Optimal. Leaf size=116 \[ -\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2}}{825 (5 x+3)^{3/2}}+\frac{343 \sqrt{5 x+3} (1-2 x)^{3/2}}{18150}+\frac{343 \sqrt{5 x+3} \sqrt{1-2 x}}{5500}+\frac{343 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{500 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.142511, 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 \[ -\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2}}{825 (5 x+3)^{3/2}}+\frac{343 \sqrt{5 x+3} (1-2 x)^{3/2}}{18150}+\frac{343 \sqrt{5 x+3} \sqrt{1-2 x}}{5500}+\frac{343 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{500 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(2 + 3*x)^2)/(3 + 5*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 12.059, size = 105, normalized size = 0.91 \[ - \frac{388 \left (- 2 x + 1\right )^{\frac{5}{2}}}{9075 \sqrt{5 x + 3}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{825 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{343 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{18150} + \frac{343 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5500} + \frac{343 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{5000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(2+3*x)**2/(3+5*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.171384, size = 65, normalized size = 0.56 \[ \frac{\sqrt{1-2 x} \left (-2700 x^3+1845 x^2+3610 x+901\right )}{1500 (5 x+3)^{3/2}}-\frac{343 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{500 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^2)/(3 + 5*x)^(5/2),x]
[Out]
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Maple [A] time = 0.018, size = 130, normalized size = 1.1 \[{\frac{1}{30000} \left ( 25725\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-54000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+30870\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+36900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+9261\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +72200\,x\sqrt{-10\,{x}^{2}-x+3}+18020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(2+3*x)^2/(3+5*x)^(5/2),x)
[Out]
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Maxima [A] time = 1.52232, size = 208, normalized size = 1.79 \[ \frac{343}{10000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{297}{2500} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{375 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{250 \,{\left (5 \, x + 3\right )}} - \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{1875 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{116 \, \sqrt{-10 \, x^{2} - x + 3}}{375 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2*(-2*x + 1)^(3/2)/(5*x + 3)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21818, size = 120, normalized size = 1.03 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (2700 \, x^{3} - 1845 \, x^{2} - 3610 \, x - 901\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1029 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{30000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2*(-2*x + 1)^(3/2)/(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)**(3/2)*(2+3*x)**2/(3+5*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.299718, size = 238, normalized size = 2.05 \[ -\frac{3}{12500} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 149 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{150000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{343}{5000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{127 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{12500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{381 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{9375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^2*(-2*x + 1)^(3/2)/(5*x + 3)^(5/2),x, algorithm="giac")
[Out]