Optimal. Leaf size=93 \[ \frac{3 \sqrt{1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}-\frac{41 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)}-\frac{451 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]
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Rubi [A] time = 0.123221, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 \sqrt{1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}-\frac{41 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)}-\frac{451 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[3 + 5*x]/(Sqrt[1 - 2*x]*(2 + 3*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 10.3984, size = 83, normalized size = 0.89 \[ - \frac{41 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{196 \left (3 x + 2\right )} + \frac{3 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{14 \left (3 x + 2\right )^{2}} - \frac{451 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1372} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(1/2)/(2+3*x)**3/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0744831, size = 72, normalized size = 0.77 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} (87 x+44)}{(3 x+2)^2}-451 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2744} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[3 + 5*x]/(Sqrt[1 - 2*x]*(2 + 3*x)^3),x]
[Out]
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Maple [B] time = 0.02, size = 154, normalized size = 1.7 \[{\frac{1}{2744\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4059\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+5412\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1804\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1218\,x\sqrt{-10\,{x}^{2}-x+3}+616\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50505, size = 103, normalized size = 1.11 \[ \frac{451}{2744} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{29 \, \sqrt{-10 \, x^{2} - x + 3}}{196 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)^3*sqrt(-2*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221353, size = 107, normalized size = 1.15 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (87 \, x + 44\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 451 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2744 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)^3*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)**(1/2)/(2+3*x)**3/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.301326, size = 347, normalized size = 3.73 \[ \frac{451}{27440} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (41 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 7000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)^3*sqrt(-2*x + 1)),x, algorithm="giac")
[Out]