Optimal. Leaf size=184 \[ -\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac {329 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{96000 (2 x+3)^6}+\frac {329 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1536000 (2 x+3)^4}-\frac {329 (8 x+7) \sqrt {3 x^2+5 x+2}}{20480000 (2 x+3)^2}+\frac {329 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40960000 \sqrt {5}} \]
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Rubi [A] time = 0.10, antiderivative size = 184, 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 = {806, 720, 724, 206} \begin {gather*} -\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac {329 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{96000 (2 x+3)^6}+\frac {329 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{1536000 (2 x+3)^4}-\frac {329 (8 x+7) \sqrt {3 x^2+5 x+2}}{20480000 (2 x+3)^2}+\frac {329 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40960000 \sqrt {5}} \end {gather*}
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 )^{7/2}}{(3+2 x)^{10}} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {47}{10} \int \frac {\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx\\ &=\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac {329 \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{1600}\\ &=-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {329 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{38400}\\ &=\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac {329 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1024000}\\ &=-\frac {329 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {329 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{40960000}\\ &=-\frac {329 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}-\frac {329 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{20480000}\\ &=-\frac {329 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000 (3+2 x)^2}+\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1536000 (3+2 x)^4}-\frac {329 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{96000 (3+2 x)^6}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{45 (3+2 x)^9}+\frac {329 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40960000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 185, normalized size = 1.01 \begin {gather*} -\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800 (2 x+3)^8}-\frac {329 \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 )}{3072000} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.86, size = 101, normalized size = 0.55 \begin {gather*} \frac {329 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{20480000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (28394496 x^8+2848109952 x^7+15895201728 x^6+38558367264 x^5+51825176720 x^4+41530110824 x^3+19810691268 x^2+5201574542 x+578701331\right )}{184320000 (2 x+3)^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 200, normalized size = 1.09 \begin {gather*} \frac {2961 \, \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 (28394496 \, x^{8} + 2848109952 \, x^{7} + 15895201728 \, x^{6} + 38558367264 \, x^{5} + 51825176720 \, x^{4} + 41530110824 \, x^{3} + 19810691268 \, x^{2} + 5201574542 \, x + 578701331\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3686400000 \, {\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 = 0.38, size = 563, normalized size = 3.06 \begin {gather*} \frac {329}{204800000} \, \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 {14930678016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 204061569408 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} + 3866707486848 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 14840812733760 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 114102022608000 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 198779998219488 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 649357338634272 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 207317438979984 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 2217334591351040 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 5247913396815000 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 20151247122371016 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 17924557725783828 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 35125577732048328 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 16953161853593070 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 17752204726475250 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 4253745315948057 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 1882391465118753 \, \sqrt {3} x - 129047626217736 \, \sqrt {3} + 1882391465118753 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{184320000 \, {\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 [B] time = 0.11, size = 369, normalized size = 2.01 \begin {gather*} -\frac {329 \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 )}{204800000}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{51200 \left (x +\frac {3}{2}\right )^{8}}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{32000 \left (x +\frac {3}{2}\right )^{7}}-\frac {90287 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{16000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {2621237 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{200000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {259393 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{30000000 \left (x +\frac {3}{2}\right )^{3}}+\frac {491479 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{50000000}-\frac {191149 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{200000000}-\frac {491479 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{25000000 \left (x +\frac {3}{2}\right )}+\frac {9541 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{96000000}-\frac {329 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{25600000}+\frac {329 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{204800000}+\frac {329 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{384000000}+\frac {329 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{800000000}+\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{200000000}-\frac {1457 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{400000 \left (x +\frac {3}{2}\right )^{5}}-\frac {893 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{384000 \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 {9}{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 = 513, normalized size = 2.79 \begin {gather*} \frac {7863711}{200000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{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 {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{200 \, {\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 {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {893 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{6000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {1457 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {90287 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {259393 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{3750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2621237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {573447}{100000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {3822651}{800000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {491479 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{10000000 \, {\left (2 \, x + 3\right )}} + \frac {9541}{16000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {191149}{384000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {987}{12800000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {329}{204800000} \, \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 {6251}{102400000} \, \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 = -0.01 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\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 {40 \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 {292 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 {870 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 {1339 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 {1090 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 \left (- \frac {396 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}\right )\, dx - \int \frac {27 x^{7} \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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