Optimal. Leaf size=129 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{21 \sqrt{3 x+2}}-\frac{205}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{41}{189} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{974}{189} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.247893, 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 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{21 \sqrt{3 x+2}}-\frac{205}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{41}{189} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{974}{189} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 24.8275, size = 114, normalized size = 0.88 \[ - \frac{205 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{189} + \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{21 \sqrt{3 x + 2}} - \frac{974 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{567} - \frac{451 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{6615} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(5/2)/(2+3*x)**(3/2)/(1-2*x)**(1/2),x)
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Mathematica [A] time = 0.221964, size = 97, normalized size = 0.75 \[ \frac{-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} (525 x+356)}{\sqrt{3 x+2}}-595 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1948 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1134} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)),x]
[Out]
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Maple [C] time = 0.025, size = 164, normalized size = 1.3 \[{\frac{1}{34020\,{x}^{3}+26082\,{x}^{2}-7938\,x-6804}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 595\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1948\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -31500\,{x}^{3}-24510\,{x}^{2}+7314\,x+6408 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(5/2)/(2+3*x)^(3/2)/(1-2*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^(3/2)*sqrt(-2*x + 1)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^(3/2)*sqrt(-2*x + 1)),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)**(5/2)/(2+3*x)**(3/2)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^(3/2)*sqrt(-2*x + 1)),x, algorithm="giac")
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