Optimal. Leaf size=158 \[ \frac{38 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{8 \sqrt{1-2 x} \sqrt{5 x+3}}{49 (3 x+2)^{3/2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{212 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}-\frac{38}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.345145, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{38 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{8 \sqrt{1-2 x} \sqrt{5 x+3}}{49 (3 x+2)^{3/2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{212 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}-\frac{38}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[3 + 5*x]/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 31.5641, size = 143, normalized size = 0.91 \[ \frac{38 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{343 \sqrt{3 x + 2}} - \frac{8 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{49 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{38 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1029} - \frac{212 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11319} + \frac{2 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(1/2)/(1-2*x)**(3/2)/(2+3*x)**(5/2),x)
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Mathematica [A] time = 0.188817, size = 99, normalized size = 0.63 \[ \frac{2 \left (\sqrt{2} \left (140 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+19 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{3 \sqrt{5 x+3} \left (114 x^2-37 x-59\right )}{\sqrt{1-2 x} (3 x+2)^{3/2}}\right )}{1029} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[3 + 5*x]/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)),x]
[Out]
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Maple [C] time = 0.033, size = 267, normalized size = 1.7 \[ -{\frac{2}{10290\,{x}^{2}+1029\,x-3087}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 420\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+57\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+280\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +38\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1710\,{x}^{3}-471\,{x}^{2}+1218\,x+531 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)^(5/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{5 \, x + 3}}{{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)^(5/2)*(-2*x + 1)^(3/2)),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((3+5*x)**(1/2)/(1-2*x)**(3/2)/(2+3*x)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)^(5/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]