Optimal. Leaf size=94 \[ \frac{9}{40} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{49 (5 x+3)^{3/2}}{22 \sqrt{1-2 x}}+\frac{17951 \sqrt{1-2 x} \sqrt{5 x+3}}{1760}-\frac{17951 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{160 \sqrt{10}} \]
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Rubi [A] time = 0.0241907, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 80, 50, 54, 216} \[ \frac{9}{40} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{49 (5 x+3)^{3/2}}{22 \sqrt{1-2 x}}+\frac{17951 \sqrt{1-2 x} \sqrt{5 x+3}}{1760}-\frac{17951 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{160 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 80
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2 \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac{49 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}-\frac{1}{22} \int \frac{\sqrt{3+5 x} \left (\frac{853}{2}+99 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{49 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{9}{40} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{17951}{880} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=\frac{17951 \sqrt{1-2 x} \sqrt{3+5 x}}{1760}+\frac{49 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{9}{40} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{17951}{320} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{17951 \sqrt{1-2 x} \sqrt{3+5 x}}{1760}+\frac{49 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{9}{40} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{17951 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{160 \sqrt{5}}\\ &=\frac{17951 \sqrt{1-2 x} \sqrt{3+5 x}}{1760}+\frac{49 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{9}{40} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{17951 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{160 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0294591, size = 64, normalized size = 0.68 \[ \frac{17951 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (360 x^2+1518 x-2809\right )}{1600 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 106, normalized size = 1.1 \begin{align*} -{\frac{1}{6400\,x-3200} \left ( 35902\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-7200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-17951\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -30360\,x\sqrt{-10\,{x}^{2}-x+3}+56180\,\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 = 2.57909, size = 88, normalized size = 0.94 \begin{align*} -\frac{17951}{3200} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{8} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{849}{160} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{4 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85273, size = 248, normalized size = 2.64 \begin{align*} \frac{17951 \, \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 (360 \, x^{2} + 1518 \, x - 2809\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3200 \,{\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} \sqrt{5 x + 3}}{\left (1 - 2 x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.67173, size = 96, normalized size = 1.02 \begin{align*} -\frac{17951}{1600} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 181 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 17951 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4000 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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