Optimal. Leaf size=116 \[ -\frac{2 (1-2 x)^{7/2}}{165 (5 x+3)^{3/2}}-\frac{182 (1-2 x)^{5/2}}{825 \sqrt{5 x+3}}-\frac{91}{825} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{91}{250} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{1001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{250 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.118545, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{2 (1-2 x)^{7/2}}{165 (5 x+3)^{3/2}}-\frac{182 (1-2 x)^{5/2}}{825 \sqrt{5 x+3}}-\frac{91}{825} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{91}{250} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{1001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{250 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(2 + 3*x))/(3 + 5*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 11.3422, size = 107, normalized size = 0.92 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{7}{2}}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{182 \left (- 2 x + 1\right )^{\frac{5}{2}}}{825 \sqrt{5 x + 3}} - \frac{91 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{825} - \frac{91 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{250} - \frac{1001 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2500} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(2+3*x)/(3+5*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.156415, size = 65, normalized size = 0.56 \[ \frac{\frac{10 \sqrt{1-2 x} \left (900 x^3-2715 x^2-7970 x-3707\right )}{(5 x+3)^{3/2}}+3003 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{7500} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(2 + 3*x))/(3 + 5*x)^(5/2),x]
[Out]
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Maple [A] time = 0.016, size = 130, normalized size = 1.1 \[ -{\frac{1}{15000} \left ( 75075\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-18000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+90090\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+54300\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+27027\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +159400\,x\sqrt{-10\,{x}^{2}-x+3}+74140\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(2+3*x)/(3+5*x)^(5/2),x)
[Out]
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Maxima [A] time = 1.50813, size = 251, normalized size = 2.16 \[ -\frac{1001}{5000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{25 \,{\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{50 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac{11 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{150 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{100 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{121 \, \sqrt{-10 \, x^{2} - x + 3}}{750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{2959 \, \sqrt{-10 \, x^{2} - x + 3}}{1500 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224189, size = 120, normalized size = 1.03 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (900 \, x^{3} - 2715 \, x^{2} - 7970 \, x - 3707\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3003 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{15000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)*(-2*x + 1)^(5/2)/(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((1-2*x)**(5/2)*(2+3*x)/(3+5*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.300117, size = 238, normalized size = 2.05 \[ \frac{1}{6250} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 289 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{150000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{1001}{2500} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{627 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{12500 \, \sqrt{5 \, x + 3}} + \frac{11 \,{\left (\frac{171 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{9375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2),x, algorithm="giac")
[Out]