Optimal. Leaf size=93 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac{11 \sqrt{1-2 x} \sqrt{5 x+3}}{28 (3 x+2)}-\frac{121 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
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Rubi [A] time = 0.0215352, 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} \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac{11 \sqrt{1-2 x} \sqrt{5 x+3}}{28 (3 x+2)}-\frac{121 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^3} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac{11}{4} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{11 \sqrt{1-2 x} \sqrt{3+5 x}}{28 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac{121}{56} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{11 \sqrt{1-2 x} \sqrt{3+5 x}}{28 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}+\frac{121}{28} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{11 \sqrt{1-2 x} \sqrt{3+5 x}}{28 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}-\frac{121 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{28 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0360224, size = 69, normalized size = 0.74 \[ \frac{1}{196} \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (37 x+20)}{(3 x+2)^2}-121 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 154, normalized size = 1.7 \begin{align*}{\frac{1}{392\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1089\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1452\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+484\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +518\,x\sqrt{-10\,{x}^{2}-x+3}+280\,\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 = 3.4633, size = 122, normalized size = 1.31 \begin{align*} \frac{121}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{5}{21} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{37 \, \sqrt{-10 \, x^{2} - x + 3}}{84 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80934, size = 250, normalized size = 2.69 \begin{align*} -\frac{121 \, \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 (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{392 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{1 - 2 x} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.77022, size = 343, normalized size = 3.69 \begin{align*} \frac{121}{3920} \, \sqrt{5}{\left (\sqrt{70} \sqrt{2}{\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{280 \, \sqrt{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 )}^{3} - \frac{280 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{1120 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\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}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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