Optimal. Leaf size=106 \[ \frac{(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}+\frac{3 (4097 x+2943) \sqrt{3 x^2+2}}{19600 (2 x+3)^2}-\frac{39663 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
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Rubi [A] time = 0.0618945, antiderivative size = 106, normalized size of antiderivative = 1., 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 = {811, 844, 215, 725, 206} \[ \frac{(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}+\frac{3 (4097 x+2943) \sqrt{3 x^2+2}}{19600 (2 x+3)^2}-\frac{39663 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
Antiderivative was successfully verified.
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Rule 811
Rule 844
Rule 215
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx &=\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac{\int \frac{(-936+840 x) \sqrt{2+3 x^2}}{(3+2 x)^3} \, dx}{1120}\\ &=\frac{3 (2943+4097 x) \sqrt{2+3 x^2}}{19600 (3+2 x)^2}+\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}+\frac{\int \frac{105408-352800 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{627200}\\ &=\frac{3 (2943+4097 x) \sqrt{2+3 x^2}}{19600 (3+2 x)^2}+\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac{9}{32} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{39663 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{39200}\\ &=\frac{3 (2943+4097 x) \sqrt{2+3 x^2}}{19600 (3+2 x)^2}+\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{39663 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{39200}\\ &=\frac{3 (2943+4097 x) \sqrt{2+3 x^2}}{19600 (3+2 x)^2}+\frac{(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{39663 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{39200 \sqrt{35}}\\ \end{align*}
Mathematica [A] time = 0.151156, size = 90, normalized size = 0.85 \[ \frac{\frac{70 \sqrt{3 x^2+2} \left (250602 x^3+559764 x^2+718441 x+245943\right )}{(2 x+3)^4}-118989 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{4116000}-\frac{3}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 194, normalized size = 1.8 \begin{align*} -{\frac{211}{117600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{999}{686000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{5779}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{13221}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{7227\,x}{686000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{3\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{39663}{1372000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{39663\,\sqrt{35}}{1372000}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{17337\,x}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{13}{2240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.5419, size = 247, normalized size = 2.33 \begin{align*} \frac{2997}{686000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{211 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{14700 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{999 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{171500 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{7227}{686000} \, \sqrt{3 \, x^{2} + 2} x - \frac{3}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{39663}{1372000} \, \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{39663}{686000} \, \sqrt{3 \, x^{2} + 2} - \frac{5779 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{686000 \,{\left (2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24925, size = 490, normalized size = 4.62 \begin{align*} \frac{385875 \, \sqrt{3}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 118989 \, \sqrt{35}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \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 ) + 140 \,{\left (250602 \, x^{3} + 559764 \, x^{2} + 718441 \, x + 245943\right )} \sqrt{3 \, x^{2} + 2}}{8232000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37664, size = 333, normalized size = 3.14 \begin{align*} -\frac{39663}{1372000} \, \sqrt{35} \log \left (-9 \, \sqrt{35} + 35 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{35 \, \sqrt{35}}{2 \, x + 3}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + \frac{3}{32} \, \sqrt{3} \log \left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{35}}{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{3} + \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{470400} \,{\left (\frac{35 \,{\left (\frac{35 \,{\left (\frac{1365 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 1193 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 16227 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 125301 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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