Optimal. Leaf size=164 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{36 (3 x+2)}+\frac{41}{18} (5 x+3)^{3/2} \sqrt{1-2 x}-\frac{1649}{108} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{6829 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{162 \sqrt{10}}-\frac{1945}{324} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.0673162, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{36 (3 x+2)}+\frac{41}{18} (5 x+3)^{3/2} \sqrt{1-2 x}-\frac{1649}{108} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{6829 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{162 \sqrt{10}}-\frac{1945}{324} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}-\frac{1}{18} \int \frac{\left (-\frac{1335}{4}-1230 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{2+3 x} \, dx\\ &=\frac{41}{18} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}-\frac{1}{540} \int \frac{\left (\frac{2115}{2}-49470 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{1649}{108} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{41}{18} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}+\frac{\int \frac{-68505-204870 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{3240}\\ &=-\frac{1649}{108} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{41}{18} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}+\frac{13615}{648} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx-\frac{6829}{324} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{1649}{108} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{41}{18} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}+\frac{13615}{324} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{6829 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{162 \sqrt{5}}\\ &=-\frac{1649}{108} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{41}{18} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{6 (2+3 x)^2}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{36 (2+3 x)}-\frac{6829 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{162 \sqrt{10}}-\frac{1945}{324} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [A] time = 0.15825, size = 131, normalized size = 0.8 \[ \frac{-15 \sqrt{5 x+3} \left (720 x^4-2820 x^3-5712 x^2+215 x+1628\right )+6829 \sqrt{10-20 x} (3 x+2)^2 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-9725 \sqrt{7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1620 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 225, normalized size = 1.4 \begin{align*} -{\frac{1}{3240\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 61461\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-87525\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-10800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+81948\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-116700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+36900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+27316\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -38900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +104130\,x\sqrt{-10\,{x}^{2}-x+3}+48840\,\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 = 3.85964, size = 176, normalized size = 1.07 \begin{align*} \frac{5}{9} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{205}{18} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{6829}{3240} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1945}{648} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{911}{108} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{5 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59615, size = 440, normalized size = 2.68 \begin{align*} -\frac{9725 \, \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\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 ) - 6829 \, \sqrt{10}{\left (9 \, x^{2} + 12 \, x + 4\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 ) - 30 \,{\left (360 \, x^{3} - 1230 \, x^{2} - 3471 \, x - 1628\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3240 \,{\left (9 \, x^{2} + 12 \, x + 4\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 = 3.20841, size = 481, normalized size = 2.93 \begin{align*} \frac{389}{1296} \, \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{1}{270} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 107 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{6829}{3240} \, \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{77 \,{\left (41 \, \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} + 17640 \, \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 )}\right )}}{54 \,{\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 )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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