Optimal. Leaf size=92 \[ \frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}+\frac {1}{16} (455-123 x) \sqrt {3 x^2+2}-\frac {455}{32} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {1529}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {815, 844, 215, 725, 206} \begin {gather*} \frac {1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}+\frac {1}{16} (455-123 x) \sqrt {3 x^2+2}-\frac {455}{32} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {1529}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 725
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx &=\frac {1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}+\frac {1}{48} \int \frac {(516-1476 x) \sqrt {2+3 x^2}}{3+2 x} \, dx\\ &=\frac {1}{16} (455-123 x) \sqrt {2+3 x^2}+\frac {1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}+\frac {\int \frac {77904-330264 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1152}\\ &=\frac {1}{16} (455-123 x) \sqrt {2+3 x^2}+\frac {1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}-\frac {4587}{32} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {15925}{32} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {1}{16} (455-123 x) \sqrt {2+3 x^2}+\frac {1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}-\frac {1529}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {15925}{32} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=\frac {1}{16} (455-123 x) \sqrt {2+3 x^2}+\frac {1}{24} (26-3 x) \left (2+3 x^2\right )^{3/2}-\frac {1529}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {455}{32} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 80, normalized size = 0.87 \begin {gather*} \frac {1}{96} \left (-1365 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-2 \sqrt {3 x^2+2} \left (18 x^3-156 x^2+381 x-1469\right )-4587 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 109, normalized size = 1.18 \begin {gather*} \frac {1529}{32} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )+\frac {455}{16} \sqrt {35} \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )+\frac {1}{48} \sqrt {3 x^2+2} \left (-18 x^3+156 x^2-381 x+1469\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 102, normalized size = 1.11 \begin {gather*} -\frac {1}{48} \, {\left (18 \, x^{3} - 156 \, x^{2} + 381 \, x - 1469\right )} \sqrt {3 \, x^{2} + 2} + \frac {1529}{64} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac {455}{64} \, \sqrt {35} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 116, normalized size = 1.26 \begin {gather*} -\frac {1}{48} \, {\left (3 \, {\left (2 \, {\left (3 \, x - 26\right )} x + 127\right )} x - 1469\right )} \sqrt {3 \, x^{2} + 2} + \frac {1529}{32} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {455}{32} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \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 = 117, normalized size = 1.27 \begin {gather*} -\frac {\left (3 x^{2}+2\right )^{\frac {3}{2}} x}{8}-\frac {3 \sqrt {3 x^{2}+2}\, x}{8}-\frac {117 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{16}-\frac {1529 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{32}-\frac {455 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{32}+\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{12}+\frac {455 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 93, normalized size = 1.01 \begin {gather*} -\frac {1}{8} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {13}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {123}{16} \, \sqrt {3 \, x^{2} + 2} x - \frac {1529}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {455}{32} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {455}{16} \, \sqrt {3 \, x^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 76, normalized size = 0.83 \begin {gather*} \frac {\sqrt {35}\,\left (31850\,\ln \left (x+\frac {3}{2}\right )-31850\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{2240}-\frac {1529\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {9\,x^3}{8}-\frac {39\,x^2}{4}+\frac {381\,x}{16}-\frac {1469}{16}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {10 \sqrt {3 x^{2} + 2}}{2 x + 3}\right )\, dx - \int \frac {2 x \sqrt {3 x^{2} + 2}}{2 x + 3}\, dx - \int \left (- \frac {15 x^{2} \sqrt {3 x^{2} + 2}}{2 x + 3}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 2}}{2 x + 3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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