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

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {63, 217, 203} \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-b x-3}}{\sqrt {b x+2}}\right )}{b} \end {gather*}

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 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])

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]

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*}

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Mathematica [B]  time = 0.02, size = 53, normalized size = 2.04 \begin {gather*} -\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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.05, size = 26, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-b x-3}}{\sqrt {b x+2}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.82, size = 44, normalized size = 1.69 \begin {gather*} -\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 {gather*}

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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giac [C]  time = 1.07, size = 23, normalized size = 0.88 \begin {gather*} \frac {2 i \, \log \left (\sqrt {b x + 3} - \sqrt {b x + 2}\right )}{b} \end {gather*}

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*log(sqrt(b*x + 3) - sqrt(b*x + 2))/b

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maple [B]  time = 0.01, size = 66, normalized size = 2.54 \begin {gather*} \frac {\sqrt {\left (-b x -3\right ) \left (b x +2\right )}\, \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {5}{2 b}\right )}{\sqrt {-b^{2} x^{2}-5 b x -6}}\right )}{\sqrt {-b x -3}\, \sqrt {b x +2}\, \sqrt {b^{2}}} \end {gather*}

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 [A]  time = 3.01, size = 21, normalized size = 0.81 \begin {gather*} -\frac {\arcsin \left (-\frac {2 \, b^{2} x + 5 \, b}{b}\right )}{b} \end {gather*}

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]

-arcsin(-(2*b^2*x + 5*b)/b)/b

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mupad [B]  time = 0.30, size = 47, normalized size = 1.81 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {-b\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {b\,x+2}\right )\,\sqrt {b^2}}\right )}{\sqrt {b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*atan((b*(3^(1/2)*1i - (- b*x - 3)^(1/2)))/((2^(1/2) - (b*x + 2)^(1/2))*(b^2)^(1/2))))/(b^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- b x - 3} \sqrt {b x + 2}}\, dx \end {gather*}

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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