Optimal. Leaf size=156 \[ -\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac {(4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}}{4536}+\frac {4507 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{15552}-\frac {22535 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{746496}+\frac {22535 (6 x+5) \sqrt {3 x^2+5 x+2}}{5971968}-\frac {22535 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{11943936 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 156, 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 = {832, 779, 612, 621, 206} \begin {gather*} -\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac {(4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}}{4536}+\frac {4507 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{15552}-\frac {22535 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{746496}+\frac {22535 (6 x+5) \sqrt {3 x^2+5 x+2}}{5971968}-\frac {22535 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{11943936 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{27} \int (3+2 x) \left (\frac {931}{2}+307 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}+\frac {4507}{432} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{31104}\\ &=-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}+\frac {22535 \int \sqrt {2+5 x+3 x^2} \, dx}{497664}\\ &=\frac {22535 (5+6 x) \sqrt {2+5 x+3 x^2}}{5971968}-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{11943936}\\ &=\frac {22535 (5+6 x) \sqrt {2+5 x+3 x^2}}{5971968}-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{5971968}\\ &=\frac {22535 (5+6 x) \sqrt {2+5 x+3 x^2}}{5971968}-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{11943936 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 92, normalized size = 0.59 \begin {gather*} \frac {-157745 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-6 \sqrt {3 x^2+5 x+2} \left (167215104 x^8+268240896 x^7-3275873280 x^6-15455860992 x^5-30355761024 x^4-32476001904 x^3-19762157208 x^2-6434937470 x-871825317\right )}{250822656} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.83, size = 94, normalized size = 0.60 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-167215104 x^8-268240896 x^7+3275873280 x^6+15455860992 x^5+30355761024 x^4+32476001904 x^3+19762157208 x^2+6434937470 x+871825317\right )}{41803776}-\frac {22535 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{5971968 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 93, normalized size = 0.60 \begin {gather*} -\frac {1}{41803776} \, {\left (167215104 \, x^{8} + 268240896 \, x^{7} - 3275873280 \, x^{6} - 15455860992 \, x^{5} - 30355761024 \, x^{4} - 32476001904 \, x^{3} - 19762157208 \, x^{2} - 6434937470 \, x - 871825317\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {22535}{71663616} \, \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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giac [A] time = 0.44, size = 89, normalized size = 0.57 \begin {gather*} -\frac {1}{41803776} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, x + 77\right )} x - 13165\right )} x - 2236091\right )} x - 26350487\right )} x - 225527791\right )} x - 823423217\right )} x - 3217468735\right )} x - 871825317\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {22535}{35831808} \, \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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maple [A] time = 0.05, size = 134, normalized size = 0.86 \begin {gather*} -\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{2}}{27}+\frac {163 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x}{324}-\frac {22535 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{35831808}+\frac {8699 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{4536}+\frac {4507 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{15552}-\frac {22535 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{746496}+\frac {22535 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{5971968} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 162, normalized size = 1.04 \begin {gather*} -\frac {4}{27} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {163}{324} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {8699}{4536} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {4507}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {22535}{15552} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {22535}{124416} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {112675}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {22535}{995328} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {22535}{35831808} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {112675}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \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 )}^{5/2} \,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 (- 1104 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2717 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3381 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2151 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 551 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 48 x^{6} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 36 x^{7} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 180 \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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