Optimal. Leaf size=113 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^3}{11 \sqrt{1-2 x}}+\frac{243}{220} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{9 \sqrt{1-2 x} \sqrt{5 x+3} (11316 x+27269)}{7040}-\frac{184641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640 \sqrt{10}} \]
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Rubi [A] time = 0.188224, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^3}{11 \sqrt{1-2 x}}+\frac{243}{220} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{9 \sqrt{1-2 x} \sqrt{5 x+3} (11316 x+27269)}{7040}-\frac{184641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^4/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 19.4104, size = 105, normalized size = 0.93 \[ \frac{243 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{220} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{1909575 x}{2} + \frac{18406575}{8}\right )}{66000} - \frac{184641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{6400} + \frac{7 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{11 \sqrt{- 2 x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0961245, size = 69, normalized size = 0.61 \[ \frac{2031051 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (19008 x^3+78408 x^2+196614 x-312365\right )}{70400 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^4/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]
[Out]
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Maple [A] time = 0.019, size = 123, normalized size = 1.1 \[ -{\frac{1}{-140800+281600\,x} \left ( -380160\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4062102\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1568160\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-2031051\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3932280\,x\sqrt{-10\,{x}^{2}-x+3}+6247300\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50258, size = 111, normalized size = 0.98 \[ \frac{27}{20} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{184641}{12800} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{999}{160} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{2187}{128} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2401 \, \sqrt{-10 \, x^{2} - x + 3}}{88 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/(sqrt(5*x + 3)*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238541, size = 107, normalized size = 0.95 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (19008 \, x^{3} + 78408 \, x^{2} + 196614 \, x - 312365\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2031051 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{140800 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/(sqrt(5*x + 3)*(-2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{4}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.23141, size = 113, normalized size = 1. \[ -\frac{184641}{6400} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (594 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 93 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 5179 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 50776531 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4400000 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/(sqrt(5*x + 3)*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]