Optimal. Leaf size=178 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{871 \sqrt{1-2 x} (5 x+3)^{3/2}}{6048 (3 x+2)^2}-\frac{77269 \sqrt{1-2 x} \sqrt{5 x+3}}{254016 (3 x+2)}+\frac{100}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1922677 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{762048 \sqrt{7}} \]
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Rubi [A] time = 0.063871, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{871 \sqrt{1-2 x} (5 x+3)^{3/2}}{6048 (3 x+2)^2}-\frac{77269 \sqrt{1-2 x} \sqrt{5 x+3}}{254016 (3 x+2)}+\frac{100}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1922677 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{762048 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac{1}{108} \int \frac{\left (-\frac{1511}{4}-240 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac{\int \frac{\left (-\frac{166869}{8}-16800 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{4536}\\ &=-\frac{77269 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)}-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac{\int \frac{-\frac{8194677}{16}-588000 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{95256}\\ &=-\frac{77269 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)}-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1922677 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1524096}+\frac{500}{243} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{77269 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)}-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1922677 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{762048}+\frac{1}{243} \left (200 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{77269 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)}-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{100}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{1922677 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{762048 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.188224, size = 131, normalized size = 0.74 \[ \frac{-21 \sqrt{5 x+3} \left (26580294 x^4+33080973 x^3+3682800 x^2-8266660 x-2583760\right )-2195200 \sqrt{10-20 x} (3 x+2)^4 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-1922677 \sqrt{7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5334336 \sqrt{1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 315, normalized size = 1.8 \begin{align*}{\frac{1}{10668672\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 155736837\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+177811200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{4}+415298232\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+474163200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+415298232\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+474163200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+558186174\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+184576992\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+210739200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+973793520\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+30762832\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +35123200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +564235560\,x\sqrt{-10\,{x}^{2}-x+3}+108517920\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.00873, size = 266, normalized size = 1.49 \begin{align*} \frac{27065}{148176} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{169 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1176 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{5413 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32928 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{528205}{296352} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{50}{243} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1922677}{10668672} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{802877}{1778112} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3667 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{197568 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60707, size = 564, normalized size = 3.17 \begin{align*} -\frac{1922677 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 2195200 \, \sqrt{10}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \,{\left (13290147 \, x^{3} + 23185560 \, x^{2} + 13434180 \, x + 2583760\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{10668672 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = 3.36465, size = 602, normalized size = 3.38 \begin{align*} \frac{1922677}{106686720} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{50}{243} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (77269 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 81002040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 31057924800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 8580356288000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{127008 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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