Optimal. Leaf size=154 \[ -\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{600 (2 x+3)^6}-\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^4}+\frac {47 (8 x+7) \sqrt {3 x^2+5 x+2}}{128000 (2 x+3)^2}-\frac {47 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{256000 \sqrt {5}} \]
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Rubi [A] time = 0.08, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {806, 720, 724, 206} \[ -\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{600 (2 x+3)^6}-\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^4}+\frac {47 (8 x+7) \sqrt {3 x^2+5 x+2}}{128000 (2 x+3)^2}-\frac {47 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{256000 \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rubi steps
\begin {align*} \int \frac {(5-x) \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}}{35 (3+2 x)^7}+\frac {47}{10} \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}-\frac {47}{240} \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=-\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^4}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}+\frac {47 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{6400}\\ &=\frac {47 (7+8 x) \sqrt {2+5 x+3 x^2}}{128000 (3+2 x)^2}-\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^4}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}-\frac {47 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{256000}\\ &=\frac {47 (7+8 x) \sqrt {2+5 x+3 x^2}}{128000 (3+2 x)^2}-\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^4}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}+\frac {47 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{128000}\\ &=\frac {47 (7+8 x) \sqrt {2+5 x+3 x^2}}{128000 (3+2 x)^2}-\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^4}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}-\frac {47 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{256000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 154, normalized size = 1.00 \[ -\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{600 (2 x+3)^6}-\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^4}+\frac {47 \left (\frac {10 \sqrt {3 x^2+5 x+2} (8 x+7)}{(2 x+3)^2}+\sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )}{1280000} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 171, normalized size = 1.11 \[ \frac {987 \, \sqrt {5} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (1089792 \, x^{6} + 22620128 \, x^{5} + 81951440 \, x^{4} + 127557120 \, x^{3} + 100711840 \, x^{2} + 39981058 \, x + 6404247\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{53760000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 461, normalized size = 2.99 \[ -\frac {47}{1280000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {72512832 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 651952224 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 6898276448 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 8494566864 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 58878767920 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 326450774496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 2207907445056 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 3147944405424 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 9314774279636 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 6492162811470 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 9472821206534 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 3070624865553 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 1792565462541 \, \sqrt {3} x - 158637115728 \, \sqrt {3} + 1792565462541 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{2688000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 290, normalized size = 1.88 \[ \frac {47 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{1280000}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{9600 \left (x +\frac {3}{2}\right )^{6}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{4480 \left (x +\frac {3}{2}\right )^{7}}-\frac {2867 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{150000 \left (x +\frac {3}{2}\right )^{3}}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{6000 \left (x +\frac {3}{2}\right )^{5}}-\frac {87373 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{3000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {27307 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1250000}-\frac {1363 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{600000}-\frac {27307 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{625000 \left (x +\frac {3}{2}\right )}+\frac {47 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{160000}-\frac {47 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1280000}-\frac {987 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{80000 \left (x +\frac {3}{2}\right )^{4}}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{2400000}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{5000000} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 367, normalized size = 2.38 \[ \frac {87373}{1000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{35 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {94 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{375 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {987 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {2867 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{18750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {87373 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{750000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1363}{100000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {27307}{2400000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {27307 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{250000 \, {\left (2 \, x + 3\right )}} + \frac {141}{80000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {47}{1280000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {893}{640000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^8} \,d x \]
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 \[ - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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