Optimal. Leaf size=204 \[ -\frac {1321 \left (3 x^2+5 x+2\right )^{7/2}}{5250 (2 x+3)^7}-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{1800 (2 x+3)^8}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}+\frac {6167 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{120000 (2 x+3)^6}-\frac {6167 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1920000 (2 x+3)^4}+\frac {6167 (8 x+7) \sqrt {3 x^2+5 x+2}}{25600000 (2 x+3)^2}-\frac {6167 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{51200000 \sqrt {5}} \]
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Rubi [A] time = 0.13, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \begin {gather*} -\frac {1321 \left (3 x^2+5 x+2\right )^{7/2}}{5250 (2 x+3)^7}-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{1800 (2 x+3)^8}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}+\frac {6167 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{120000 (2 x+3)^6}-\frac {6167 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1920000 (2 x+3)^4}+\frac {6167 (8 x+7) \sqrt {3 x^2+5 x+2}}{25600000 (2 x+3)^2}-\frac {6167 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{51200000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {1}{45} \int \frac {\left (-\frac {293}{2}+78 x\right ) \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}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}+\frac {\int \frac {\left (\frac {11109}{2}-1581 x\right ) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx}{1800}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}+\frac {6167 \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{2000}\\ &=\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}-\frac {6167 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{48000}\\ &=-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}+\frac {6167 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1280000}\\ &=\frac {6167 (7+8 x) \sqrt {2+5 x+3 x^2}}{25600000 (3+2 x)^2}-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}-\frac {6167 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{51200000}\\ &=\frac {6167 (7+8 x) \sqrt {2+5 x+3 x^2}}{25600000 (3+2 x)^2}-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}+\frac {6167 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{25600000}\\ &=\frac {6167 (7+8 x) \sqrt {2+5 x+3 x^2}}{25600000 (3+2 x)^2}-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}-\frac {6167 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{51200000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 207, normalized size = 1.01 \begin {gather*} \frac {1}{45} \left (-\frac {3963 \left (3 x^2+5 x+2\right )^{7/2}}{350 (2 x+3)^7}-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{40 (2 x+3)^8}-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^9}+\frac {18501 \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 )}{256000}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.86, size = 101, normalized size = 0.50 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (333241344 x^8+4204480128 x^7+23288995392 x^6+76435267296 x^5+149661252080 x^4+173974546136 x^3+117870367452 x^2+43246799138 x+6706847909\right )}{1612800000 (2 x+3)^9}-\frac {6167 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{25600000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 201, normalized size = 0.99 \begin {gather*} \frac {388521 \, \sqrt {5} {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\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 (333241344 \, x^{8} + 4204480128 \, x^{7} + 23288995392 \, x^{6} + 76435267296 \, x^{5} + 149661252080 \, x^{4} + 173974546136 \, x^{3} + 117870367452 \, x^{2} + 43246799138 \, x + 6706847909\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{32256000000 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 563, normalized size = 2.76 \begin {gather*} -\frac {6167}{256000000} \, \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 {99461376 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 2536265088 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 83954355072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 341000936640 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 17778066768000 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 177356386111968 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 2399974462831392 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 6844601123556624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 41172892580130560 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 60936872688585000 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 204498063708405624 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 174436297943297292 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 339439601929212792 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 164994557892929730 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 174936772514694750 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 42504221165006223 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 19065836258759367 \, \sqrt {3} x + 1323473153587704 \, \sqrt {3} - 19065836258759367 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{1612800000 \, {\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 )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 332, normalized size = 1.63 \begin {gather*} \frac {6167 \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 )}{256000000}-\frac {6167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1920000 \left (x +\frac {3}{2}\right )^{6}}-\frac {1321 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{672000 \left (x +\frac {3}{2}\right )^{7}}-\frac {376187 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{30000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {6167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1200000 \left (x +\frac {3}{2}\right )^{5}}-\frac {11464453 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{600000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {3583027 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{250000000}-\frac {178843 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{120000000}-\frac {3583027 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{125000000 \left (x +\frac {3}{2}\right )}+\frac {6167 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{32000000}-\frac {6167 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{256000000}-\frac {129507 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{16000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {6167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{480000000}-\frac {6167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1000000000}-\frac {527 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{460800 \left (x +\frac {3}{2}\right )^{8}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{23040 \left (x +\frac {3}{2}\right )^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 484, normalized size = 2.37 \begin {gather*} \frac {11464453}{200000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{45 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} - \frac {527 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1800 \, {\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 {1321 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5250 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {6167 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{30000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {6167 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{37500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {129507 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {376187 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{3750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {11464453 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {178843}{20000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {3583027}{480000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {3583027 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50000000 \, {\left (2 \, x + 3\right )}} + \frac {18501}{16000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {6167}{256000000} \, \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 {117173}{128000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \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 = -0.00 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{10}} \,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 \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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