Optimal. Leaf size=154 \[ -\frac {1}{33} \left (3 x^2+2\right )^{7/2} (2 x+3)^4+\frac {49}{165} \left (3 x^2+2\right )^{7/2} (2 x+3)^3+\frac {6433 \left (3 x^2+2\right )^{7/2} (2 x+3)^2}{4455}+\frac {2 (62244 x+181243) \left (3 x^2+2\right )^{7/2}}{13365}+\frac {4991}{90} x \left (3 x^2+2\right )^{5/2}+\frac {4991}{36} x \left (3 x^2+2\right )^{3/2}+\frac {4991}{12} x \sqrt {3 x^2+2}+\frac {4991 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
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Rubi [A] time = 0.08, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac {1}{33} \left (3 x^2+2\right )^{7/2} (2 x+3)^4+\frac {49}{165} \left (3 x^2+2\right )^{7/2} (2 x+3)^3+\frac {6433 \left (3 x^2+2\right )^{7/2} (2 x+3)^2}{4455}+\frac {2 (62244 x+181243) \left (3 x^2+2\right )^{7/2}}{13365}+\frac {4991}{90} x \left (3 x^2+2\right )^{5/2}+\frac {4991}{36} x \left (3 x^2+2\right )^{3/2}+\frac {4991}{12} x \sqrt {3 x^2+2}+\frac {4991 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
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 \left (2+3 x^2\right )^{5/2} \, dx &=-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {1}{33} \int (3+2 x)^3 (511+294 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {1}{990} \int (3+2 x)^2 (42462+38598 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac {49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {\int (3+2 x) (3130638+2987712 x) \left (2+3 x^2\right )^{5/2} \, dx}{26730}\\ &=\frac {6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac {49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac {4991}{15} \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac {6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac {49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac {4991}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {4991}{36} x \left (2+3 x^2\right )^{3/2}+\frac {4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac {6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac {49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac {4991}{6} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {4991}{12} x \sqrt {2+3 x^2}+\frac {4991}{36} x \left (2+3 x^2\right )^{3/2}+\frac {4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac {6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac {49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac {4991}{6} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {4991}{12} x \sqrt {2+3 x^2}+\frac {4991}{36} x \left (2+3 x^2\right )^{3/2}+\frac {4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac {6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac {49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac {4991 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 85, normalized size = 0.55 \[ \frac {\sqrt {3 x^2+2} \left (-699840 x^{10}-769824 x^9+12921120 x^8+50615928 x^7+93646260 x^6+129966606 x^5+150762600 x^4+127123425 x^3+92160240 x^2+64370295 x+19537120\right )}{53460}+\frac {4991 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 90, normalized size = 0.58 \[ -\frac {1}{53460} \, {\left (699840 \, x^{10} + 769824 \, x^{9} - 12921120 \, x^{8} - 50615928 \, x^{7} - 93646260 \, x^{6} - 129966606 \, x^{5} - 150762600 \, x^{4} - 127123425 \, x^{3} - 92160240 \, x^{2} - 64370295 \, x - 19537120\right )} \sqrt {3 \, x^{2} + 2} + \frac {4991}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 82, normalized size = 0.53 \[ -\frac {1}{53460} \, {\left (3 \, {\left ({\left (9 \, {\left (2 \, {\left ({\left (2 \, {\left (6 \, {\left (4 \, {\left (27 \, {\left (10 \, x + 11\right )} x - 4985\right )} x - 78111\right )} x - 867095\right )} x - 2406789\right )} x - 2791900\right )} x - 4708275\right )} x - 30720080\right )} x - 21456765\right )} x - 19537120\right )} \sqrt {3 \, x^{2} + 2} - \frac {4991}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 115, normalized size = 0.75 \[ -\frac {16 \left (3 x^{2}+2\right )^{\frac {7}{2}} x^{4}}{33}-\frac {8 \left (3 x^{2}+2\right )^{\frac {7}{2}} x^{3}}{15}+\frac {8840 \left (3 x^{2}+2\right )^{\frac {7}{2}} x^{2}}{891}+\frac {542 \left (3 x^{2}+2\right )^{\frac {7}{2}} x}{15}+\frac {4991 \left (3 x^{2}+2\right )^{\frac {5}{2}} x}{90}+\frac {4991 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{36}+\frac {4991 \sqrt {3 x^{2}+2}\, x}{12}+\frac {4991 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{18}+\frac {122107 \left (3 x^{2}+2\right )^{\frac {7}{2}}}{2673} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 114, normalized size = 0.74 \[ -\frac {16}{33} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{4} - \frac {8}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{3} + \frac {8840}{891} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{2} + \frac {542}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {122107}{2673} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {4991}{90} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {4991}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {4991}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {4991}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.75, size = 75, normalized size = 0.49 \[ \frac {4991\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {432\,x^{10}}{11}-\frac {216\,x^9}{5}+\frac {7976\,x^8}{11}+\frac {14202\,x^7}{5}+\frac {173419\,x^6}{33}+\frac {72933\,x^5}{10}+\frac {279190\,x^4}{33}+\frac {28535\,x^3}{4}+\frac {1536004\,x^2}{297}+\frac {14449\,x}{4}+\frac {976856}{891}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 113.01, size = 199, normalized size = 1.29 \[ - \frac {144 x^{10} \sqrt {3 x^{2} + 2}}{11} - \frac {72 x^{9} \sqrt {3 x^{2} + 2}}{5} + \frac {7976 x^{8} \sqrt {3 x^{2} + 2}}{33} + \frac {4734 x^{7} \sqrt {3 x^{2} + 2}}{5} + \frac {173419 x^{6} \sqrt {3 x^{2} + 2}}{99} + \frac {24311 x^{5} \sqrt {3 x^{2} + 2}}{10} + \frac {279190 x^{4} \sqrt {3 x^{2} + 2}}{99} + \frac {28535 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {1536004 x^{2} \sqrt {3 x^{2} + 2}}{891} + \frac {14449 x \sqrt {3 x^{2} + 2}}{12} + \frac {976856 \sqrt {3 x^{2} + 2}}{2673} + \frac {4991 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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