Optimal. Leaf size=79 \[ \frac{2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{2 \sqrt{5 x+3}}{49 \sqrt{1-2 x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
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Rubi [A] time = 0.0181559, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{2 \sqrt{5 x+3}}{49 \sqrt{1-2 x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\frac{2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{1}{7} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=-\frac{2 \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}+\frac{1}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}+\frac{2}{49} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{2 \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{49 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0402798, size = 71, normalized size = 0.9 \[ \frac{2 \left (7 \sqrt{5 x+3} (41 x+18)+3 \sqrt{7-14 x} (2 x-1) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{1029 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 154, normalized size = 2. \begin{align*}{\frac{1}{1029\, \left ( 2\,x-1 \right ) ^{2}} \left ( 12\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-12\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +574\,x\sqrt{-10\,{x}^{2}-x+3}+252\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\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 = 1.8452, size = 140, normalized size = 1.77 \begin{align*} \frac{1}{343} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{205 \, x}{147 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{125 \, x^{2}}{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{37}{588 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1385 \, x}{84 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{67}{28 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51716, size = 246, normalized size = 3.11 \begin{align*} -\frac{3 \, \sqrt{7}{\left (4 \, x^{2} - 4 \, x + 1\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 ) - 14 \,{\left (41 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1029 \,{\left (4 \, x^{2} - 4 \, x + 1\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 [A] time = 2.2288, size = 153, normalized size = 1.94 \begin{align*} \frac{1}{3430} \, \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 \,{\left (41 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{3675 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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