Optimal. Leaf size=159 \[ -\frac{8}{27} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{25}{12} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{3065 \sqrt{1-2 x} \sqrt{5 x+3}}{1296}-\frac{43 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3888}-\frac{181}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.0645757, antiderivative size = 159, 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, 154, 157, 54, 216, 93, 204} \[ -\frac{8}{27} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{25}{12} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{3065 \sqrt{1-2 x} \sqrt{5 x+3}}{1296}-\frac{43 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3888}-\frac{181}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
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)^2} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{1}{3} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac{8}{27} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{1}{135} \int \frac{\left (\frac{1835}{2}-3375 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{25}{12} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac{\int \frac{\left (-2655-\frac{45975 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx}{1620}\\ &=-\frac{3065 \sqrt{1-2 x} \sqrt{3+5 x}}{1296}+\frac{25}{12} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{\int \frac{\frac{49605}{2}-\frac{3225 x}{4}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{9720}\\ &=-\frac{3065 \sqrt{1-2 x} \sqrt{3+5 x}}{1296}+\frac{25}{12} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac{215 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{7776}+\frac{1267}{486} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{3065 \sqrt{1-2 x} \sqrt{3+5 x}}{1296}+\frac{25}{12} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{1267}{243} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{\left (43 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{3888}\\ &=-\frac{3065 \sqrt{1-2 x} \sqrt{3+5 x}}{1296}+\frac{25}{12} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac{43 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{3888}-\frac{181}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [A] time = 0.136733, size = 127, normalized size = 0.8 \[ \frac{6 \sqrt{5 x+3} \left (14400 x^4-10920 x^3-5166 x^2+4973 x-730\right )+43 \sqrt{10-20 x} (3 x+2) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-5792 \sqrt{7-14 x} (3 x+2) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7776 \sqrt{1-2 x} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 180, normalized size = 1.1 \begin{align*} -{\frac{1}{31104+46656\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 86400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+129\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-17376\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-22320\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+86\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -11584\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -42156\,x\sqrt{-10\,{x}^{2}-x+3}+8760\,\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 = 3.47104, size = 140, normalized size = 0.88 \begin{align*} \frac{5}{27} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{245}{108} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{43}{15552} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{181}{486} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1301}{1296} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{9 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58357, size = 420, normalized size = 2.64 \begin{align*} \frac{43 \, \sqrt{5} \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 5792 \, \sqrt{7}{\left (3 \, x + 2\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 ) - 12 \,{\left (7200 \, x^{3} - 1860 \, x^{2} - 3513 \, x + 730\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{15552 \,{\left (3 \, x + 2\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 = 3.09687, size = 412, normalized size = 2.59 \begin{align*} \frac{181}{4860} \, \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}{2160} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 85 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 835 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{43}{15552} \, \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{154 \, \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 )}}{81 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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