Optimal. Leaf size=168 \[ \frac{\sqrt{5 x+3} (3 x+2)^5}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^4+\frac{10389 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3}{1600}+\frac{847637 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{32000}+\frac{49 \sqrt{1-2 x} \sqrt{5 x+3} (36265980 x+87394471)}{5120000}-\frac{35439958001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120000 \sqrt{10}} \]
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Rubi [A] time = 0.316331, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{5 x+3} (3 x+2)^5}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^4+\frac{10389 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3}{1600}+\frac{847637 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{32000}+\frac{49 \sqrt{1-2 x} \sqrt{5 x+3} (36265980 x+87394471)}{5120000}-\frac{35439958001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)^5*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 33.1101, size = 156, normalized size = 0.93 \[ \frac{33 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4} \sqrt{5 x + 3}}{20} + \frac{10389 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{1600} + \frac{847637 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{32000} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{33319369125 x}{8} + \frac{321174680925}{32}\right )}{12000000} - \frac{35439958001 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{51200000} + \frac{\left (3 x + 2\right )^{5} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**5*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.126889, size = 79, normalized size = 0.47 \[ \frac{35439958001 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (124416000 x^5+613267200 x^4+1429191360 x^3+2297649240 x^2+3810769458 x-5389783159\right )}{51200000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)^5*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]
[Out]
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Maple [A] time = 0.023, size = 157, normalized size = 0.9 \[ -{\frac{1}{-102400000+204800000\,x} \left ( -2488320000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-12265344000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-28583827200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+70879916002\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-45952984800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-35439958001\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -76215389160\,x\sqrt{-10\,{x}^{2}-x+3}+107795663180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^5*(3+5*x)^(1/2)/(1-2*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.51282, size = 150, normalized size = 0.89 \[ -\frac{243}{200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{103599}{16000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{35439958001}{102400000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1086219}{64000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{80155719}{256000} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{2961355719}{5120000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{16807 \, \sqrt{-10 \, x^{2} - x + 3}}{32 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^5/(-2*x + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23427, size = 120, normalized size = 0.71 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (124416000 \, x^{5} + 613267200 \, x^{4} + 1429191360 \, x^{3} + 2297649240 \, x^{2} + 3810769458 \, x - 5389783159\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 35439958001 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{102400000 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^5/(-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((2+3*x)**5*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.239403, size = 149, normalized size = 0.89 \[ -\frac{35439958001}{51200000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6 \,{\left (12 \,{\left (8 \,{\left (36 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} + 463 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 140711 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 10847547 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1789896455 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 177199790005 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{640000000 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^5/(-2*x + 1)^(3/2),x, algorithm="giac")
[Out]