Optimal. Leaf size=108 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x}}+\frac{505}{84} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{475}{36} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{63 \sqrt{7}} \]
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Rubi [A] time = 0.0395885, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {98, 154, 157, 54, 216, 93, 204} \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x}}+\frac{505}{84} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{475}{36} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{63 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}-\frac{1}{7} \int \frac{\sqrt{3+5 x} \left (168+\frac{505 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{505}{84} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}+\frac{1}{42} \int \frac{-\frac{5543}{2}-\frac{16625 x}{4}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{505}{84} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}-\frac{1}{63} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx-\frac{2375}{72} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{505}{84} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}-\frac{2}{63} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{1}{36} \left (475 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=\frac{505}{84} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x}}-\frac{475}{36} \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 )}{63 \sqrt{7}}\\ \end{align*}
Mathematica [C] time = 0.0550931, size = 116, normalized size = 1.07 \[ \frac{8085 \sqrt{22} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{5}{11} (1-2 x)\right )-924 \sqrt{5 x+3}-490 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+8 \sqrt{7-14 x} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1764 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 146, normalized size = 1.4 \begin{align*} -{\frac{1}{14112\,x-7056} \left ( 46550\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+32\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-23275\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -16\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -29400\,x\sqrt{-10\,{x}^{2}-x+3}+75684\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\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 = 1.73091, size = 116, normalized size = 1.07 \begin{align*} -\frac{125 \, x^{2}}{6 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{475}{144} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{441} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3455 \, x}{84 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{901}{28 \, \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 = 1.9479, size = 389, normalized size = 3.6 \begin{align*} \frac{23275 \, \sqrt{5} \sqrt{2}{\left (2 \, x - 1\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 ) + 16 \, \sqrt{7}{\left (2 \, x - 1\right )} \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 ) + 84 \,{\left (350 \, x - 901\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{7056 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [B] time = 2.8324, size = 243, normalized size = 2.25 \begin{align*} -\frac{1}{4410} \, \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{475}{144} \, \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{{\left (70 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1111 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{420 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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