Optimal. Leaf size=116 \[ -\frac{938 (5 x+3)^{5/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{40787 \sqrt{1-2 x} (5 x+3)^{3/2}}{5808}-\frac{40787}{704} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{40787 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.141227, antiderivative size = 116, 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 \[ -\frac{938 (5 x+3)^{5/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{40787 \sqrt{1-2 x} (5 x+3)^{3/2}}{5808}-\frac{40787}{704} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{40787 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)^2*(3 + 5*x)^(3/2))/(1 - 2*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 12.3842, size = 105, normalized size = 0.91 \[ - \frac{40787 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{5808} - \frac{40787 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{704} + \frac{40787 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{640} - \frac{938 \left (5 x + 3\right )^{\frac{5}{2}}}{363 \sqrt{- 2 x + 1}} + \frac{49 \left (5 x + 3\right )^{\frac{5}{2}}}{66 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**2*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.148764, size = 74, normalized size = 0.64 \[ \frac{122361 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (2160 x^3+12780 x^2-52256 x+18351\right )}{1920 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)^2*(3 + 5*x)^(3/2))/(1 - 2*x)^(5/2),x]
[Out]
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Maple [A] time = 0.019, size = 137, normalized size = 1.2 \[{\frac{1}{3840\, \left ( -1+2\,x \right ) ^{2}} \left ( 489444\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-43200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-489444\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-255600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+122361\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1045120\,x\sqrt{-10\,{x}^{2}-x+3}-367020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^2*(3+5*x)^(3/2)/(1-2*x)^(5/2),x)
[Out]
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Maxima [A] time = 1.5333, size = 208, normalized size = 1.79 \[ \frac{40787}{1280} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{297}{64} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{24 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (2 \, x - 1\right )}} + \frac{539 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{5873 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(3*x + 2)^2/(-2*x + 1)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226328, size = 120, normalized size = 1.03 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (2160 \, x^{3} + 12780 \, x^{2} - 52256 \, x + 18351\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 122361 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3840 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(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)**(3/2)/(1-2*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.23517, size = 113, normalized size = 0.97 \[ \frac{40787}{640} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 247 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 81574 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1345971 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{24000 \,{\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(3*x + 2)^2/(-2*x + 1)^(5/2),x, algorithm="giac")
[Out]