Optimal. Leaf size=189 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}-\frac{458 \sqrt{1-2 x} (3 x+2)^{5/2}}{825 \sqrt{5 x+3}}+\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6875}+\frac{2719 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{34375}-\frac{523 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}-\frac{47342 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625 \sqrt{33}} \]
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Rubi [A] time = 0.41411, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}-\frac{458 \sqrt{1-2 x} (3 x+2)^{5/2}}{825 \sqrt{5 x+3}}+\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6875}+\frac{2719 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{34375}-\frac{523 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}-\frac{47342 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625 \sqrt{33}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(3 + 5*x)^(5/2),x]
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Rubi in Sympy [A] time = 39.2212, size = 172, normalized size = 0.91 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{458 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{825 \sqrt{5 x + 3}} + \frac{2818 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6875} + \frac{2719 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{34375} - \frac{47342 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{515625} - \frac{5753 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{546875} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(7/2)*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)
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Mathematica [A] time = 0.411594, size = 107, normalized size = 0.57 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (222750 x^3+398475 x^2+221200 x+37273\right )}{(5 x+3)^{3/2}}+95165 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+94684 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1031250} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(3 + 5*x)^(5/2),x]
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Maple [C] time = 0.027, size = 277, normalized size = 1.5 \[ -{\frac{1}{6187500\,{x}^{2}+1031250\,x-2062500} \left ( 475825\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+473420\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+285495\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +284052\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -13365000\,{x}^{5}-26136000\,{x}^{4}-12801750\,{x}^{3}+3521120\,{x}^{2}+4051270\,x+745460 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(7/2)*(1-2*x)^(1/2)/(3+5*x)^(5/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(7/2)*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. \[{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(7/2)*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)**(7/2)*(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(7/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="giac")
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