Optimal. Leaf size=96 \[ -\frac{2 (1-2 x)^{5/2}}{165 (5 x+3)^{3/2}}-\frac{38 (1-2 x)^{3/2}}{165 \sqrt{5 x+3}}-\frac{38}{275} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{19}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
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Rubi [A] time = 0.0206144, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {78, 47, 50, 54, 216} \[ -\frac{2 (1-2 x)^{5/2}}{165 (5 x+3)^{3/2}}-\frac{38 (1-2 x)^{3/2}}{165 \sqrt{5 x+3}}-\frac{38}{275} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{19}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}+\frac{19}{33} \int \frac{(1-2 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{38 (1-2 x)^{3/2}}{165 \sqrt{3+5 x}}-\frac{38}{55} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{38 (1-2 x)^{3/2}}{165 \sqrt{3+5 x}}-\frac{38}{275} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{19}{25} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{38 (1-2 x)^{3/2}}{165 \sqrt{3+5 x}}-\frac{38}{275} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{38 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{38 (1-2 x)^{3/2}}{165 \sqrt{3+5 x}}-\frac{38}{275} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{19}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}
Mathematica [C] time = 0.0323211, size = 59, normalized size = 0.61 \[ -\frac{2 (1-2 x)^{5/2} \left (19 \sqrt{22} (5 x+3)^{3/2} \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{5}{11} (1-2 x)\right )+121\right )}{19965 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 113, normalized size = 1.2 \begin{align*} -{\frac{1}{750} \left ( 1425\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+1710\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+513\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2900\,x\sqrt{-10\,{x}^{2}-x+3}+1460\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.92077, size = 161, normalized size = 1.68 \begin{align*} -\frac{19}{250} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{75 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{25 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{283 \, \sqrt{-10 \, x^{2} - x + 3}}{375 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52198, size = 284, normalized size = 2.96 \begin{align*} \frac{57 \, \sqrt{5} \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\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 ) - 20 \,{\left (45 \, x^{2} + 145 \, x + 73\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{750 \,{\left (25 \, x^{2} + 30 \, x + 9\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 = 2.5855, size = 220, normalized size = 2.29 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{30000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{6}{625} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{19}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{61 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{183 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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