Optimal. Leaf size=140 \[ -\frac{1183 (5 x+3)^{7/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{24749 \sqrt{1-2 x} (5 x+3)^{5/2}}{2904}-\frac{123745 \sqrt{1-2 x} (5 x+3)^{3/2}}{2112}-\frac{123745}{256} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{272239}{256} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
[Out]
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Rubi [A] time = 0.167187, antiderivative size = 140, 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 \[ -\frac{1183 (5 x+3)^{7/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{24749 \sqrt{1-2 x} (5 x+3)^{5/2}}{2904}-\frac{123745 \sqrt{1-2 x} (5 x+3)^{3/2}}{2112}-\frac{123745}{256} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{272239}{256} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)^2*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 14.7156, size = 126, normalized size = 0.9 \[ - \frac{24749 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{2904} - \frac{123745 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{2112} - \frac{123745 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{256} + \frac{272239 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{512} - \frac{1183 \left (5 x + 3\right )^{\frac{7}{2}}}{363 \sqrt{- 2 x + 1}} + \frac{49 \left (5 x + 3\right )^{\frac{7}{2}}}{66 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**2*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.153539, size = 79, normalized size = 0.56 \[ \frac{816717 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-2 \sqrt{5 x+3} \left (28800 x^4+146160 x^3+497868 x^2-1713440 x+617319\right )}{1536 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)^2*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]
[Out]
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Maple [A] time = 0.019, size = 154, normalized size = 1.1 \[{\frac{1}{3072\, \left ( -1+2\,x \right ) ^{2}} \left ( -115200\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+3266868\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-584640\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-3266868\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1991472\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+816717\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +6853760\,x\sqrt{-10\,{x}^{2}-x+3}-2469276\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^2*(3+5*x)^(5/2)/(1-2*x)^(5/2),x)
[Out]
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Maxima [A] time = 1.52009, size = 333, normalized size = 2.38 \[ \frac{272239}{1024} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{49 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{5445}{256} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2695 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{96 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{1155 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{165 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{64 \,{\left (2 \, x - 1\right )}} + \frac{29645 \, \sqrt{-10 \, x^{2} - x + 3}}{192 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{104335 \, \sqrt{-10 \, x^{2} - x + 3}}{96 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^2/(-2*x + 1)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239262, size = 135, normalized size = 0.96 \[ -\frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (28800 \, x^{4} + 146160 \, x^{3} + 497868 \, x^{2} - 1713440 \, x + 617319\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 816717 \, \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3072 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^2/(-2*x + 1)^(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)**2*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.257254, size = 131, normalized size = 0.94 \[ \frac{272239}{512} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 107 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 24749 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 2722390 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 44919435 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{96000 \,{\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^2/(-2*x + 1)^(5/2),x, algorithm="giac")
[Out]