Optimal. Leaf size=209 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac{122343637 \sqrt{1-2 x} \sqrt{5 x+3}}{232339968 (3 x+2)}+\frac{958171 \sqrt{1-2 x} \sqrt{5 x+3}}{16595712 (3 x+2)^2}-\frac{71369 \sqrt{1-2 x} \sqrt{5 x+3}}{2963520 (3 x+2)^3}-\frac{149951 \sqrt{1-2 x} \sqrt{5 x+3}}{1481760 (3 x+2)^4}+\frac{503 \sqrt{1-2 x} \sqrt{5 x+3}}{26460 (3 x+2)^5}-\frac{52573169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8605184 \sqrt{7}} \]
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Rubi [A] time = 0.454823, 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 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac{122343637 \sqrt{1-2 x} \sqrt{5 x+3}}{232339968 (3 x+2)}+\frac{958171 \sqrt{1-2 x} \sqrt{5 x+3}}{16595712 (3 x+2)^2}-\frac{71369 \sqrt{1-2 x} \sqrt{5 x+3}}{2963520 (3 x+2)^3}-\frac{149951 \sqrt{1-2 x} \sqrt{5 x+3}}{1481760 (3 x+2)^4}+\frac{503 \sqrt{1-2 x} \sqrt{5 x+3}}{26460 (3 x+2)^5}-\frac{52573169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8605184 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^7),x]
[Out]
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Rubi in Sympy [A] time = 44.0773, size = 190, normalized size = 0.91 \[ \frac{122343637 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{232339968 \left (3 x + 2\right )} + \frac{958171 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{16595712 \left (3 x + 2\right )^{2}} - \frac{71369 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2963520 \left (3 x + 2\right )^{3}} - \frac{149951 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1481760 \left (3 x + 2\right )^{4}} + \frac{503 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{26460 \left (3 x + 2\right )^{5}} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{126 \left (3 x + 2\right )^{6}} - \frac{52573169 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{60236288} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(5/2)/(2+3*x)**7/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.134551, size = 92, normalized size = 0.44 \[ \frac{\frac{378 \sqrt{1-2 x} \sqrt{5 x+3} \left (16516390995 x^5+55658284380 x^4+74931979536 x^3+50261760608 x^2+16771747280 x+2225100096\right )}{(3 x+2)^6}-21292133445 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{48791393280} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^7),x]
[Out]
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Maple [B] time = 0.023, size = 346, normalized size = 1.7 \[{\frac{1}{1807088640\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 574887603015\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+2299550412060\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+3832584020100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+231229473930\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+3406741351200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+779215981320\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+1703370675600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1049047713504\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+454232180160\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+703664648512\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+50470242240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +234804461920\,x\sqrt{-10\,{x}^{2}-x+3}+31151401344\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(5/2)/(2+3*x)^7/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.51971, size = 311, normalized size = 1.49 \[ \frac{52573169}{120472576} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{378 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{853 \, \sqrt{-10 \, x^{2} - x + 3}}{26460 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac{149951 \, \sqrt{-10 \, x^{2} - x + 3}}{1481760 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{71369 \, \sqrt{-10 \, x^{2} - x + 3}}{2963520 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{958171 \, \sqrt{-10 \, x^{2} - x + 3}}{16595712 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{122343637 \, \sqrt{-10 \, x^{2} - x + 3}}{232339968 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^7*sqrt(-2*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230771, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (16516390995 \, x^{5} + 55658284380 \, x^{4} + 74931979536 \, x^{3} + 50261760608 \, x^{2} + 16771747280 \, x + 2225100096\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 788597535 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1807088640 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^7*sqrt(-2*x + 1)),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((3+5*x)**(5/2)/(2+3*x)**7/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.644462, size = 676, normalized size = 3.23 \[ \frac{52573169}{1204725760} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1331 \,{\left (118497 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 188015240 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 122630175360 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 17238395059200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 3670540357120000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 197895383347200000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{12907776 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^7*sqrt(-2*x + 1)),x, algorithm="giac")
[Out]