Optimal. Leaf size=116 \[ -\frac{3}{40} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{23}{96} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{253 \sqrt{5 x+3} (1-2 x)^{3/2}}{1920}+\frac{2783 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{30613 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]
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Rubi [A] time = 0.0291132, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{3}{40} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{23}{96} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{253 \sqrt{5 x+3} (1-2 x)^{3/2}}{1920}+\frac{2783 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{30613 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x} \, dx &=-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{23}{16} \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\\ &=-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{253}{192} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{253 (1-2 x)^{3/2} \sqrt{3+5 x}}{1920}-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{2783 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1280}\\ &=\frac{2783 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{253 (1-2 x)^{3/2} \sqrt{3+5 x}}{1920}-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{30613 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{12800}\\ &=\frac{2783 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{253 (1-2 x)^{3/2} \sqrt{3+5 x}}{1920}-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{30613 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{6400 \sqrt{5}}\\ &=\frac{2783 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{253 (1-2 x)^{3/2} \sqrt{3+5 x}}{1920}-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{30613 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{6400 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0336978, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (-28800 x^3-6880 x^2+23420 x+1959\right )-91839 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{192000} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{384000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-137600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+91839\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +468400\,x\sqrt{-10\,{x}^{2}-x+3}+39180\,\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 = 4.18823, size = 95, normalized size = 0.82 \begin{align*} \frac{3}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1}{48} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{253}{320} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{30613}{128000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{253}{6400} \, \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 = 1.55923, size = 246, normalized size = 2.12 \begin{align*} -\frac{1}{19200} \,{\left (28800 \, x^{3} + 6880 \, x^{2} - 23420 \, x - 1959\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{30613}{128000} \, \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 = 28.6448, size = 316, normalized size = 2.72 \begin{align*} \frac{22 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{121} + \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{62 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (- \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{1936} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} - \frac{12 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (- \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{3872} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.45594, size = 220, normalized size = 1.9 \begin{align*} -\frac{1}{320000} \, \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}{24000} \, \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{1}{200} \, \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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