Optimal. Leaf size=138 \[ -\frac {1}{27} \left (3 x^2+2\right )^{5/2} (2 x+3)^4+\frac {13}{36} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac {4421 \left (3 x^2+2\right )^{5/2} (2 x+3)^2}{2268}+\frac {(226755 x+661583) \left (3 x^2+2\right )^{5/2}}{17010}+\frac {2777}{36} x \left (3 x^2+2\right )^{3/2}+\frac {2777}{12} x \sqrt {3 x^2+2}+\frac {2777 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 138, 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} \begin {gather*} -\frac {1}{27} \left (3 x^2+2\right )^{5/2} (2 x+3)^4+\frac {13}{36} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac {4421 \left (3 x^2+2\right )^{5/2} (2 x+3)^2}{2268}+\frac {(226755 x+661583) \left (3 x^2+2\right )^{5/2}}{17010}+\frac {2777}{36} x \left (3 x^2+2\right )^{3/2}+\frac {2777}{12} x \sqrt {3 x^2+2}+\frac {2777 \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)^4 \left (2+3 x^2\right )^{3/2} \, dx &=-\frac {1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac {1}{27} \int (3+2 x)^3 (421+234 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac {1}{648} \int (3+2 x)^2 (27504+26526 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac {13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac {\int (3+2 x) (1520544+1632636 x) \left (2+3 x^2\right )^{3/2} \, dx}{13608}\\ &=\frac {4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac {13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac {(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac {2777}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {2777}{36} x \left (2+3 x^2\right )^{3/2}+\frac {4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac {13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac {(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac {2777}{6} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {2777}{12} x \sqrt {2+3 x^2}+\frac {2777}{36} x \left (2+3 x^2\right )^{3/2}+\frac {4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac {13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac {(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac {2777}{6} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {2777}{12} x \sqrt {2+3 x^2}+\frac {2777}{36} x \left (2+3 x^2\right )^{3/2}+\frac {4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac {13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac {(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac {2777 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 75, normalized size = 0.54 \begin {gather*} \frac {\sqrt {3 x^2+2} \left (-181440 x^8-204120 x^7+3676320 x^6+14492520 x^5+24490404 x^4+27468315 x^3+27537072 x^2+19683405 x+8598544\right )}{34020}+\frac {2777 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 86, normalized size = 0.62 \begin {gather*} \frac {\sqrt {3 x^2+2} \left (-181440 x^8-204120 x^7+3676320 x^6+14492520 x^5+24490404 x^4+27468315 x^3+27537072 x^2+19683405 x+8598544\right )}{34020}-\frac {2777 \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 = 0.43, size = 80, normalized size = 0.58 \begin {gather*} -\frac {1}{34020} \, {\left (181440 \, x^{8} + 204120 \, x^{7} - 3676320 \, x^{6} - 14492520 \, x^{5} - 24490404 \, x^{4} - 27468315 \, x^{3} - 27537072 \, x^{2} - 19683405 \, x - 8598544\right )} \sqrt {3 \, x^{2} + 2} + \frac {2777}{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 = 0.19, size = 72, normalized size = 0.52 \begin {gather*} -\frac {1}{34020} \, {\left (3 \, {\left ({\left (9 \, {\left (4 \, {\left (10 \, {\left ({\left (21 \, {\left (8 \, x + 9\right )} x - 3404\right )} x - 13419\right )} x - 226763\right )} x - 1017345\right )} x - 9179024\right )} x - 6561135\right )} x - 8598544\right )} \sqrt {3 \, x^{2} + 2} - \frac {2777}{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 = 0.05, size = 103, normalized size = 0.75 \begin {gather*} -\frac {16 \left (3 x^{2}+2\right )^{\frac {5}{2}} x^{4}}{27}-\frac {2 \left (3 x^{2}+2\right )^{\frac {5}{2}} x^{3}}{3}+\frac {7256 \left (3 x^{2}+2\right )^{\frac {5}{2}} x^{2}}{567}+\frac {434 \left (3 x^{2}+2\right )^{\frac {5}{2}} x}{9}+\frac {2777 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{36}+\frac {2777 \sqrt {3 x^{2}+2}\, x}{12}+\frac {2777 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{18}+\frac {537409 \left (3 x^{2}+2\right )^{\frac {5}{2}}}{8505} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 102, normalized size = 0.74 \begin {gather*} -\frac {16}{27} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{4} - \frac {2}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{3} + \frac {7256}{567} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{2} + \frac {434}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {537409}{8505} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {2777}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {2777}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {2777}{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 = 2.10, size = 65, normalized size = 0.47 \begin {gather*} \frac {2777\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-16\,x^8-18\,x^7+\frac {6808\,x^6}{21}+1278\,x^5+\frac {226763\,x^4}{105}+\frac {9689\,x^3}{4}+\frac {2294756\,x^2}{945}+\frac {6943\,x}{4}+\frac {2149636}{2835}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.52, size = 162, normalized size = 1.17 \begin {gather*} - \frac {16 x^{8} \sqrt {3 x^{2} + 2}}{3} - 6 x^{7} \sqrt {3 x^{2} + 2} + \frac {6808 x^{6} \sqrt {3 x^{2} + 2}}{63} + 426 x^{5} \sqrt {3 x^{2} + 2} + \frac {226763 x^{4} \sqrt {3 x^{2} + 2}}{315} + \frac {9689 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {2294756 x^{2} \sqrt {3 x^{2} + 2}}{2835} + \frac {6943 x \sqrt {3 x^{2} + 2}}{12} + \frac {2149636 \sqrt {3 x^{2} + 2}}{8505} + \frac {2777 \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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