Optimal. Leaf size=122 \[ -\frac {1}{21} \left (3 x^2+2\right )^{3/2} (2 x+3)^4+\frac {29}{63} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {923}{315} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac {2}{405} (4599 x+13781) \left (3 x^2+2\right )^{3/2}+\frac {2341}{18} x \sqrt {3 x^2+2}+\frac {2341 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac {1}{21} \left (3 x^2+2\right )^{3/2} (2 x+3)^4+\frac {29}{63} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {923}{315} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac {2}{405} (4599 x+13781) \left (3 x^2+2\right )^{3/2}+\frac {2341}{18} x \sqrt {3 x^2+2}+\frac {2341 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 780
Rule 833
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx &=-\frac {1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac {1}{21} \int (3+2 x)^3 (331+174 x) \sqrt {2+3 x^2} \, dx\\ &=\frac {29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac {1}{378} \int (3+2 x)^2 (15786+16614 x) \sqrt {2+3 x^2} \, dx\\ &=\frac {923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac {\int (3+2 x) (577458+772632 x) \sqrt {2+3 x^2} \, dx}{5670}\\ &=\frac {923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac {2}{405} (13781+4599 x) \left (2+3 x^2\right )^{3/2}+\frac {2341}{9} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {2341}{18} x \sqrt {2+3 x^2}+\frac {923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac {2}{405} (13781+4599 x) \left (2+3 x^2\right )^{3/2}+\frac {2341}{9} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {2341}{18} x \sqrt {2+3 x^2}+\frac {923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac {2}{405} (13781+4599 x) \left (2+3 x^2\right )^{3/2}+\frac {2341 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 65, normalized size = 0.53 \[ \frac {\sqrt {3 x^2+2} \left (-12960 x^6-15120 x^5+297648 x^4+1222200 x^3+1956174 x^2+1558935 x+1167988\right )}{5670}+\frac {2341 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 70, normalized size = 0.57 \[ -\frac {1}{5670} \, {\left (12960 \, x^{6} + 15120 \, x^{5} - 297648 \, x^{4} - 1222200 \, x^{3} - 1956174 \, x^{2} - 1558935 \, x - 1167988\right )} \sqrt {3 \, x^{2} + 2} + \frac {2341}{54} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 64, normalized size = 0.52 \[ -\frac {1}{5670} \, {\left (3 \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (5 \, {\left (6 \, x + 7\right )} x - 689\right )} x - 16975\right )} x - 326029\right )} x - 519645\right )} x - 1167988\right )} \sqrt {3 \, x^{2} + 2} - \frac {2341}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 91, normalized size = 0.75 \[ -\frac {16 \left (3 x^{2}+2\right )^{\frac {3}{2}} x^{4}}{21}-\frac {8 \left (3 x^{2}+2\right )^{\frac {3}{2}} x^{3}}{9}+\frac {5672 \left (3 x^{2}+2\right )^{\frac {3}{2}} x^{2}}{315}+\frac {652 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{9}+\frac {2341 \sqrt {3 x^{2}+2}\, x}{18}+\frac {2341 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{27}+\frac {291997 \left (3 x^{2}+2\right )^{\frac {3}{2}}}{2835} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 90, normalized size = 0.74 \[ -\frac {16}{21} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{4} - \frac {8}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{3} + \frac {5672}{315} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + \frac {652}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {291997}{2835} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {2341}{18} \, \sqrt {3 \, x^{2} + 2} x + \frac {2341}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 55, normalized size = 0.45 \[ \frac {2341\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {48\,x^6}{7}-8\,x^5+\frac {5512\,x^4}{35}+\frac {1940\,x^3}{3}+\frac {326029\,x^2}{315}+\frac {4949\,x}{6}+\frac {583994}{945}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.13, size = 131, normalized size = 1.07 \[ - \frac {16 x^{6} \sqrt {3 x^{2} + 2}}{7} - \frac {8 x^{5} \sqrt {3 x^{2} + 2}}{3} + \frac {5512 x^{4} \sqrt {3 x^{2} + 2}}{105} + \frac {1940 x^{3} \sqrt {3 x^{2} + 2}}{9} + \frac {326029 x^{2} \sqrt {3 x^{2} + 2}}{945} + \frac {4949 x \sqrt {3 x^{2} + 2}}{18} + \frac {583994 \sqrt {3 x^{2} + 2}}{2835} + \frac {2341 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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