Optimal. Leaf size=174 \[ -\frac {4892 \left (3 x^2+5 x+2\right )^{5/2}}{13125 (2 x+3)^5}-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{1050 (2 x+3)^6}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}+\frac {4663 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{60000 (2 x+3)^4}-\frac {4663 (8 x+7) \sqrt {3 x^2+5 x+2}}{800000 (2 x+3)^2}+\frac {4663 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1600000 \sqrt {5}} \]
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Rubi [A] time = 0.11, antiderivative size = 174, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {4892 \left (3 x^2+5 x+2\right )^{5/2}}{13125 (2 x+3)^5}-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{1050 (2 x+3)^6}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}+\frac {4663 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{60000 (2 x+3)^4}-\frac {4663 (8 x+7) \sqrt {3 x^2+5 x+2}}{800000 (2 x+3)^2}+\frac {4663 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1600000 \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 )^{3/2}}{(3+2 x)^8} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {1}{35} \int \frac {\left (-\frac {199}{2}+78 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}+\frac {\int \frac {\left (\frac {5887}{2}-1299 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx}{1050}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac {4663 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{1500}\\ &=\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}-\frac {4663 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{40000}\\ &=-\frac {4663 (7+8 x) \sqrt {2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac {4663 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{1600000}\\ &=-\frac {4663 (7+8 x) \sqrt {2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}-\frac {4663 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{800000}\\ &=-\frac {4663 (7+8 x) \sqrt {2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac {4663 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1600000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 157, normalized size = 0.90 \begin {gather*} \frac {1}{35} \left (-\frac {4892 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}+\frac {32641 \left (\frac {10 \sqrt {3 x^2+5 x+2} \left (864 x^3+2068 x^2+1572 x+371\right )}{(2 x+3)^4}-3 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )}{4800000}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.72, size = 91, normalized size = 0.52 \begin {gather*} \frac {4663 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{800000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (191232 x^6+2893088 x^5+16376240 x^4+55403520 x^3+64140640 x^2+15759118 x-6554463\right )}{16800000 (2 x+3)^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 170, normalized size = 0.98 \begin {gather*} \frac {97923 \, \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 (191232 \, x^{6} + 2893088 \, x^{5} + 16376240 \, x^{4} + 55403520 \, x^{3} + 64140640 \, x^{2} + 15759118 \, x - 6554463\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{336000000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 461, normalized size = 2.65 \begin {gather*} \frac {4663}{8000000} \, \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 {6267072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 122207904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 3852187808 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 18344551344 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 131374293680 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 134399090784 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 264419126976 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 1446858601104 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 6675760646156 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 5954681858370 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 10149146991914 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 3640765552263 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 2268672558411 \, \sqrt {3} x - 208833935688 \, \sqrt {3} + 2268672558411 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{16800000 \, {\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 253, normalized size = 1.45 \begin {gather*} -\frac {4663 \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 )}{8000000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{4480 \left (x +\frac {3}{2}\right )^{7}}-\frac {433 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{67200 \left (x +\frac {3}{2}\right )^{6}}-\frac {1223 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{105000 \left (x +\frac {3}{2}\right )^{5}}-\frac {4663 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{240000 \left (x +\frac {3}{2}\right )^{4}}-\frac {4663 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{150000 \left (x +\frac {3}{2}\right )^{3}}-\frac {144553 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{3000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {135227 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1875000 \left (x +\frac {3}{2}\right )}+\frac {4663 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{15000000}-\frac {4663 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{1000000}+\frac {4663 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{8000000}+\frac {135227 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{3750000} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.30, size = 338, normalized size = 1.94 \begin {gather*} \frac {144553}{1000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{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 {433 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{1050 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {4892 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{13125 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {4663 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{15000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {4663 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{18750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {144553 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{750000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {13989}{500000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {4663}{8000000} \, \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 {88597}{4000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {135227 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{750000 \, {\left (2 \, x + 3\right )}} \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 )}^{3/2}}{{\left (2\,x+3\right )}^8} \,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 {10 \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 {23 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 {10 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 \frac {3 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}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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