Optimal. Leaf size=94 \[ -\frac{12 (1-2 x)^{3/2}}{275 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2}}{825 (5 x+3)^{3/2}}+\frac{3}{55} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]
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Rubi [A] time = 0.0219834, 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, 78, 50, 54, 216} \[ -\frac{12 (1-2 x)^{3/2}}{275 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2}}{825 (5 x+3)^{3/2}}+\frac{3}{55} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}+\frac{2}{825} \int \frac{\sqrt{1-2 x} \left (\frac{1089}{2}+\frac{1485 x}{2}\right )}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}-\frac{12 (1-2 x)^{3/2}}{275 \sqrt{3+5 x}}+\frac{3}{11} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}-\frac{12 (1-2 x)^{3/2}}{275 \sqrt{3+5 x}}+\frac{3}{55} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{3}{10} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}-\frac{12 (1-2 x)^{3/2}}{275 \sqrt{3+5 x}}+\frac{3}{55} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{5 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}-\frac{12 (1-2 x)^{3/2}}{275 \sqrt{3+5 x}}+\frac{3}{55} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{5 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0445937, size = 78, normalized size = 0.83 \[ \frac{10 \left (-594 x^3-259 x^2+160 x+59\right )-99 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1650 \sqrt{1-2 x} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 113, normalized size = 1.2 \begin{align*}{\frac{1}{3300} \left ( 2475\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+2970\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+5940\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+891\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +5560\,x\sqrt{-10\,{x}^{2}-x+3}+1180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71765, size = 269, normalized size = 2.86 \begin{align*} -\frac{99 \, \sqrt{10}{\left (25 \, x^{2} + 30 \, x + 9\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 (297 \, x^{2} + 278 \, x + 59\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3300 \,{\left (25 \, x^{2} + 30 \, x + 9\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 )^{2}}{\left (5 x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.92101, size = 220, normalized size = 2.34 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{330000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{9}{625} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{3}{50} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{131 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{27500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{393 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{20625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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