Optimal. Leaf size=209 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{31603880465 \sqrt{5 x+3} \sqrt{1-2 x}}{4741632 (3 x+2)}+\frac{302171615 \sqrt{5 x+3} \sqrt{1-2 x}}{338688 (3 x+2)^2}+\frac{1729615 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^3}+\frac{21199 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}+\frac{497 \sqrt{5 x+3} \sqrt{1-2 x}}{108 (3 x+2)^5}-\frac{13391796605 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]
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Rubi [A] time = 0.448089, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{31603880465 \sqrt{5 x+3} \sqrt{1-2 x}}{4741632 (3 x+2)}+\frac{302171615 \sqrt{5 x+3} \sqrt{1-2 x}}{338688 (3 x+2)^2}+\frac{1729615 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^3}+\frac{21199 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}+\frac{497 \sqrt{5 x+3} \sqrt{1-2 x}}{108 (3 x+2)^5}-\frac{13391796605 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(5/2)/((2 + 3*x)^7*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 42.4404, size = 192, normalized size = 0.92 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{18 \left (3 x + 2\right )^{6}} + \frac{31603880465 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4741632 \left (3 x + 2\right )} + \frac{302171615 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{338688 \left (3 x + 2\right )^{2}} + \frac{1729615 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12096 \left (3 x + 2\right )^{3}} + \frac{21199 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{864 \left (3 x + 2\right )^{4}} + \frac{497 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{108 \left (3 x + 2\right )^{5}} - \frac{13391796605 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1229312} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)/(2+3*x)**7/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.158378, size = 112, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (31603880465 (3 x+2)^5+4230402610 (3 x+2)^4+678009080 (3 x+2)^3+116340112 (3 x+2)^2+20590976 (3 x+2)+4302592\right )}{(3 x+2)^6}-361578508335 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{66382848} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^7*Sqrt[3 + 5*x]),x]
[Out]
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Maple [B] time = 0.022, size = 346, normalized size = 1.7 \[{\frac{1}{7375872\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 29287859175135\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+117151436700540\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+195252394500900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+11946266815770\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+173557684000800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+40353920114760\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+86778842000400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+54544410839520\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+23141024533440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+36876342922048\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2571224948160\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +12470758445152\,x\sqrt{-10\,{x}^{2}-x+3}+1687693053312\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50612, size = 311, normalized size = 1.49 \[ \frac{13391796605}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{54 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{469 \, \sqrt{-10 \, x^{2} - x + 3}}{108 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{21199 \, \sqrt{-10 \, x^{2} - x + 3}}{864 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1729615 \, \sqrt{-10 \, x^{2} - x + 3}}{12096 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{302171615 \, \sqrt{-10 \, x^{2} - x + 3}}{338688 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{31603880465 \, \sqrt{-10 \, x^{2} - x + 3}}{4741632 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228641, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (853304772555 \, x^{5} + 2882422865340 \, x^{4} + 3896029345680 \, x^{3} + 2634024494432 \, x^{2} + 890768460368 \, x + 120549503808\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 40175389815 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7375872 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^7),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)/(2+3*x)**7/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.595288, size = 676, normalized size = 3.23 \[ \frac{2678359321}{4917248} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{6655 \,{\left (20305527 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 17887837240 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 7599643632000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 1749282956467200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 210267345272320000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 10389680589926400000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{263424 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^7),x, algorithm="giac")
[Out]