Optimal. Leaf size=218 \[ -\frac{1}{11} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{329683 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{34650}-\frac{43624697 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{623700}-\frac{43624697 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{283500 \sqrt{33}}-\frac{725140729 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{141750 \sqrt{33}} \]
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Rubi [A] time = 0.495028, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{1}{11} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{34}{99} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{1053}{770} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{329683 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{34650}-\frac{43624697 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{623700}-\frac{43624697 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{283500 \sqrt{33}}-\frac{725140729 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{141750 \sqrt{33}} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]
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Rubi in Sympy [A] time = 49.1627, size = 201, normalized size = 0.92 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{11} - \frac{170 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{297} - \frac{9001 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{4158} - \frac{156944 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{10395} - \frac{41741369 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{623700} - \frac{725140729 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{4677750} - \frac{43624697 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9355500} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(5/2)*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)
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Mathematica [A] time = 0.375741, size = 110, normalized size = 0.5 \[ \frac{2900562916 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (12757500 x^4+48384000 x^3+81985950 x^2+86822370 x+75000749\right )+292189583 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{9355500 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]
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Maple [C] time = 0.018, size = 184, normalized size = 0.8 \[{\frac{1}{561330000\,{x}^{3}+430353000\,{x}^{2}-130977000\,x-112266000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -11481750000\,{x}^{7}-52348275000\,{x}^{6}+1460947915\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2900562916\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -104493240000\,{x}^{5}-122253448500\,{x}^{4}-101481939900\,{x}^{3}-18760348110\,{x}^{2}+31378183890\,x+13500134820 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(5/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)/sqrt(-2*x + 1),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{\sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)/sqrt(-2*x + 1),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)**(5/2)*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(3*x + 2)^(5/2)/sqrt(-2*x + 1),x, algorithm="giac")
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