Optimal. Leaf size=139 \[ \frac{\sqrt{5 x+3} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{27}{16} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3+\frac{2203}{320} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{\sqrt{1-2 x} \sqrt{5 x+3} (4618500 x+11129753)}{51200}-\frac{92108287 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200 \sqrt{10}} \]
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Rubi [A] time = 0.244564, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{5 x+3} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{27}{16} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3+\frac{2203}{320} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{\sqrt{1-2 x} \sqrt{5 x+3} (4618500 x+11129753)}{51200}-\frac{92108287 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)^4*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 25.8782, size = 129, normalized size = 0.93 \[ \frac{27 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{16} + \frac{2203 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{320} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{86596875 x}{4} + \frac{834731475}{16}\right )}{240000} - \frac{92108287 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{512000} + \frac{\left (3 x + 2\right )^{4} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**4*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)
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Mathematica [A] time = 0.114539, size = 74, normalized size = 0.53 \[ \frac{92108287 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (518400 x^4+2283840 x^3+5020200 x^2+9587886 x-14050073\right )}{512000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)^4*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]
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Maple [A] time = 0.019, size = 140, normalized size = 1. \[ -{\frac{1}{-1024000+2048000\,x} \left ( -10368000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-45676800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+184216574\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-100404000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-92108287\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -191757720\,x\sqrt{-10\,{x}^{2}-x+3}+281001460\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^4*(3+5*x)^(1/2)/(1-2*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.50924, size = 127, normalized size = 0.91 \[ -\frac{81}{160} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{92108287}{1024000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1557}{640} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{154953}{2560} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{6740553}{51200} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2401 \, \sqrt{-10 \, x^{2} - x + 3}}{16 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^4/(-2*x + 1)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.229259, size = 113, normalized size = 0.81 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (518400 \, x^{4} + 2283840 \, x^{3} + 5020200 \, x^{2} + 9587886 \, x - 14050073\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 92108287 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1024000 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(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((2+3*x)**4*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.240353, size = 131, normalized size = 0.94 \[ -\frac{92108287}{512000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6 \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 361 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 28181 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4651913 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 460541435 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{6400000 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^4/(-2*x + 1)^(3/2),x, algorithm="giac")
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