Optimal. Leaf size=150 \[ \frac{1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1080}+\frac{7093 (5 x+3)^{3/2} \sqrt{1-2 x}}{21600}-\frac{390869 \sqrt{5 x+3} \sqrt{1-2 x}}{259200}+\frac{1922677 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{777600 \sqrt{10}}-\frac{98}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.063693, antiderivative size = 150, 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 = {101, 154, 157, 54, 216, 93, 204} \[ \frac{1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1080}+\frac{7093 (5 x+3)^{3/2} \sqrt{1-2 x}}{21600}-\frac{390869 \sqrt{5 x+3} \sqrt{1-2 x}}{259200}+\frac{1922677 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{777600 \sqrt{10}}-\frac{98}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx &=\frac{1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{12} \int \frac{\left (-51-\frac{181 x}{2}\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{2+3 x} \, dx\\ &=\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac{1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{540} \int \frac{\left (-\frac{5133}{2}-\frac{21279 x}{4}\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{2+3 x} \, dx\\ &=\frac{7093 \sqrt{1-2 x} (3+5 x)^{3/2}}{21600}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac{1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{\int \frac{\left (-\frac{116469}{4}-\frac{1172607 x}{8}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx}{16200}\\ &=-\frac{390869 \sqrt{1-2 x} \sqrt{3+5 x}}{259200}+\frac{7093 \sqrt{1-2 x} (3+5 x)^{3/2}}{21600}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac{1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{\int \frac{\frac{3020277}{8}+\frac{5768031 x}{16}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{97200}\\ &=-\frac{390869 \sqrt{1-2 x} \sqrt{3+5 x}}{259200}+\frac{7093 \sqrt{1-2 x} (3+5 x)^{3/2}}{21600}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac{1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{1922677 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1555200}+\frac{343}{243} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{390869 \sqrt{1-2 x} \sqrt{3+5 x}}{259200}+\frac{7093 \sqrt{1-2 x} (3+5 x)^{3/2}}{21600}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac{1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{686}{243} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{1922677 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{777600 \sqrt{5}}\\ &=-\frac{390869 \sqrt{1-2 x} \sqrt{3+5 x}}{259200}+\frac{7093 \sqrt{1-2 x} (3+5 x)^{3/2}}{21600}+\frac{181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac{1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{1922677 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{777600 \sqrt{10}}-\frac{98}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0789448, size = 110, normalized size = 0.73 \[ \frac{-30 \sqrt{5 x+3} \left (864000 x^4-1646400 x^3+1069080 x^2-111742 x-59599\right )-1922677 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-3136000 \sqrt{7-14 x} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7776000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 132, normalized size = 0.9 \begin{align*}{\frac{1}{15552000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 25920000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-36432000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1922677\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +3136000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +13856400\,x\sqrt{-10\,{x}^{2}-x+3}+3575940\,\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 = 2.26574, size = 132, normalized size = 0.88 \begin{align*} -\frac{1}{6} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{271}{1080} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{7093}{4320} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{1922677}{15552000} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{49}{243} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{135521}{259200} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57896, size = 387, normalized size = 2.58 \begin{align*} \frac{1}{259200} \,{\left (432000 \, x^{3} - 607200 \, x^{2} + 230940 \, x + 59599\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{49}{243} \, \sqrt{7} \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 ) - \frac{1922677}{15552000} \, \sqrt{10} \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 ) \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 [A] time = 2.7621, size = 269, normalized size = 1.79 \begin{align*} \frac{49}{2430} \, \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{1}{1296000} \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} - 577 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 23769 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 390869 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{1922677}{15552000} \, \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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