Optimal. Leaf size=132 \[ -\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}+\frac {91}{270} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac {(4977 x+15244) \left (3 x^2+2\right )^{7/2}}{1620}+\frac {3731}{180} x \left (3 x^2+2\right )^{5/2}+\frac {3731}{72} x \left (3 x^2+2\right )^{3/2}+\frac {3731}{24} x \sqrt {3 x^2+2}+\frac {3731 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac {1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}+\frac {91}{270} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac {(4977 x+15244) \left (3 x^2+2\right )^{7/2}}{1620}+\frac {3731}{180} x \left (3 x^2+2\right )^{5/2}+\frac {3731}{72} x \left (3 x^2+2\right )^{3/2}+\frac {3731}{24} x \sqrt {3 x^2+2}+\frac {3731 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \]
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)^3 \left (2+3 x^2\right )^{5/2} \, dx &=-\frac {1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac {1}{30} \int (3+2 x)^2 (462+273 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac {1}{810} \int (3+2 x) (35238+29862 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac {(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac {3731}{30} \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac {91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac {(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac {3731}{18} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {3731}{72} x \left (2+3 x^2\right )^{3/2}+\frac {3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac {91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac {(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac {3731}{12} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {3731}{24} x \sqrt {2+3 x^2}+\frac {3731}{72} x \left (2+3 x^2\right )^{3/2}+\frac {3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac {91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac {(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac {3731}{12} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {3731}{24} x \sqrt {2+3 x^2}+\frac {3731}{72} x \left (2+3 x^2\right )^{3/2}+\frac {3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac {91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac {(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac {3731 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 80, normalized size = 0.61 \[ \frac {335790 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (23328 x^9-12960 x^8-418446 x^7-1035720 x^6-1503522 x^5-2036880 x^4-1922805 x^3-1350240 x^2-1245915 x-299200\right )}{3240} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 85, normalized size = 0.64 \[ -\frac {1}{3240} \, {\left (23328 \, x^{9} - 12960 \, x^{8} - 418446 \, x^{7} - 1035720 \, x^{6} - 1503522 \, x^{5} - 2036880 \, x^{4} - 1922805 \, x^{3} - 1350240 \, x^{2} - 1245915 \, x - 299200\right )} \sqrt {3 \, x^{2} + 2} + \frac {3731}{72} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 76, normalized size = 0.58 \[ -\frac {1}{3240} \, {\left (3 \, {\left ({\left (9 \, {\left (2 \, {\left ({\left ({\left (3 \, {\left (16 \, {\left (9 \, x - 5\right )} x - 2583\right )} x - 19180\right )} x - 27843\right )} x - 37720\right )} x - 71215\right )} x - 450080\right )} x - 415305\right )} x - 299200\right )} \sqrt {3 \, x^{2} + 2} - \frac {3731}{36} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 101, normalized size = 0.77 \[ -\frac {4 \left (3 x^{2}+2\right )^{\frac {7}{2}} x^{3}}{15}+\frac {4 \left (3 x^{2}+2\right )^{\frac {7}{2}} x^{2}}{27}+\frac {319 \left (3 x^{2}+2\right )^{\frac {7}{2}} x}{60}+\frac {3731 \left (3 x^{2}+2\right )^{\frac {5}{2}} x}{180}+\frac {3731 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{72}+\frac {3731 \sqrt {3 x^{2}+2}\, x}{24}+\frac {3731 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{36}+\frac {935 \left (3 x^{2}+2\right )^{\frac {7}{2}}}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 100, normalized size = 0.76 \[ -\frac {4}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{3} + \frac {4}{27} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{2} + \frac {319}{60} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {935}{81} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {3731}{180} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {3731}{72} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {3731}{24} \, \sqrt {3 \, x^{2} + 2} x + \frac {3731}{36} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.75, size = 70, normalized size = 0.53 \[ \frac {3731\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{36}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {108\,x^9}{5}+12\,x^8+\frac {7749\,x^7}{20}+959\,x^6+\frac {27843\,x^5}{20}+1886\,x^4+\frac {14243\,x^3}{8}+\frac {11252\,x^2}{9}+\frac {9229\,x}{8}+\frac {7480}{27}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 77.88, size = 180, normalized size = 1.36 \[ - \frac {36 x^{9} \sqrt {3 x^{2} + 2}}{5} + 4 x^{8} \sqrt {3 x^{2} + 2} + \frac {2583 x^{7} \sqrt {3 x^{2} + 2}}{20} + \frac {959 x^{6} \sqrt {3 x^{2} + 2}}{3} + \frac {9281 x^{5} \sqrt {3 x^{2} + 2}}{20} + \frac {1886 x^{4} \sqrt {3 x^{2} + 2}}{3} + \frac {14243 x^{3} \sqrt {3 x^{2} + 2}}{24} + \frac {11252 x^{2} \sqrt {3 x^{2} + 2}}{27} + \frac {9229 x \sqrt {3 x^{2} + 2}}{24} + \frac {7480 \sqrt {3 x^{2} + 2}}{81} + \frac {3731 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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