Optimal. Leaf size=86 \[ -\frac{22 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}+\frac{4}{15} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{14}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.0305258, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 157, 54, 216, 93, 204} \[ -\frac{22 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}+\frac{4}{15} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{14}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx &=-\frac{22 \sqrt{1-2 x}}{5 \sqrt{3+5 x}}-\frac{2}{5} \int \frac{\frac{79}{2}-2 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{22 \sqrt{1-2 x}}{5 \sqrt{3+5 x}}+\frac{4}{15} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx-\frac{49}{3} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{22 \sqrt{1-2 x}}{5 \sqrt{3+5 x}}-\frac{98}{3} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{8 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{15 \sqrt{5}}\\ &=-\frac{22 \sqrt{1-2 x}}{5 \sqrt{3+5 x}}+\frac{4}{15} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )+\frac{14}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [A] time = 0.106163, size = 112, normalized size = 1.3 \[ \frac{330 \sqrt{5 x+3} (2 x-1)-4 \sqrt{10-20 x} (5 x+3) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+350 \sqrt{7-14 x} (5 x+3) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{75 \sqrt{1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 124, normalized size = 1.4 \begin{align*}{\frac{1}{75} \left ( 10\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-875\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+6\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -525\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -330\,\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 = 1.69883, size = 93, normalized size = 1.08 \begin{align*} \frac{2}{75} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{7}{3} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{44 \, x}{5 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{22}{5 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53558, size = 366, normalized size = 4.26 \begin{align*} -\frac{2 \, \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 ) - 175 \, \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 ) + 330 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{75 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (1 - 2 x\right )^{\frac{3}{2}}}{\left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.29602, size = 270, normalized size = 3.14 \begin{align*} -\frac{7}{30} \, \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{2}{75} \, \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{11}{50} \, \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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