Optimal. Leaf size=74 \[ -\frac{2 (1-2 x)^{3/2}}{165 (5 x+3)^{3/2}}-\frac{6 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{6}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
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Rubi [A] time = 0.0158633, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {78, 47, 54, 216} \[ -\frac{2 (1-2 x)^{3/2}}{165 (5 x+3)^{3/2}}-\frac{6 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{6}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}+\frac{3}{5} \int \frac{\sqrt{1-2 x}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac{6 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}-\frac{6}{25} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac{6 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}-\frac{12 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac{6 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}-\frac{6}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0780631, size = 73, normalized size = 0.99 \[ \frac{10 \left (970 x^2+119 x-302\right )+198 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4125 \sqrt{1-2 x} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 96, normalized size = 1.3 \begin{align*} -{\frac{1}{4125} \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+891\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +4850\,x\sqrt{-10\,{x}^{2}-x+3}+3020\,\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 [A] time = 2.34406, size = 65, normalized size = 0.88 \begin{align*} -\frac{4 \, \sqrt{-10 \, x^{2} - x + 3}}{15 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{8 \, \sqrt{-10 \, x^{2} - x + 3}}{165 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74078, size = 274, normalized size = 3.7 \begin{align*} \frac{99 \, \sqrt{5} \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 10 \,{\left (485 \, x + 302\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{4125 \,{\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 )}{\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 = 2.14845, size = 194, normalized size = 2.62 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{66000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{6}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{13 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1100 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{195 \, \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}}}{4125 \,{\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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