3.1542 \(\int \frac{1}{\sqrt{-3-b x} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3}}{\sqrt{b x+2}}\right )}{b} \]

[Out]

(-2*ArcTan[Sqrt[-3 - b*x]/Sqrt[2 + b*x]])/b

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Rubi [A]  time = 0.0137541, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {63, 217, 203} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3}}{\sqrt{b x+2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-3 - b*x]*Sqrt[2 + b*x]),x]

[Out]

(-2*ArcTan[Sqrt[-3 - b*x]/Sqrt[2 + b*x]])/b

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

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

Rule 203

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

Rubi steps

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

Mathematica [B]  time = 0.0153942, size = 53, normalized size = 2.04 \[ -\frac{2 \sqrt{-b x-3} \sqrt{-b x-2} \sin ^{-1}\left (\sqrt{b x+3}\right )}{b \sqrt{b x+2} \sqrt{b x+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-3 - b*x]*Sqrt[2 + b*x]),x]

[Out]

(-2*Sqrt[-3 - b*x]*Sqrt[-2 - b*x]*ArcSin[Sqrt[3 + b*x]])/(b*Sqrt[2 + b*x]*Sqrt[3 + b*x])

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Maple [B]  time = 0.006, size = 66, normalized size = 2.5 \begin{align*}{\sqrt{ \left ( -bx-3 \right ) \left ( bx+2 \right ) }\arctan \left ({\sqrt{{b}^{2}} \left ( x+{\frac{5}{2\,b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-5\,bx-6}}}} \right ){\frac{1}{\sqrt{-bx-3}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-b*x-3)*(b*x+2))^(1/2)/(-b*x-3)^(1/2)/(b*x+2)^(1/2)/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+5/2/b)/(-b^2*x^2-5*b*x
-6)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.04065, size = 107, normalized size = 4.12 \begin{align*} -\frac{\arctan \left (\frac{{\left (2 \, b x + 5\right )} \sqrt{b x + 2} \sqrt{-b x - 3}}{2 \,{\left (b^{2} x^{2} + 5 \, b x + 6\right )}}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(1/2*(2*b*x + 5)*sqrt(b*x + 2)*sqrt(-b*x - 3)/(b^2*x^2 + 5*b*x + 6))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- b x - 3} \sqrt{b x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x-3)**(1/2)/(b*x+2)**(1/2),x)

[Out]

Integral(1/(sqrt(-b*x - 3)*sqrt(b*x + 2)), x)

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Giac [C]  time = 1.06312, size = 20, normalized size = 0.77 \begin{align*} -\frac{2 i \, \arcsin \left (i \, \sqrt{b x + 2}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*arcsin(I*sqrt(b*x + 2))/b