Optimal. Leaf size=91 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)}+\frac{5}{9} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{103 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{63 \sqrt{7}} \]
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Rubi [A] time = 0.175931, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)}+\frac{5}{9} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{103 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{63 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 16.2451, size = 80, normalized size = 0.88 \[ \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21 \left (3 x + 2\right )} + \frac{5 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{9} + \frac{103 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{441} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(3/2)/(2+3*x)**2/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.13841, size = 102, normalized size = 1.12 \[ \frac{1}{882} \left (\frac{42 \sqrt{1-2 x} \sqrt{5 x+3}}{3 x+2}+103 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+245 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^2),x]
[Out]
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Maple [A] time = 0.019, size = 131, normalized size = 1.4 \[ -{\frac{1}{1764+2646\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 309\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-735\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+206\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -490\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -42\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(3/2)/(2+3*x)^2/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.5099, size = 82, normalized size = 0.9 \[ \frac{5}{18} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{103}{882} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{21 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^2*sqrt(-2*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229806, size = 138, normalized size = 1.52 \[ \frac{\sqrt{7}{\left (35 \, \sqrt{10} \sqrt{7}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 103 \,{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{882 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^2*sqrt(-2*x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)**(3/2)/(2+3*x)**2/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.295393, size = 351, normalized size = 3.86 \[ -\frac{103}{8820} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{5}{18} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{22 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{21 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^2*sqrt(-2*x + 1)),x, algorithm="giac")
[Out]