Optimal. Leaf size=179 \[ -\frac{107 \left (3 x^2+5 x+2\right )^{7/2}}{350 (2 x+3)^7}-\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}+\frac{1517 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{24000 (2 x+3)^6}-\frac{1517 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{384000 (2 x+3)^4}+\frac{1517 (8 x+7) \sqrt{3 x^2+5 x+2}}{5120000 (2 x+3)^2}-\frac{1517 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{10240000 \sqrt{5}} \]
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Rubi [A] time = 0.102482, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {834, 806, 720, 724, 206} \[ -\frac{107 \left (3 x^2+5 x+2\right )^{7/2}}{350 (2 x+3)^7}-\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}+\frac{1517 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{24000 (2 x+3)^6}-\frac{1517 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{384000 (2 x+3)^4}+\frac{1517 (8 x+7) \sqrt{3 x^2+5 x+2}}{5120000 (2 x+3)^2}-\frac{1517 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{10240000 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 834
Rule 806
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac{1}{40} \int \frac{\left (-\frac{311}{2}+39 x\right ) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac{107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}+\frac{1517}{400} \int \frac{\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac{107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}-\frac{1517 \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{9600}\\ &=-\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac{107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}+\frac{1517 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{256000}\\ &=\frac{1517 (7+8 x) \sqrt{2+5 x+3 x^2}}{5120000 (3+2 x)^2}-\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac{107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}-\frac{1517 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{10240000}\\ &=\frac{1517 (7+8 x) \sqrt{2+5 x+3 x^2}}{5120000 (3+2 x)^2}-\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac{107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}+\frac{1517 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{5120000}\\ &=\frac{1517 (7+8 x) \sqrt{2+5 x+3 x^2}}{5120000 (3+2 x)^2}-\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{384000 (3+2 x)^4}+\frac{1517 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{24000 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{40 (3+2 x)^8}-\frac{107 \left (2+5 x+3 x^2\right )^{7/2}}{350 (3+2 x)^7}-\frac{1517 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{10240000 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.183597, size = 182, normalized size = 1.02 \[ \frac{1}{40} \left (-\frac{428 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}-\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^8}+\frac{1517 \left (\frac{32 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}-\frac{2 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}+\frac{3 (8 x+7) \sqrt{3 x^2+5 x+2}}{20 (2 x+3)^2}+\frac{3 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40 \sqrt{5}}\right )}{19200}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 311, normalized size = 1.7 \begin{align*} -{\frac{1517}{384000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{1517}{240000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{31857}{3200000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{92537}{6000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{2820103}{120000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{4406885+5288262\,x}{50000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{881377}{25000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{219965+263958\,x}{24000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{7585+9102\,x}{6400000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{1517\,\sqrt{5}}{51200000}{\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{1517}{200000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{1517}{96000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{1517}{51200000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13}{10240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-8}}-{\frac{107}{44800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.89446, size = 571, normalized size = 3.19 \begin{align*} \frac{2820103}{40000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{40 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac{107 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{350 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{1517 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{6000 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{1517 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{7500 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{31857 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{200000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{92537 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{750000 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2820103 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{30000000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{43993}{4000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{881377}{96000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{881377 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{10000000 \,{\left (2 \, x + 3\right )}} + \frac{4551}{3200000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{1517}{51200000} \, \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{28823}{25600000} \, \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 = 1.45968, size = 639, normalized size = 3.57 \begin{align*} \frac{31857 \, \sqrt{5}{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\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 (35495424 \, x^{7} + 395685952 \, x^{6} + 2141523904 \, x^{5} + 5486222160 \, x^{4} + 7363989440 \, x^{3} + 5395613996 \, x^{2} + 2061624348 \, x + 325079151\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{2150400000 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\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 = 1.23764, size = 691, normalized size = 3.86 \begin{align*} -\frac{1517}{51200000} \, \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{4077696 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 2811291840 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 54242130880 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 23829496320 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 4407279220960 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 22617729467088 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 195051199819760 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 377875254407040 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 1580087388997720 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 1627784736400620 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 3742975645158764 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2115026806109280 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 2573382759804010 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 709918795444635 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 358308332266605 \, \sqrt{3} x + 27766562618088 \, \sqrt{3} - 358308332266605 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{107520000 \,{\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 )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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