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

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 = 0.0123619, 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, Rules used = {1958, 50, 54, 215} \[ \frac{1}{3} \sqrt{x-1} \sqrt{3 x+5}-\frac{8 \sinh ^{-1}\left (\frac{1}{2} \sqrt{\frac{3}{2}} \sqrt{x-1}\right )}{3 \sqrt{3}} \]

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])

Rule 1958

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[(u*(e*(a + b*x
^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \sqrt{\frac{-1+x}{5+3 x}} \, dx &=\int \frac{\sqrt{-1+x}}{\sqrt{5+3 x}} \, dx\\ &=\frac{1}{3} \sqrt{-1+x} \sqrt{5+3 x}-\frac{4}{3} \int \frac{1}{\sqrt{-1+x} \sqrt{5+3 x}} \, dx\\ &=\frac{1}{3} \sqrt{-1+x} \sqrt{5+3 x}-\frac{8}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{8+3 x^2}} \, dx,x,\sqrt{-1+x}\right )\\ &=\frac{1}{3} \sqrt{-1+x} \sqrt{5+3 x}-\frac{8 \sinh ^{-1}\left (\frac{1}{2} \sqrt{\frac{3}{2}} \sqrt{-1+x}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0455404, size = 76, normalized size = 1.55 \[ \frac{3 (x-1) \sqrt{3 x+5}-8 \sqrt{3} \sqrt{x-1} \sinh ^{-1}\left (\frac{1}{2} \sqrt{\frac{3}{2}} \sqrt{x-1}\right )}{9 \sqrt{\frac{x-1}{3 x+5}} \sqrt{3 x+5}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-1 + x)/(5 + 3*x)],x]

[Out]

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

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Maple [B]  time = 0.01, size = 76, normalized size = 1.6 \begin{align*} -{\frac{5+3\,x}{9}\sqrt{{\frac{x-1}{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 ( x-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x-1)/(5+3*x))^(1/2),x)

[Out]

-1/9*((x-1)/(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)*(x-1))^(1/2)

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Maxima [B]  time = 1.47067, size = 108, normalized size = 2.2 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1+x)/(5+3*x))^(1/2),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 = 1.85112, size = 149, normalized size = 3.04 \begin{align*} \frac{1}{3} \,{\left (3 \, x + 5\right )} \sqrt{\frac{x - 1}{3 \, x + 5}} + \frac{4}{9} \, \sqrt{3} \log \left (\sqrt{3}{\left (3 \, x + 5\right )} \sqrt{\frac{x - 1}{3 \, x + 5}} - 3 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1+x)/(5+3*x))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{x - 1}{3 x + 5}}\, dx \end{align*}

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 [B]  time = 1.16954, size = 100, normalized size = 2.04 \begin{align*} -\frac{8}{9} \, \sqrt{3} \log \left (2\right ) \mathrm{sgn}\left (3 \, x + 5\right ) + \frac{4}{9} \, \sqrt{3} \log \left ({\left | -\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2 \, x - 5}\right )} - 1 \right |}\right ) \mathrm{sgn}\left (3 \, x + 5\right ) + \frac{1}{3} \, \sqrt{3 \, x^{2} + 2 \, x - 5} \mathrm{sgn}\left (3 \, x + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1+x)/(5+3*x))^(1/2),x, algorithm="giac")

[Out]

-8/9*sqrt(3)*log(2)*sgn(3*x + 5) + 4/9*sqrt(3)*log(abs(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2*x - 5)) - 1))*sgn(
3*x + 5) + 1/3*sqrt(3*x^2 + 2*x - 5)*sgn(3*x + 5)