Optimal. Leaf size=93 \[ \frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}+\frac{33 \sqrt{5 x+3} \sqrt{1-2 x}}{4 (3 x+2)}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]
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Rubi [A] time = 0.0205677, antiderivative size = 93, 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} \[ \frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}+\frac{33 \sqrt{5 x+3} \sqrt{1-2 x}}{4 (3 x+2)}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 \sqrt{3+5 x}} \, dx &=\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{2 (2+3 x)^2}+\frac{33}{4} \int \frac{\sqrt{1-2 x}}{(2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{2 (2+3 x)^2}+\frac{33 \sqrt{1-2 x} \sqrt{3+5 x}}{4 (2+3 x)}+\frac{363}{8} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{2 (2+3 x)^2}+\frac{33 \sqrt{1-2 x} \sqrt{3+5 x}}{4 (2+3 x)}+\frac{363}{4} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{2 (2+3 x)^2}+\frac{33 \sqrt{1-2 x} \sqrt{3+5 x}}{4 (2+3 x)}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{4 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0365639, size = 69, normalized size = 0.74 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} (95 x+68)}{4 (3 x+2)^2}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 154, normalized size = 1.7 \begin{align*}{\frac{1}{56\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3267\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+4356\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1452\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1330\,x\sqrt{-10\,{x}^{2}-x+3}+952\,\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 = 1.58938, size = 103, normalized size = 1.11 \begin{align*} \frac{363}{56} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{6 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{95 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55027, size = 248, normalized size = 2.67 \begin{align*} -\frac{363 \, \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\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 (95 \, x + 68\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{56 \,{\left (9 \, x^{2} + 12 \, x + 4\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 = 2.29638, size = 346, normalized size = 3.72 \begin{align*} \frac{363}{560} \, \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{605 \,{\left (\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 )}^{3} + 168 \, \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 )}\right )}}{2 \,{\left ({\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 )}^{2} + 280\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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