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 = 0.366443, 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 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{36 (3 x+2)}+\frac{41}{18} (5 x+3)^{3/2} \sqrt{1-2 x}-\frac{1649}{108} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{6829 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{162 \sqrt{10}}-\frac{1945}{324} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 37.2018, size = 150, normalized size = 0.91 \[ - \frac{115 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{252 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{6 \left (3 x + 2\right )^{2}} - \frac{85 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{126} - \frac{74 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{27} - \frac{6829 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1620} - \frac{1945 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{324} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**3,x)
[Out]
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Mathematica [A] time = 0.182596, size = 117, normalized size = 0.71 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (360 x^3-1230 x^2-3471 x-1628\right )}{(3 x+2)^2}-9725 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-6829 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{3240} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^3,x]
[Out]
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Maple [A] time = 0.018, size = 225, normalized size = 1.4 \[{\frac{1}{3240\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 87525\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-61461\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+10800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+116700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-81948\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-36900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+38900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -27316\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -104130\,x\sqrt{-10\,{x}^{2}-x+3}-48840\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^3,x)
[Out]
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Maxima [A] time = 1.50573, size = 176, normalized size = 1.07 \[ \frac{5}{9} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{205}{18} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{6829}{3240} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1945}{648} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{911}{108} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{5 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236003, size = 178, normalized size = 1.09 \[ \frac{\sqrt{10}{\left (1945 \, \sqrt{10} \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{10}{\left (360 \, x^{3} - 1230 \, x^{2} - 3471 \, x - 1628\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 13658 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{6480 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.456115, size = 481, normalized size = 2.93 \[ \frac{389}{1296} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{270} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 107 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{6829}{3240} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{77 \,{\left (41 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 17640 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{54 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^3,x, algorithm="giac")
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