Optimal. Leaf size=138 \[ \frac{9}{80} \sqrt{1-2 x} (5 x+3)^{7/2}+\frac{49 (5 x+3)^{7/2}}{22 \sqrt{1-2 x}}+\frac{25397 \sqrt{1-2 x} (5 x+3)^{5/2}}{3520}+\frac{25397}{512} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{838101 \sqrt{1-2 x} \sqrt{5 x+3}}{2048}-\frac{9219111 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048 \sqrt{10}} \]
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Rubi [A] time = 0.0422166, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 80, 50, 54, 216} \[ \frac{9}{80} \sqrt{1-2 x} (5 x+3)^{7/2}+\frac{49 (5 x+3)^{7/2}}{22 \sqrt{1-2 x}}+\frac{25397 \sqrt{1-2 x} (5 x+3)^{5/2}}{3520}+\frac{25397}{512} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{838101 \sqrt{1-2 x} \sqrt{5 x+3}}{2048}-\frac{9219111 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}-\frac{1}{22} \int \frac{(3+5 x)^{5/2} \left (\frac{1833}{2}+99 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{76191 \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx}{1760}\\ &=\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{25397}{128} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{25397}{512} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{838101 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{1024}\\ &=\frac{838101 \sqrt{1-2 x} \sqrt{3+5 x}}{2048}+\frac{25397}{512} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{9219111 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{4096}\\ &=\frac{838101 \sqrt{1-2 x} \sqrt{3+5 x}}{2048}+\frac{25397}{512} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{9219111 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{2048 \sqrt{5}}\\ &=\frac{838101 \sqrt{1-2 x} \sqrt{3+5 x}}{2048}+\frac{25397}{512} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{9219111 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{2048 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0406542, size = 74, normalized size = 0.54 \[ \frac{9219111 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (57600 x^4+243520 x^3+517096 x^2+966014 x-1405233\right )}{20480 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 140, normalized size = 1. \begin{align*} -{\frac{1}{81920\,x-40960} \left ( -1152000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4870400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+18438222\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-10341920\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-9219111\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -19320280\,x\sqrt{-10\,{x}^{2}-x+3}+28104660\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\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 = 1.75005, size = 147, normalized size = 1.07 \begin{align*} -\frac{1125 \, x^{5}}{8 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{21725 \, x^{4}}{32 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{414505 \, x^{3}}{256 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3190679 \, x^{2}}{1024 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{9219111}{40960} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{4128123 \, x}{2048 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4215699}{2048 \, \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 = 1.77008, size = 297, normalized size = 2.15 \begin{align*} \frac{9219111 \, \sqrt{10}{\left (2 \, x - 1\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 ) + 20 \,{\left (57600 \, x^{4} + 243520 \, x^{3} + 517096 \, x^{2} + 966014 \, x - 1405233\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{40960 \,{\left (2 \, x - 1\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 = 2.47542, size = 131, normalized size = 0.95 \begin{align*} -\frac{9219111}{20480} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 329 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 25397 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1396835 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 46095555 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{256000 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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