Optimal. Leaf size=72 \[ \frac{9}{20} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{49 \sqrt{5 x+3}}{22 \sqrt{1-2 x}}-\frac{321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]
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Rubi [A] time = 0.0174986, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {89, 80, 54, 216} \[ \frac{9}{20} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{49 \sqrt{5 x+3}}{22 \sqrt{1-2 x}}-\frac{321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 80
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{49 \sqrt{3+5 x}}{22 \sqrt{1-2 x}}-\frac{1}{22} \int \frac{\frac{363}{2}+99 x}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{49 \sqrt{3+5 x}}{22 \sqrt{1-2 x}}+\frac{9}{20} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{321}{40} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{49 \sqrt{3+5 x}}{22 \sqrt{1-2 x}}+\frac{9}{20} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{321 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{20 \sqrt{5}}\\ &=\frac{49 \sqrt{3+5 x}}{22 \sqrt{1-2 x}}+\frac{9}{20} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{20 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0233546, size = 59, normalized size = 0.82 \[ \frac{10 \sqrt{5 x+3} (589-198 x)+3531 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2200 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 89, normalized size = 1.2 \begin{align*} -{\frac{1}{8800\,x-4400} \left ( 7062\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3531\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3960\,x\sqrt{-10\,{x}^{2}-x+3}+11780\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,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 = 3.10343, size = 68, normalized size = 0.94 \begin{align*} -\frac{321}{400} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{20} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{22 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76242, size = 231, normalized size = 3.21 \begin{align*} \frac{3531 \, \sqrt{10}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (198 \, x - 589\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{4400 \,{\left (2 \, x - 1\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 (3 x + 2\right )^{2}}{\left (1 - 2 x\right )^{\frac{3}{2}} \sqrt{5 x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.01226, size = 78, normalized size = 1.08 \begin{align*} -\frac{321}{200} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (198 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3539 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{5500 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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