \, x^{2} + 213122626 \, x + 72189541\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{204800 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27964, size = 563, normalized size = 2.86 \begin{align*} -\frac{3}{512} \,{\left (2 \,{\left (12 \, x - 157\right )} x + 2067\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{2345091}{102400} \, \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{30275}{1024} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{60397264 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 739203704 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 11836231432 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 36096211012 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 207702455456 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 259725515674 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 635418284542 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 326158305587 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 287216072451 \, \sqrt{3} x + 36785380096 \, \sqrt{3} - 287216072451 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{10240 \,{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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