Optimal. Leaf size=129 \[ -\frac{22 \sqrt{3 x+2} (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{388}{225} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1196 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}+\frac{5594 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125} \]
[Out]
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Rubi [A] time = 0.257039, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{22 \sqrt{3 x+2} (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{388}{225} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1196 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}+\frac{5594 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(5/2)/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 26.397, size = 114, normalized size = 0.88 \[ - \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}}{5 \sqrt{5 x + 3}} - \frac{388 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{225} + \frac{5594 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3375} + \frac{1196 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3375} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)/(3+5*x)**(3/2)/(2+3*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.462311, size = 97, normalized size = 0.75 \[ \frac{2 \left (\frac{15 \sqrt{1-2 x} \sqrt{3 x+2} (20 x-1077)}{\sqrt{5 x+3}}-7070 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2797 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{3375} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(5/2)/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)),x]
[Out]
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Maple [C] time = 0.026, size = 164, normalized size = 1.3 \[{\frac{2}{101250\,{x}^{3}+77625\,{x}^{2}-23625\,x-20250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 7070\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2797\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1800\,{x}^{3}-96630\,{x}^{2}-16755\,x+32310 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*sqrt(3*x + 2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*sqrt(3*x + 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)/(3+5*x)**(3/2)/(2+3*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*sqrt(3*x + 2)),x, algorithm="giac")
[Out]