Optimal. Leaf size=137 \[ \frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}+\frac {(1528 x+2087) \sqrt {3 x^2+5 x+2}}{3200 (2 x+3)^2}-\frac {3}{32} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {2359 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{6400 \sqrt {5}} \]
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Rubi [A] time = 0.08, antiderivative size = 137, 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 = {810, 843, 621, 206, 724} \begin {gather*} \frac {(352 x+333) \left (3 x^2+5 x+2\right )^{3/2}}{240 (2 x+3)^4}+\frac {(1528 x+2087) \sqrt {3 x^2+5 x+2}}{3200 (2 x+3)^2}-\frac {3}{32} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {2359 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{6400 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 810
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx &=\frac {(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac {1}{160} \int \frac {(139+120 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx\\ &=\frac {(2087+1528 x) \sqrt {2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac {(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}+\frac {\int \frac {-6082-7200 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{12800}\\ &=\frac {(2087+1528 x) \sqrt {2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac {(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac {9}{32} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {2359 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{6400}\\ &=\frac {(2087+1528 x) \sqrt {2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac {(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac {9}{16} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {2359 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{3200}\\ &=\frac {(2087+1528 x) \sqrt {2+5 x+3 x^2}}{3200 (3+2 x)^2}+\frac {(333+352 x) \left (2+5 x+3 x^2\right )^{3/2}}{240 (3+2 x)^4}-\frac {3}{32} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {2359 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{6400 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 110, normalized size = 0.80 \begin {gather*} \frac {-7077 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-9000 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )+\frac {10 \sqrt {3 x^2+5 x+2} \left (60576 x^3+190412 x^2+211148 x+82989\right )}{(2 x+3)^4}}{96000} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.61, size = 111, normalized size = 0.81 \begin {gather*} -\frac {3}{16} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )+\frac {2359 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{3200 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (60576 x^3+190412 x^2+211148 x+82989\right )}{9600 (2 x+3)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 183, normalized size = 1.34 \begin {gather*} \frac {9000 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 7077 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 (60576 \, x^{3} + 190412 \, x^{2} + 211148 \, x + 82989\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{192000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 106, normalized size = 0.77 \begin {gather*} -\frac {1}{19200} \, {\left (\frac {5 \, {\left (\frac {10 \, {\left (\frac {195 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 488 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 4109 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 7572 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {631}{1600} \, \sqrt {3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 221, normalized size = 1.61 \begin {gather*} -\frac {2359 \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 )}{32000}-\frac {3 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{32}-\frac {17 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{300 \left (x +\frac {3}{2}\right )^{3}}-\frac {1129 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{12000 \left (x +\frac {3}{2}\right )^{2}}-\frac {911 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{7500 \left (x +\frac {3}{2}\right )}+\frac {2359 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{60000}-\frac {109 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{4000}+\frac {2359 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{32000}+\frac {911 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{15000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{320 \left (x +\frac {3}{2}\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, size = 227, normalized size = 1.66 \begin {gather*} \frac {1129}{4000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {34 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{75 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1129 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{3000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {327}{2000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {3}{32} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {2359}{32000} \, \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 {179}{16000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {911 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{3000 \, {\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 )}^5} \,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}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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