Optimal. Leaf size=218 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{9/2}}{15 (5 x+3)^{3/2}}-\frac{118 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1925}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{48125}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{240625}-\frac{31288 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{1473539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750 \sqrt{33}} \]
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Rubi [A] time = 0.495995, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{9/2}}{15 (5 x+3)^{3/2}}-\frac{118 \sqrt{1-2 x} (3 x+2)^{7/2}}{165 \sqrt{5 x+3}}+\frac{958 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{1925}+\frac{5153 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{48125}-\frac{12601 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{240625}-\frac{31288 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{109375 \sqrt{33}}-\frac{1473539 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{218750 \sqrt{33}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2))/(3 + 5*x)^(5/2),x]
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Rubi in Sympy [A] time = 47.1336, size = 201, normalized size = 0.92 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{9}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{118 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}}{165 \sqrt{5 x + 3}} + \frac{958 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1925} + \frac{5153 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{48125} - \frac{12601 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{240625} - \frac{1473539 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{7218750} - \frac{31288 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{3828125} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(9/2)*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)
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Mathematica [A] time = 0.459184, size = 112, normalized size = 0.51 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (3341250 x^4+8575875 x^3+6882975 x^2+1854575 x+54083\right )}{(5 x+3)^{3/2}}-441035 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1473539 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{7218750} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2))/(3 + 5*x)^(5/2),x]
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Maple [C] time = 0.052, size = 282, normalized size = 1.3 \[{\frac{1}{43312500\,{x}^{2}+7218750\,x-14437500} \left ( 2205175\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7367695\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+200475000\,{x}^{6}+1323105\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4420617\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +547965000\,{x}^{5}+431912250\,{x}^{4}+8586750\,{x}^{3}-115868770\,{x}^{2}-36550670\,x-1081660 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(9/2)*(1-2*x)^(1/2)/(3+5*x)^(5/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="fricas")
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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((2+3*x)**(9/2)*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(9/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="giac")
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