Optimal. Leaf size=49 \[ \frac{1}{3} \sqrt{x-1} \sqrt{3 x+5}-\frac{8 \sinh ^{-1}\left (\frac{1}{2} \sqrt{\frac{3}{2}} \sqrt{x-1}\right )}{3 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0366841, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{3} \sqrt{x-1} \sqrt{3 x+5}-\frac{8 \sinh ^{-1}\left (\frac{1}{2} \sqrt{\frac{3}{2}} \sqrt{x-1}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(-1 + x)/(5 + 3*x)],x]
[Out]
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Rubi in Sympy [A] time = 2.16425, size = 51, normalized size = 1.04 \[ \frac{8 \sqrt{\frac{x - 1}{3 x + 5}}}{3 \left (- \frac{3 \left (x - 1\right )}{3 x + 5} + 1\right )} - \frac{8 \sqrt{3} \operatorname{atanh}{\left (\sqrt{3} \sqrt{\frac{x - 1}{3 x + 5}} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((-1+x)/(5+3*x))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0675081, size = 71, normalized size = 1.45 \[ \frac{\sqrt{\frac{x-1}{3 x+5}} \left (3 \sqrt{x-1} (3 x+5)-8 \sqrt{9 x+15} \sinh ^{-1}\left (\frac{1}{2} \sqrt{\frac{3}{2}} \sqrt{x-1}\right )\right )}{9 \sqrt{x-1}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(-1 + x)/(5 + 3*x)],x]
[Out]
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Maple [B] time = 0.015, size = 76, normalized size = 1.6 \[ -{\frac{5+3\,x}{9}\sqrt{{\frac{-1+x}{5+3\,x}}} \left ( 4\,\ln \left ( x\sqrt{3}+1/3\,\sqrt{3}+\sqrt{3\,{x}^{2}+2\,x-5} \right ) \sqrt{3}-3\,\sqrt{3\,{x}^{2}+2\,x-5} \right ){\frac{1}{\sqrt{ \left ( 5+3\,x \right ) \left ( -1+x \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((-1+x)/(5+3*x))^(1/2),x)
[Out]
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Maxima [A] time = 0.803245, size = 108, normalized size = 2.2 \[ \frac{4}{9} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - 3 \, \sqrt{\frac{x - 1}{3 \, x + 5}}}{\sqrt{3} + 3 \, \sqrt{\frac{x - 1}{3 \, x + 5}}}\right ) - \frac{8 \, \sqrt{\frac{x - 1}{3 \, x + 5}}}{3 \,{\left (\frac{3 \,{\left (x - 1\right )}}{3 \, x + 5} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x - 1)/(3*x + 5)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285896, size = 84, normalized size = 1.71 \[ \frac{1}{9} \, \sqrt{3}{\left (\sqrt{3}{\left (3 \, x + 5\right )} \sqrt{\frac{x - 1}{3 \, x + 5}} + 4 \, \log \left (-\sqrt{3}{\left (3 \, x + 1\right )} + 3 \,{\left (3 \, x + 5\right )} \sqrt{\frac{x - 1}{3 \, x + 5}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x - 1)/(3*x + 5)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x - 1}{3 x + 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((-1+x)/(5+3*x))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.275127, size = 100, normalized size = 2.04 \[ -\frac{4}{9} \, \sqrt{3}{\rm ln}\left (4\right ){\rm sign}\left (3 \, x + 5\right ) + \frac{4}{9} \, \sqrt{3}{\rm ln}\left ({\left | -\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2 \, x - 5}\right )} - 1 \right |}\right ){\rm sign}\left (3 \, x + 5\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 2 \, x - 5}{\rm sign}\left (3 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x - 1)/(3*x + 5)),x, algorithm="giac")
[Out]