Optimal. Leaf size=113 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^3}{11 \sqrt{1-2 x}}+\frac{243}{220} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{9 \sqrt{1-2 x} \sqrt{5 x+3} (11316 x+27269)}{7040}-\frac{184641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640 \sqrt{10}} \]
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Rubi [A] time = 0.0315155, antiderivative size = 113, 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 = {98, 153, 147, 54, 216} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^3}{11 \sqrt{1-2 x}}+\frac{243}{220} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{9 \sqrt{1-2 x} \sqrt{5 x+3} (11316 x+27269)}{7040}-\frac{184641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 153
Rule 147
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^4}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^3 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{1}{11} \int \frac{(2+3 x)^2 \left (222+\frac{729 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{243}{220} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{7 (2+3 x)^3 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{1}{330} \int \frac{\left (-\frac{39033}{2}-\frac{127305 x}{4}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{243}{220} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{7 (2+3 x)^3 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} \sqrt{3+5 x} (27269+11316 x)}{7040}-\frac{184641 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1280}\\ &=\frac{243}{220} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{7 (2+3 x)^3 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} \sqrt{3+5 x} (27269+11316 x)}{7040}-\frac{184641 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{640 \sqrt{5}}\\ &=\frac{243}{220} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{7 (2+3 x)^3 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} \sqrt{3+5 x} (27269+11316 x)}{7040}-\frac{184641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{640 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0379385, size = 69, normalized size = 0.61 \[ \frac{2031051 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (19008 x^3+78408 x^2+196614 x-312365\right )}{70400 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 123, normalized size = 1.1 \begin{align*} -{\frac{1}{281600\,x-140800} \left ( -380160\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4062102\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1568160\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-2031051\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3932280\,x\sqrt{-10\,{x}^{2}-x+3}+6247300\,\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 = 2.88895, size = 111, normalized size = 0.98 \begin{align*} \frac{27}{20} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{184641}{12800} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{999}{160} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{2187}{128} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2401 \, \sqrt{-10 \, x^{2} - x + 3}}{88 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83656, size = 278, normalized size = 2.46 \begin{align*} \frac{2031051 \, \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 (19008 \, x^{3} + 78408 \, x^{2} + 196614 \, x - 312365\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{140800 \,{\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 )^{4}}{\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 = 1.53166, size = 113, normalized size = 1. \begin{align*} -\frac{184641}{6400} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (594 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 93 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 5179 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 50776531 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4400000 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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