Optimal. Leaf size=179 \[ -\frac {39389 (5+6 x) \sqrt {2+5 x+3 x^2}}{143327232}+\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac {5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac {39389 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{286654464 \sqrt {3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {846, 793, 626,
635, 212} \begin {gather*} -\frac {1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac {(20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}}{26730}+\frac {5627 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{25920}-\frac {39389 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1866240}+\frac {39389 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{17915904}-\frac {39389 (6 x+5) \sqrt {3 x^2+5 x+2}}{143327232}+\frac {39389 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{286654464 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 793
Rule 846
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx &=-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {1}{33} \int (3+2 x) \left (\frac {1141}{2}+377 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac {5627}{540} \int \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac {5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}-\frac {39389 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{51840}\\ &=-\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac {5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac {39389 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{746496}\\ &=\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac {5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}-\frac {39389 \int \sqrt {2+5 x+3 x^2} \, dx}{11943936}\\ &=-\frac {39389 (5+6 x) \sqrt {2+5 x+3 x^2}}{143327232}+\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac {5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac {39389 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{286654464}\\ &=-\frac {39389 (5+6 x) \sqrt {2+5 x+3 x^2}}{143327232}+\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac {5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac {39389 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{143327232}\\ &=-\frac {39389 (5+6 x) \sqrt {2+5 x+3 x^2}}{143327232}+\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{17915904}-\frac {39389 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1866240}+\frac {5627 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{25920}-\frac {1}{33} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(47425+20358 x) \left (2+5 x+3 x^2\right )^{9/2}}{26730}+\frac {39389 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{286654464 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.76, size = 101, normalized size = 0.56 \begin {gather*} \frac {-3 \sqrt {2+5 x+3 x^2} \left (-254668717065-2519542755670 x-10882383306360 x^2-26847121235760 x^3-41472321125760 x^4-41190616509696 x^5-25723491978240 x^6-9116575930368 x^7-1156531322880 x^8+261858852864 x^9+77396705280 x^{10}\right )+2166395 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{23648993280} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 153, normalized size = 0.85
method | result | size |
risch | \(-\frac {\left (77396705280 x^{10}+261858852864 x^{9}-1156531322880 x^{8}-9116575930368 x^{7}-25723491978240 x^{6}-41190616509696 x^{5}-41472321125760 x^{4}-26847121235760 x^{3}-10882383306360 x^{2}-2519542755670 x -254668717065\right ) \sqrt {3 x^{2}+5 x +2}}{7882997760}+\frac {39389 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{859963392}\) | \(95\) |
trager | \(\left (-\frac {108}{11} x^{10}-\frac {1827}{55} x^{9}+\frac {9683}{66} x^{8}+\frac {18318737}{15840} x^{7}+\frac {20675389}{6336} x^{6}+\frac {5959290583}{1140480} x^{5}+\frac {7200055751}{1368576} x^{4}+\frac {37287668383}{10948608} x^{3}+\frac {90686527553}{65691648} x^{2}+\frac {251954275567}{788299776} x +\frac {16977914471}{525533184}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {39389 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{859963392}\) | \(106\) |
default | \(\frac {39389 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{17915904}-\frac {39389 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{1866240}+\frac {5627 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{25920}+\frac {8027 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{5346}-\frac {4 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{33}+\frac {197 x \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{495}+\frac {39389 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{859963392}-\frac {39389 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{143327232}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 191, normalized size = 1.07 \begin {gather*} -\frac {4}{33} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{2} + \frac {197}{495} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x + \frac {8027}{5346} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {5627}{4320} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {5627}{5184} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {39389}{311040} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {39389}{373248} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {39389}{2985984} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {196945}{17915904} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {39389}{23887872} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {39389}{859963392} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {196945}{143327232} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.85, size = 103, normalized size = 0.58 \begin {gather*} -\frac {1}{7882997760} \, {\left (77396705280 \, x^{10} + 261858852864 \, x^{9} - 1156531322880 \, x^{8} - 9116575930368 \, x^{7} - 25723491978240 \, x^{6} - 41190616509696 \, x^{5} - 41472321125760 \, x^{4} - 26847121235760 \, x^{3} - 10882383306360 \, x^{2} - 2519542755670 \, x - 254668717065\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {39389}{1719926784} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \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 (- 3108 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11494 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 23659 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 29358 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 22000 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 9112 x^{6} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1341 x^{7} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 324 x^{8} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 108 x^{9} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 360 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.50, size = 99, normalized size = 0.55 \begin {gather*} -\frac {1}{7882997760} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, {\left (54 \, {\left (60 \, x + 203\right )} x - 48415\right )} x - 18318737\right )} x - 103376945\right )} x - 5959290583\right )} x - 36000278755\right )} x - 186438341915\right )} x - 453432637765\right )} x - 1259771377835\right )} x - 254668717065\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {39389}{859963392} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\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 {\left (2\,x+3\right )}^2\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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