Optimal. Leaf size=166 \[ -\frac{36657025 \sqrt{1-2 x}}{332024 \sqrt{5 x+3}}-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{6525}{392 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{37}{28 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{1}{7 \sqrt{1-2 x} (3 x+2)^3 \sqrt{5 x+3}}+\frac{2079585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.421736, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{36657025 \sqrt{1-2 x}}{332024 \sqrt{5 x+3}}-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{6525}{392 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{37}{28 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{1}{7 \sqrt{1-2 x} (3 x+2)^3 \sqrt{5 x+3}}+\frac{2079585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 36.2737, size = 153, normalized size = 0.92 \[ - \frac{36657025 \sqrt{- 2 x + 1}}{332024 \sqrt{5 x + 3}} + \frac{2079585 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} - \frac{73435}{15092 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{6525}{392 \sqrt{- 2 x + 1} \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{37}{28 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{1}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.139571, size = 87, normalized size = 0.52 \[ \frac{1979479350 x^4+2925598635 x^3+622325745 x^2-723664682 x-283149136}{332024 \sqrt{1-2 x} (3 x+2)^3 \sqrt{5 x+3}}+\frac{2079585 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{5488 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.024, size = 305, normalized size = 1.8 \[ -{\frac{1}{4648336\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) }\sqrt{1-2\,x} \left ( 67940041950\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+142674088095\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+83792718405\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+27712710900\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-11574970110\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+40958380890\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-25162978500\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+8712560430\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-6039114840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -10131305548\,x\sqrt{-10\,{x}^{2}-x+3}-3964087904\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.50658, size = 285, normalized size = 1.72 \[ -\frac{2079585}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{36657025 \, x}{166012 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{38272595}{332024 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{7 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{37}{28 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{6525}{392 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242611, size = 167, normalized size = 1.01 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1979479350 \, x^{4} + 2925598635 \, x^{3} + 622325745 \, x^{2} - 723664682 \, x - 283149136\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 251629785 \,{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{4648336 \,{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*(-2*x + 1)^(3/2)),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/(1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.490994, size = 544, normalized size = 3.28 \[ -\frac{415917}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{625}{242} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{64 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1452605 \,{\left (2 \, x - 1\right )}} - \frac{297 \,{\left (37841 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 16959040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 2009470400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9604 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]