Optimal. Leaf size=117 \[ -\frac{(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac{1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}-\frac{7}{16} (775-243 x) \sqrt{3 x^2+2}+\frac{5425}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )+\frac{18543}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
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Rubi [A] time = 0.0783686, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {813, 815, 844, 215, 725, 206} \[ -\frac{(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac{1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}-\frac{7}{16} (775-243 x) \sqrt{3 x^2+2}+\frac{5425}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )+\frac{18543}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
Antiderivative was successfully verified.
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Rule 813
Rule 815
Rule 844
Rule 215
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx &=-\frac{(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac{1}{8} \int \frac{(8-408 x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=-\frac{1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac{1}{384} \int \frac{(15456-163296 x) \sqrt{2+3 x^2}}{3+2 x} \, dx\\ &=-\frac{7}{16} (775-243 x) \sqrt{2+3 x^2}-\frac{1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}-\frac{\int \frac{6620544-32042304 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{9216}\\ &=-\frac{7}{16} (775-243 x) \sqrt{2+3 x^2}-\frac{1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac{55629}{32} \int \frac{1}{\sqrt{2+3 x^2}} \, dx-\frac{189875}{32} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{7}{16} (775-243 x) \sqrt{2+3 x^2}-\frac{1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac{18543}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{189875}{32} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=-\frac{7}{16} (775-243 x) \sqrt{2+3 x^2}-\frac{1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac{(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac{18543}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{5425}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.136162, size = 97, normalized size = 0.83 \[ \frac{1}{480} \left (-\frac{2 \sqrt{3 x^2+2} \left (216 x^5-1836 x^4+5118 x^3-19458 x^2+89521 x+265989\right )}{2 x+3}+81375 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )+278145 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 164, normalized size = 1.4 \begin{align*} -{\frac{31}{35} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{51\,x}{8} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{1701\,x}{16}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{18543\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{155}{12} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{5425}{32}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{5425\,\sqrt{35}}{32}{\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{13}{70} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{39\,x}{70} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54035, size = 165, normalized size = 1.41 \begin{align*} -\frac{1}{20} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{51}{8} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x - \frac{155}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{4 \,{\left (2 \, x + 3\right )}} + \frac{1701}{16} \, \sqrt{3 \, x^{2} + 2} x + \frac{18543}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{5425}{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{5425}{16} \, \sqrt{3 \, x^{2} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9677, size = 378, normalized size = 3.23 \begin{align*} \frac{278145 \, \sqrt{3}{\left (2 \, x + 3\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 81375 \, \sqrt{35}{\left (2 \, x + 3\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 ) - 4 \,{\left (216 \, x^{5} - 1836 \, x^{4} + 5118 \, x^{3} - 19458 \, x^{2} + 89521 \, x + 265989\right )} \sqrt{3 \, x^{2} + 2}}{960 \,{\left (2 \, x + 3\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 = 2.01361, size = 898, normalized size = 7.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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