Optimal. Leaf size=113 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^2}{1815 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x} (1051875 x+627641)}{399300 \sqrt{5 x+3}}-\frac{621 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]
[Out]
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Rubi [A] time = 0.197289, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^2}{1815 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x} (1051875 x+627641)}{399300 \sqrt{5 x+3}}-\frac{621 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 19.1783, size = 107, normalized size = 0.95 \[ - \frac{107 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{1815 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{2 \sqrt{- 2 x + 1} \left (\frac{5259375 x}{8} + \frac{3138205}{8}\right )}{499125 \sqrt{5 x + 3}} - \frac{621 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1000} + \frac{7 \left (3 x + 2\right )^{3}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.205712, size = 65, normalized size = 0.58 \[ \frac{-3234330 x^3+6746215 x^2+11581424 x+3821563}{399300 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{621 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{100 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.026, size = 151, normalized size = 1.3 \[ -{\frac{1}{-7986000+15972000\,x}\sqrt{1-2\,x} \left ( 123982650\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+86787855\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-64686600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-29755836\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+134924300\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-22316877\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +231628480\,x\sqrt{-10\,{x}^{2}-x+3}+76431260\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(5/2),x)
[Out]
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Maxima [A] time = 1.50871, size = 128, normalized size = 1.13 \[ -\frac{621}{2000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{81 \, x^{2}}{50 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{8686813 \, x}{1996500 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{31846681}{9982500 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{20625 \,{\left (5 \, \sqrt{-10 \, x^{2} - x + 3} x + 3 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23162, size = 134, normalized size = 1.19 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (3234330 \, x^{3} - 6746215 \, x^{2} - 11581424 \, x - 3821563\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2479653 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7986000 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{4}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265721, size = 247, normalized size = 2.19 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{39930000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{621}{1000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (215622 \, \sqrt{5}{\left (5 \, x + 3\right )} - 4187171 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{16637500 \,{\left (2 \, x - 1\right )}} - \frac{271 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{3327500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{813 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{2495625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^4/((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]