Optimal. Leaf size=124 \[ -\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac {141 (8 x+7) \sqrt {3 x^2+5 x+2}}{16000 (2 x+3)^2}+\frac {141 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{32000 \sqrt {5}} \]
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Rubi [A] time = 0.06, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, 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 )^{5/2}}{25 (2 x+3)^5}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac {141 (8 x+7) \sqrt {3 x^2+5 x+2}}{16000 (2 x+3)^2}+\frac {141 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{32000 \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 )^{3/2}}{(3+2 x)^6} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {47}{10} \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}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}-\frac {141}{800} \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx\\ &=-\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {141 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{32000}\\ &=-\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}-\frac {141 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{16000}\\ &=-\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {141 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{32000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 128, normalized size = 1.03 \begin {gather*} -\frac {83200 \left (3 x^2+5 x+2\right )^{5/2}-47 (2 x+3) \left (-30 (8 x+7) \sqrt {3 x^2+5 x+2} (2 x+3)^2+400 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}-3 \sqrt {5} (2 x+3)^4 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )}{160000 (2 x+3)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 81, normalized size = 0.65 \begin {gather*} \frac {141 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{16000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (6336 x^4+66616 x^3+131516 x^2+90126 x+19031\right )}{16000 (2 x+3)^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 140, normalized size = 1.13 \begin {gather*} \frac {141 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (6336 \, x^{4} + 66616 \, x^{3} + 131516 \, x^{2} + 90126 \, x + 19031\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{320000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 359, normalized size = 2.90 \begin {gather*} \frac {141}{160000} \, \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 {146256 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 654456 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 415048 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 15455452 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 140042336 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 207568854 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 544555762 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 286352757 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 252454821 \, \sqrt {3} x - 31985676 \, \sqrt {3} + 252454821 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{16000 \, {\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 )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 211, normalized size = 1.70 \begin {gather*} -\frac {141 \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 )}{160000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1600 \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 {5}{2}}}{1000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1457 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20000 \left (x +\frac {3}{2}\right )^{2}}-\frac {1363 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{12500 \left (x +\frac {3}{2}\right )}+\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{100000}-\frac {141 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{20000}+\frac {141 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{160000}+\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}}}{25000} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.29, size = 241, normalized size = 1.94 \begin {gather*} \frac {4371}{20000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{100 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1457 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{5000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {423}{10000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {141}{160000} \, \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 {2679}{80000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1363 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5000 \, {\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 )}^6} \,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}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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