Optimal. Leaf size=113 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{15 (5 x+3)^{3/2}}-\frac{392 \sqrt{1-2 x} (3 x+2)^2}{825 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1740 x+1243)}{11000}+\frac{1071 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1000 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.191253, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{15 (5 x+3)^{3/2}}-\frac{392 \sqrt{1-2 x} (3 x+2)^2}{825 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1740 x+1243)}{11000}+\frac{1071 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(3 + 5*x)^(5/2),x]
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Rubi in Sympy [A] time = 19.0309, size = 104, normalized size = 0.92 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{392 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{825 \sqrt{5 x + 3}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (45675 x + \frac{130515}{4}\right )}{41250} + \frac{1071 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{10000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**3*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.174709, size = 65, normalized size = 0.58 \[ \frac{\frac{10 \sqrt{1-2 x} \left (89100 x^3+147015 x^2+75470 x+11567\right )}{(5 x+3)^{3/2}}-35343 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{330000} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^3)/(3 + 5*x)^(5/2),x]
[Out]
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Maple [A] time = 0.018, size = 130, normalized size = 1.2 \[{\frac{1}{660000} \left ( 883575\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+1782000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1060290\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+2940300\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+318087\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1509400\,x\sqrt{-10\,{x}^{2}-x+3}+231340\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^3*(1-2*x)^(1/2)/(3+5*x)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221162, size = 120, normalized size = 1.06 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (89100 \, x^{3} + 147015 \, x^{2} + 75470 \, x + 11567\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 35343 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{660000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*sqrt(-2*x + 1)/(5*x + 3)^(5/2),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)**3*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.297112, size = 238, normalized size = 2.11 \[ \frac{27}{25000} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{1650000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{1071}{10000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{197 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{137500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{591 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{103125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="giac")
[Out]