Optimal. Leaf size=115 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt{5 x+3}}-\frac{1111 \sqrt{1-2 x}}{15 \sqrt{5 x+3}}-\frac{8}{45} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{665}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.0405006, antiderivative size = 115, 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, 150, 157, 54, 216, 93, 204} \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt{5 x+3}}-\frac{1111 \sqrt{1-2 x}}{15 \sqrt{5 x+3}}-\frac{8}{45} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{665}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx &=\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt{3+5 x}}+\frac{1}{3} \int \frac{\sqrt{1-2 x} \left (\frac{227}{2}+4 x\right )}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{1111 \sqrt{1-2 x}}{15 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt{3+5 x}}+\frac{2}{15} \int \frac{-\frac{7769}{4}-4 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{1111 \sqrt{1-2 x}}{15 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt{3+5 x}}-\frac{8}{45} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx-\frac{4655}{18} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{1111 \sqrt{1-2 x}}{15 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt{3+5 x}}-\frac{4655}{9} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{16 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{45 \sqrt{5}}\\ &=-\frac{1111 \sqrt{1-2 x}}{15 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt{3+5 x}}-\frac{8}{45} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )+\frac{665}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [C] time = 0.235379, size = 185, normalized size = 1.61 \[ \frac{121 \left (15 \sqrt{5 x+3} \left (6180 x^3+256544 x^2+34809 x-82313\right )+10477 \sqrt{10-20 x} \left (15 x^2+19 x+6\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+581875 \sqrt{7-14 x} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )-15450 \sqrt{22} (2 x-1)^3 \left (15 x^2+19 x+6\right ) \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{5}{11} (1-2 x)\right )}{952875 \sqrt{1-2 x} (3 x+2) (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 191, normalized size = 1.7 \begin{align*} -{\frac{1}{900+1350\,x} \left ( 120\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+249375\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+152\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+315875\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+48\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +99750\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +102090\,x\sqrt{-10\,{x}^{2}-x+3}+65610\,\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 = 1.78597, size = 139, normalized size = 1.21 \begin{align*} -\frac{4}{225} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{665}{18} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{6806 \, x}{45 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{10699}{135 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{27 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77791, size = 429, normalized size = 3.73 \begin{align*} \frac{8 \, \sqrt{5} \sqrt{2}{\left (15 \, x^{2} + 19 \, x + 6\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 ) + 16625 \, \sqrt{7}{\left (15 \, x^{2} + 19 \, x + 6\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 ) - 30 \,{\left (3403 \, x + 2187\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{450 \,{\left (15 \, x^{2} + 19 \, x + 6\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 = 2.90337, size = 431, normalized size = 3.75 \begin{align*} -\frac{133}{36} \, \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{4}{225} \, \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{121}{50} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{1078 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{3 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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