Optimal. Leaf size=113 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{7}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{7 (73-60 x) \sqrt{1-2 x} \sqrt{5 x+3}}{4000}+\frac{10409 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4000 \sqrt{10}} \]
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Rubi [A] time = 0.0301405, 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 = {97, 153, 147, 54, 216} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{7}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{7 (73-60 x) \sqrt{1-2 x} \sqrt{5 x+3}}{4000}+\frac{10409 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4000 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{(7-21 x) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{7}{25} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{1}{75} \int \frac{(2+3 x) \left (-63+\frac{105 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{5 \sqrt{3+5 x}}-\frac{7 (73-60 x) \sqrt{1-2 x} \sqrt{3+5 x}}{4000}+\frac{7}{25} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{10409 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{8000}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{5 \sqrt{3+5 x}}-\frac{7 (73-60 x) \sqrt{1-2 x} \sqrt{3+5 x}}{4000}+\frac{7}{25} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{10409 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{4000 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{5 \sqrt{3+5 x}}-\frac{7 (73-60 x) \sqrt{1-2 x} \sqrt{3+5 x}}{4000}+\frac{7}{25} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{10409 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{4000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.032409, size = 83, normalized size = 0.73 \[ \frac{-10 \left (14400 x^4+19080 x^3-5490 x^2-5611 x+893\right )-10409 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{40000 \sqrt{1-2 x} \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 116, normalized size = 1. \begin{align*}{\frac{1}{80000} \left ( 144000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+52045\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+262800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+31227\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +76500\,x\sqrt{-10\,{x}^{2}-x+3}-17860\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.83693, size = 107, normalized size = 0.95 \begin{align*} \frac{10409}{80000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{9}{250} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{81}{200} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{693}{20000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28403, size = 267, normalized size = 2.36 \begin{align*} -\frac{10409 \, \sqrt{10}{\left (5 \, x + 3\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 (7200 \, x^{3} + 13140 \, x^{2} + 3825 \, x - 893\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{80000 \,{\left (5 \, x + 3\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} \left (3 x + 2\right )^{3}}{\left (5 x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.83025, size = 165, normalized size = 1.46 \begin{align*} \frac{9}{100000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 463 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{10409}{40000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{6250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{3125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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