3.590 \(\int \sqrt{\frac{-1+x}{5+3 x}} \, dx\)

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]

(Sqrt[-1 + x]*Sqrt[5 + 3*x])/3 - (8*ArcSinh[(Sqrt[3/2]*Sqrt[-1 + x])/2])/(3*Sqrt
[3])

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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]

(Sqrt[-1 + x]*Sqrt[5 + 3*x])/3 - (8*ArcSinh[(Sqrt[3/2]*Sqrt[-1 + x])/2])/(3*Sqrt
[3])

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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]

8*sqrt((x - 1)/(3*x + 5))/(3*(-3*(x - 1)/(3*x + 5) + 1)) - 8*sqrt(3)*atanh(sqrt(
3)*sqrt((x - 1)/(3*x + 5)))/9

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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]

(Sqrt[(-1 + x)/(5 + 3*x)]*(3*Sqrt[-1 + x]*(5 + 3*x) - 8*Sqrt[15 + 9*x]*ArcSinh[(
Sqrt[3/2]*Sqrt[-1 + x])/2]))/(9*Sqrt[-1 + x])

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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]

-1/9*((-1+x)/(5+3*x))^(1/2)*(5+3*x)*(4*ln(x*3^(1/2)+1/3*3^(1/2)+(3*x^2+2*x-5)^(1
/2))*3^(1/2)-3*(3*x^2+2*x-5)^(1/2))/((5+3*x)*(-1+x))^(1/2)

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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]

4/9*sqrt(3)*log(-(sqrt(3) - 3*sqrt((x - 1)/(3*x + 5)))/(sqrt(3) + 3*sqrt((x - 1)
/(3*x + 5)))) - 8/3*sqrt((x - 1)/(3*x + 5))/(3*(x - 1)/(3*x + 5) - 1)

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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]

1/9*sqrt(3)*(sqrt(3)*(3*x + 5)*sqrt((x - 1)/(3*x + 5)) + 4*log(-sqrt(3)*(3*x + 1
) + 3*(3*x + 5)*sqrt((x - 1)/(3*x + 5))))

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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]

Integral(sqrt((x - 1)/(3*x + 5)), x)

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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]

-4/9*sqrt(3)*ln(4)*sign(3*x + 5) + 4/9*sqrt(3)*ln(abs(-sqrt(3)*(sqrt(3)*x - sqrt
(3*x^2 + 2*x - 5)) - 1))*sign(3*x + 5) + 1/3*sqrt(3*x^2 + 2*x - 5)*sign(3*x + 5)