Optimal. Leaf size=259 \[ -\frac {6379 \left (3 x^2+5 x+2\right )^{9/2}}{41250 (2 x+3)^9}-\frac {2067 \left (3 x^2+5 x+2\right )^{9/2}}{11000 (2 x+3)^{10}}-\frac {12 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}+\frac {25017 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800000 (2 x+3)^8}-\frac {58373 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{32000000 (2 x+3)^6}+\frac {58373 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{512000000 (2 x+3)^4}-\frac {175119 (8 x+7) \sqrt {3 x^2+5 x+2}}{20480000000 (2 x+3)^2}+\frac {175119 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40960000000 \sqrt {5}} \]
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Rubi [A] time = 0.18, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {6379 \left (3 x^2+5 x+2\right )^{9/2}}{41250 (2 x+3)^9}-\frac {2067 \left (3 x^2+5 x+2\right )^{9/2}}{11000 (2 x+3)^{10}}-\frac {12 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}+\frac {25017 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800000 (2 x+3)^8}-\frac {58373 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{32000000 (2 x+3)^6}+\frac {58373 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{512000000 (2 x+3)^4}-\frac {175119 (8 x+7) \sqrt {3 x^2+5 x+2}}{20480000000 (2 x+3)^2}+\frac {175119 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40960000000 \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 )^{7/2}}{(3+2 x)^{13}} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {1}{60} \int \frac {\left (-\frac {369}{2}+117 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}+\frac {\int \frac {\left (\frac {18045}{2}-4320 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx}{3300}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {\int \frac {\left (-\frac {869175}{2}+93015 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx}{165000}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {25017 \int \frac {\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx}{10000}\\ &=\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac {175119 \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{1600000}\\ &=-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {58373 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{12800000}\\ &=\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac {175119 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1024000000}\\ &=-\frac {175119 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {175119 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{40960000000}\\ &=-\frac {175119 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac {175119 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{20480000000}\\ &=-\frac {175119 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {175119 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40960000000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 262, normalized size = 1.01 \begin {gather*} \frac {-\frac {12758 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^9}-\frac {6201 \left (3 x^2+5 x+2\right )^{9/2}}{10 (2 x+3)^{10}}-\frac {720 \left (3 x^2+5 x+2\right )^{9/2}}{(2 x+3)^{11}}-\frac {715 \left (3 x^2+5 x+2\right )^{9/2}}{(2 x+3)^{12}}+\frac {825561 \left (\frac {2 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^8}-\frac {7 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}+\frac {7 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{960 (2 x+3)^4}-\frac {7 \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 )}{128000}\right )}{16000}}{3300} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.29, size = 116, normalized size = 0.45 \begin {gather*} \frac {175119 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{20480000000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (60734693376 x^{11}+1044584776704 x^{10}+8182662620160 x^9+38544695427840 x^8+123629135656960 x^7+273282692080768 x^6+410468875350912 x^5+412855931529440 x^4+271870111600160 x^3+111795175925940 x^2+25843081681156 x+2531527640959\right )}{675840000000 (2 x+3)^{12}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 245, normalized size = 0.95 \begin {gather*} \frac {5778927 \, \sqrt {5} {\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\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 (60734693376 \, x^{11} + 1044584776704 \, x^{10} + 8182662620160 \, x^{9} + 38544695427840 \, x^{8} + 123629135656960 \, x^{7} + 273282692080768 \, x^{6} + 410468875350912 \, x^{5} + 412855931529440 \, x^{4} + 271870111600160 \, x^{3} + 111795175925940 \, x^{2} + 25843081681156 \, x + 2531527640959\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{13516800000000 \, {\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.54, size = 716, normalized size = 2.76 \begin {gather*} \frac {175119}{204800000000} \, \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 {11835242496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{23} + 408315866112 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{22} + 20039038086144 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{21} + 535243596890112 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{20} + 13859706456921600 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{19} + 31535346744025344 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{18} - 789031961976842496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} - 7977976824329385984 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 113078650509677476096 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 358779889050339715200 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 2538162771649151164032 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} - 4660243350382625915904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} - 20499122524155108829248 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} - 24347916060701730772704 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 70788415443572756925600 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 56076083911431114398208 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 108598043564223524909928 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 56663550021725424101412 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 70668287639831997261828 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 22876037084903247115200 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 16680770211437743348146 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 2864949797863813201587 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 930278306769206446269 \, \sqrt {3} x - 47729262032858665512 \, \sqrt {3} + 930278306769206446269 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{675840000000 \, {\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 )}^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 9.36, size = 432, normalized size = 1.67 \begin {gather*} -\frac {175119 \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 )}{204800000000}-\frac {2067 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{11264000 \left (x +\frac {3}{2}\right )^{10}}-\frac {25017 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{51200000 \left (x +\frac {3}{2}\right )^{8}}-\frac {25017 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{32000000 \left (x +\frac {3}{2}\right )^{7}}-\frac {48057657 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{16000000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {1395223107 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{200000000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {46022941 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{10000000000 \left (x +\frac {3}{2}\right )^{3}}+\frac {261602769 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{50000000000}-\frac {101744139 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{200000000000}-\frac {261602769 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{25000000000 \left (x +\frac {3}{2}\right )}+\frac {1692817 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{32000000000}-\frac {175119 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{25600000000}+\frac {175119 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{204800000000}+\frac {58373 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{128000000000}+\frac {175119 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{800000000000}+\frac {25017 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{200000000000}-\frac {775527 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{400000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {158441 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{128000000 \left (x +\frac {3}{2}\right )^{6}}-\frac {3 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{28160 \left (x +\frac {3}{2}\right )^{11}}-\frac {6379 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{21120000 \left (x +\frac {3}{2}\right )^{9}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{245760 \left (x +\frac {3}{2}\right )^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 726, normalized size = 2.80
result too large to display
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 )}^{7/2}}{{\left (2\,x+3\right )}^{13}} \,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}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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