Optimal. Leaf size=116 \[ -\frac{1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{5 x+3}+\frac{1331}{512} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{512 \sqrt{10}} \]
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Rubi [A] time = 0.0289834, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac{1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{5 x+3}+\frac{1331}{512} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{512 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-2 x} (3+5 x)^{5/2} \, dx &=-\frac{1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{55}{16} \int \sqrt{1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac{55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{605}{64} \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx\\ &=-\frac{605}{256} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{6655}{512} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{1331}{512} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{14641 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1024}\\ &=\frac{1331}{512} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{14641 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{512 \sqrt{5}}\\ &=\frac{1331}{512} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{512 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0595706, size = 74, normalized size = 0.64 \[ -\frac{10 \sqrt{5 x+3} \left (19200 x^4+21440 x^3-3848 x^2-13846 x+4005\right )+43923 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15360 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{20} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}\sqrt{1-2\,x}}-{\frac{11}{240} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{121}{384} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{512}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{14641\,\sqrt{10}}{10240}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.96016, size = 95, normalized size = 0.82 \begin{align*} -\frac{5}{8} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{91}{96} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{605}{128} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{14641}{10240} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{121}{512} \, \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 = 1.85928, size = 240, normalized size = 2.07 \begin{align*} \frac{1}{1536} \,{\left (9600 \, x^{3} + 15520 \, x^{2} + 5836 \, x - 4005\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{14641}{10240} \, \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 [A] time = 17.0418, size = 272, normalized size = 2.34 \begin{align*} \begin{cases} \frac{125 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{10 x - 5}} - \frac{1925 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{24 \sqrt{10 x - 5}} - \frac{605 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{192 \sqrt{10 x - 5}} - \frac{6655 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{768 \sqrt{10 x - 5}} + \frac{14641 i \sqrt{x + \frac{3}{5}}}{512 \sqrt{10 x - 5}} - \frac{14641 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{5120} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{14641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{5120} - \frac{125 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{5 - 10 x}} + \frac{1925 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{24 \sqrt{5 - 10 x}} + \frac{605 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{192 \sqrt{5 - 10 x}} + \frac{6655 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{768 \sqrt{5 - 10 x}} - \frac{14641 \sqrt{x + \frac{3}{5}}}{512 \sqrt{5 - 10 x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.58365, size = 220, normalized size = 1.9 \begin{align*} \frac{1}{76800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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