Optimal. Leaf size=157 \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{33}{125} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{5 x+3}}{200000}+\frac{66997 (1-2 x)^{3/2} \sqrt{5 x+3}}{800000}+\frac{2210901 \sqrt{1-2 x} \sqrt{5 x+3}}{8000000}+\frac{24319911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000000 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.235221, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{33}{125} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2-\frac{9 (2127-460 x) (1-2 x)^{5/2} \sqrt{5 x+3}}{200000}+\frac{66997 (1-2 x)^{3/2} \sqrt{5 x+3}}{800000}+\frac{2210901 \sqrt{1-2 x} \sqrt{5 x+3}}{8000000}+\frac{24319911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(2 + 3*x)^3)/(3 + 5*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 24.8715, size = 144, normalized size = 0.92 \[ - \frac{\left (- 3105 x + \frac{57429}{4}\right ) \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{150000} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3}}{5 \sqrt{5 x + 3}} + \frac{33 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{125} + \frac{66997 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{800000} + \frac{2210901 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8000000} + \frac{24319911 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{80000000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(2+3*x)**3/(3+5*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.19687, size = 75, normalized size = 0.48 \[ \frac{\frac{10 \sqrt{1-2 x} \left (34560000 x^5+12528000 x^4-39487200 x^3-4101140 x^2+20337375 x+6089453\right )}{\sqrt{5 x+3}}-24319911 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{80000000} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^3)/(3 + 5*x)^(3/2),x]
[Out]
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Maple [A] time = 0.023, size = 150, normalized size = 1. \[{\frac{1}{160000000} \left ( 691200000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+250560000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-789744000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+121599555\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-82022800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+72959733\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +406747500\,x\sqrt{-10\,{x}^{2}-x+3}+121789060\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(2+3*x)^3/(3+5*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.50577, size = 170, normalized size = 1.08 \[ -\frac{216 \, x^{6}}{25 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{297 \, x^{5}}{250 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{57189 \, x^{4}}{5000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{782123 \, x^{3}}{200000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{4477589 \, x^{2}}{800000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{24319911}{160000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{8158469 \, x}{8000000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{6089453}{8000000 \, \sqrt{-10 \, x^{2} - x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-2*x + 1)^(5/2)/(5*x + 3)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229431, size = 120, normalized size = 0.76 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (34560000 \, x^{5} + 12528000 \, x^{4} - 39487200 \, x^{3} - 4101140 \, x^{2} + 20337375 \, x + 6089453\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 24319911 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{160000000 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-2*x + 1)^(5/2)/(5*x + 3)^(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((1-2*x)**(5/2)*(2+3*x)**3/(3+5*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.322291, size = 203, normalized size = 1.29 \[ \frac{1}{200000000} \,{\left (4 \,{\left (24 \,{\left (36 \,{\left (16 \, \sqrt{5}{\left (5 \, x + 3\right )} - 211 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 22859 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 969335 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 5816745 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{24319911}{80000000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{156250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{78125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-2*x + 1)^(5/2)/(5*x + 3)^(3/2),x, algorithm="giac")
[Out]