Optimal. Leaf size=57 \[ \frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.0859833, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[3 + 5*x]/((1 - 2*x)^(3/2)*(2 + 3*x)),x]
[Out]
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Rubi in Sympy [A] time = 7.81994, size = 53, normalized size = 0.93 \[ \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{49} + \frac{2 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1}} \]
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),x)
[Out]
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Mathematica [A] time = 0.0847609, size = 68, normalized size = 1.19 \[ \frac{\sqrt{7} (2 x-1) \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-14 \sqrt{1-2 x} \sqrt{5 x+3}}{98 x-49} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[3 + 5*x]/((1 - 2*x)^(3/2)*(2 + 3*x)),x]
[Out]
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Maple [B] time = 0.02, size = 108, normalized size = 1.9 \[ -{\frac{1}{-49+98\,x} \left ( 2\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-\sqrt{7}\arctan \left ({\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{14}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \right ) +14\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x),x)
[Out]
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Maxima [A] time = 1.50003, size = 78, normalized size = 1.37 \[ -\frac{1}{49} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{10 \, x}{7 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{6}{7 \, \sqrt{-10 \, x^{2} - x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22779, size = 85, normalized size = 1.49 \[ -\frac{\sqrt{7}{\left ({\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \, \sqrt{7} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{49 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 x + 3}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )}\, dx \]
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),x)
[Out]
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GIAC/XCAS [A] time = 0.248905, size = 135, normalized size = 2.37 \[ -\frac{1}{490} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{2 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{35 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)/((3*x + 2)*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]