Optimal. Leaf size=108 \[ -\frac{22 (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{128}{75} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{338}{225} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{98}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.0433073, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {98, 154, 157, 54, 216, 93, 204} \[ -\frac{22 (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{128}{75} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{338}{225} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{98}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx &=-\frac{22 (1-2 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{2}{5} \int \frac{\sqrt{1-2 x} \left (\frac{167}{2}+64 x\right )}{(2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{128}{75} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{2}{75} \int \frac{\frac{2633}{2}-169 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{128}{75} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{338}{225} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx-\frac{343}{9} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{22 (1-2 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{128}{75} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{686}{9} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{676 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{225 \sqrt{5}}\\ &=-\frac{22 (1-2 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{128}{75} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{338}{225} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )+\frac{98}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [C] time = 0.198159, size = 153, normalized size = 1.42 \[ \frac{1694 \left (165 \sqrt{5 x+3} (2 x-1)-2 \sqrt{10-20 x} (5 x+3) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+175 \sqrt{7-14 x} (5 x+3) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )-60 \sqrt{22} (2 x-1)^3 (5 x+3) \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{5}{11} (1-2 x)\right )}{27225 \sqrt{1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 139, normalized size = 1.3 \begin{align*}{\frac{1}{1125} \left ( 845\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-30625\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+507\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -18375\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +300\,x\sqrt{-10\,{x}^{2}-x+3}-10710\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\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 = 3.68349, size = 116, normalized size = 1.07 \begin{align*} -\frac{8 \, x^{2}}{15 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{169}{1125} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{49}{9} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{1448 \, x}{75 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{238}{25 \, \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 = 1.91939, size = 389, normalized size = 3.6 \begin{align*} -\frac{169 \, \sqrt{5} \sqrt{2}{\left (5 \, x + 3\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 ) - 6125 \, \sqrt{7}{\left (5 \, x + 3\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 ) - 30 \,{\left (10 \, x - 357\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1125 \,{\left (5 \, x + 3\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 = 1.86338, size = 296, normalized size = 2.74 \begin{align*} -\frac{49}{90} \, \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{169}{1125} \, \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{4}{375} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{121}{250} \, \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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