Optimal. Leaf size=84 \[ -\frac{1}{10} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (2220 x+5363)}{1600}+\frac{44437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]
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Rubi [A] time = 0.0207372, antiderivative size = 84, 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 = {100, 147, 54, 216} \[ -\frac{1}{10} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (2220 x+5363)}{1600}+\frac{44437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 100
Rule 147
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx &=-\frac{1}{10} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{1}{30} \int \frac{\left (-171-\frac{555 x}{2}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{1}{10} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (5363+2220 x)}{1600}+\frac{44437 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3200}\\ &=-\frac{1}{10} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (5363+2220 x)}{1600}+\frac{44437 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1600 \sqrt{5}}\\ &=-\frac{1}{10} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (5363+2220 x)}{1600}+\frac{44437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1600 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0910888, size = 60, normalized size = 0.71 \[ \frac{-90 \sqrt{1-2 x} \sqrt{5 x+3} \left (160 x^2+460 x+667\right )-44437 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{16000} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 87, normalized size = 1. \begin{align*}{\frac{1}{32000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -28800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+44437\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -82800\,x\sqrt{-10\,{x}^{2}-x+3}-120060\,\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 = 4.10244, size = 78, normalized size = 0.93 \begin{align*} -\frac{9}{10} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{207}{80} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{44437}{32000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{6003}{1600} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73608, size = 221, normalized size = 2.63 \begin{align*} -\frac{9}{1600} \,{\left (160 \, x^{2} + 460 \, x + 667\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{44437}{32000} \, \sqrt{10} \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 ) \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 )^{3}}{\sqrt{1 - 2 x} \sqrt{5 x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.32757, size = 73, normalized size = 0.87 \begin{align*} -\frac{1}{80000} \, \sqrt{5}{\left (18 \,{\left (4 \,{\left (40 \, x + 91\right )}{\left (5 \, x + 3\right )} + 2243\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 222185 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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