Optimal. Leaf size=86 \[ -\frac{5}{6} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{29}{18} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9 \sqrt{7}} \]
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Rubi [A] time = 0.0294624, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {102, 157, 54, 216, 93, 204} \[ -\frac{5}{6} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{29}{18} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 102
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)} \, dx &=-\frac{5}{6} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{6} \int \frac{-49-\frac{145 x}{2}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{5}{6} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{1}{9} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx+\frac{145}{36} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{5}{6} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{2}{9} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{1}{18} \left (29 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{5}{6} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{29}{18} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{9 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0208794, size = 82, normalized size = 0.95 \[ \frac{1}{252} \left (-210 \sqrt{1-2 x} \sqrt{5 x+3}-203 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-8 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 83, normalized size = 1. \begin{align*}{\frac{1}{504}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 203\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +8\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -420\,\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 = 2.24716, size = 73, normalized size = 0.85 \begin{align*} \frac{29}{72} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1}{63} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{5}{6} \, \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 = 1.68732, size = 323, normalized size = 3.76 \begin{align*} -\frac{29}{72} \, \sqrt{5} \sqrt{2} \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 ) - \frac{1}{63} \, \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac{5}{6} \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} \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 (5 x + 3\right )^{\frac{3}{2}}}{\sqrt{1 - 2 x} \left (3 x + 2\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.42347, size = 216, normalized size = 2.51 \begin{align*} \frac{1}{630} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{29}{72} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{6} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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