Optimal. Leaf size=116 \[ -\frac {1}{24} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac {71}{168} \left (3 x^2+2\right )^{5/2} (2 x+3)^2+\frac {(5405 x+16973) \left (3 x^2+2\right )^{5/2}}{1260}+\frac {1087}{36} x \left (3 x^2+2\right )^{3/2}+\frac {1087}{12} x \sqrt {3 x^2+2}+\frac {1087 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 116, 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} \begin {gather*} -\frac {1}{24} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac {71}{168} \left (3 x^2+2\right )^{5/2} (2 x+3)^2+\frac {(5405 x+16973) \left (3 x^2+2\right )^{5/2}}{1260}+\frac {1087}{36} x \left (3 x^2+2\right )^{3/2}+\frac {1087}{12} x \sqrt {3 x^2+2}+\frac {1087 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \end {gather*}
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 )^{3/2} \, dx &=-\frac {1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac {1}{24} \int (3+2 x)^2 (372+213 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac {1}{504} \int (3+2 x) (21732+19458 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac {(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac {1087}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {1087}{36} x \left (2+3 x^2\right )^{3/2}+\frac {71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac {(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac {1087}{6} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {1087}{12} x \sqrt {2+3 x^2}+\frac {1087}{36} x \left (2+3 x^2\right )^{3/2}+\frac {71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac {(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac {1087}{6} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {1087}{12} x \sqrt {2+3 x^2}+\frac {1087}{36} x \left (2+3 x^2\right )^{3/2}+\frac {71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac {(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac {1087 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 70, normalized size = 0.60 \begin {gather*} \frac {76090 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (3780 x^7-2160 x^6-75600 x^5-186012 x^4-219975 x^3-245136 x^2-226065 x-81392\right )}{1260} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 81, normalized size = 0.70 \begin {gather*} \frac {\sqrt {3 x^2+2} \left (-3780 x^7+2160 x^6+75600 x^5+186012 x^4+219975 x^3+245136 x^2+226065 x+81392\right )}{1260}-\frac {1087 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 75, normalized size = 0.65 \begin {gather*} -\frac {1}{1260} \, {\left (3780 \, x^{7} - 2160 \, x^{6} - 75600 \, x^{5} - 186012 \, x^{4} - 219975 \, x^{3} - 245136 \, x^{2} - 226065 \, x - 81392\right )} \sqrt {3 \, x^{2} + 2} + \frac {1087}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 66, normalized size = 0.57 \begin {gather*} -\frac {1}{1260} \, {\left (3 \, {\left ({\left ({\left (12 \, {\left (15 \, {\left ({\left (7 \, x - 4\right )} x - 140\right )} x - 5167\right )} x - 73325\right )} x - 81712\right )} x - 75355\right )} x - 81392\right )} \sqrt {3 \, x^{2} + 2} - \frac {1087}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 89, normalized size = 0.77 \begin {gather*} -\frac {\left (3 x^{2}+2\right )^{\frac {5}{2}} x^{3}}{3}+\frac {4 \left (3 x^{2}+2\right )^{\frac {5}{2}} x^{2}}{21}+\frac {64 \left (3 x^{2}+2\right )^{\frac {5}{2}} x}{9}+\frac {1087 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{36}+\frac {1087 \sqrt {3 x^{2}+2}\, x}{12}+\frac {1087 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{18}+\frac {5087 \left (3 x^{2}+2\right )^{\frac {5}{2}}}{315} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 88, normalized size = 0.76 \begin {gather*} -\frac {1}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{3} + \frac {4}{21} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{2} + \frac {64}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {5087}{315} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {1087}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {1087}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {1087}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 60, normalized size = 0.52 \begin {gather*} \frac {1087\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-9\,x^7+\frac {36\,x^6}{7}+180\,x^5+\frac {15501\,x^4}{35}+\frac {2095\,x^3}{4}+\frac {20428\,x^2}{35}+\frac {2153\,x}{4}+\frac {20348}{105}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.07, size = 144, normalized size = 1.24 \begin {gather*} - 3 x^{7} \sqrt {3 x^{2} + 2} + \frac {12 x^{6} \sqrt {3 x^{2} + 2}}{7} + 60 x^{5} \sqrt {3 x^{2} + 2} + \frac {5167 x^{4} \sqrt {3 x^{2} + 2}}{35} + \frac {2095 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {20428 x^{2} \sqrt {3 x^{2} + 2}}{105} + \frac {2153 x \sqrt {3 x^{2} + 2}}{12} + \frac {20348 \sqrt {3 x^{2} + 2}}{315} + \frac {1087 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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