3.956 \(\int \frac {1}{\sqrt {(a+b x) (c-d x)}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {\tan ^{-1}\left (\frac {-a d+b c-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {x (b c-a d)+a c-b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]

[Out]

-arctan(1/2*(-2*b*d*x-a*d+b*c)/b^(1/2)/d^(1/2)/(a*c+(-a*d+b*c)*x-b*d*x^2)^(1/2))/b^(1/2)/d^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1981, 621, 204} \[ -\frac {\tan ^{-1}\left (\frac {-a d+b c-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {x (b c-a d)+a c-b d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(a + b*x)*(c - d*x)],x]

[Out]

-(ArcTan[(b*c - a*d - 2*b*d*x)/(2*Sqrt[b]*Sqrt[d]*Sqrt[a*c + (b*c - a*d)*x - b*d*x^2])]/(Sqrt[b]*Sqrt[d]))

Rule 204

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

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {(a+b x) (c-d x)}} \, dx &=\int \frac {1}{\sqrt {a c+(b c-a d) x-b d x^2}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-4 b d-x^2} \, dx,x,\frac {b c-a d-2 b d x}{\sqrt {a c+(b c-a d) x-b d x^2}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {b c-a d-2 b d x}{2 \sqrt {b} \sqrt {d} \sqrt {a c+(b c-a d) x-b d x^2}}\right )}{\sqrt {b} \sqrt {d}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 94, normalized size = 1.45 \[ \frac {2 \sqrt {a+b x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d+b c}}\right )}{b \sqrt {d} \sqrt {(a+b x) (c-d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(a + b*x)*(c - d*x)],x]

[Out]

(2*Sqrt[b*c + a*d]*Sqrt[a + b*x]*Sqrt[(b*(c - d*x))/(b*c + a*d)]*ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c + a*d
]])/(b*Sqrt[d]*Sqrt[(a + b*x)*(c - d*x)])

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fricas [A]  time = 0.60, size = 202, normalized size = 3.11 \[ \left [-\frac {\sqrt {-b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 4 \, \sqrt {-b d x^{2} + a c + {\left (b c - a d\right )} x} {\left (2 \, b d x - b c + a d\right )} \sqrt {-b d} - 8 \, {\left (b^{2} c d - a b d^{2}\right )} x\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {\sqrt {-b d x^{2} + a c + {\left (b c - a d\right )} x} {\left (2 \, b d x - b c + a d\right )} \sqrt {b d}}{2 \, {\left (b^{2} d^{2} x^{2} - a b c d - {\left (b^{2} c d - a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 - 6*a*b*c*d + a^2*d^2 - 4*sqrt(-b*d*x^2 + a*c + (b*c - a*d)*x)*(2
*b*d*x - b*c + a*d)*sqrt(-b*d) - 8*(b^2*c*d - a*b*d^2)*x)/(b*d), -sqrt(b*d)*arctan(1/2*sqrt(-b*d*x^2 + a*c + (
b*c - a*d)*x)*(2*b*d*x - b*c + a*d)*sqrt(b*d)/(b^2*d^2*x^2 - a*b*c*d - (b^2*c*d - a*b*d^2)*x))/(b*d)]

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giac [A]  time = 0.74, size = 59, normalized size = 0.91 \[ -\frac {\log \left ({\left | b c - a d + 2 \, \sqrt {-b d} {\left (\sqrt {-b d} x - \sqrt {-b d x^{2} + b c x - a d x + a c}\right )} \right |}\right )}{\sqrt {-b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(b*c - a*d + 2*sqrt(-b*d)*(sqrt(-b*d)*x - sqrt(-b*d*x^2 + b*c*x - a*d*x + a*c))))/sqrt(-b*d)

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maple [A]  time = 0.01, size = 55, normalized size = 0.85 \[ \frac {\arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {-a d +b c}{2 b d}\right )}{\sqrt {-b d \,x^{2}+a c +\left (-a d +b c \right ) x}}\right )}{\sqrt {b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x+a)*(-d*x+c))^(1/2),x)

[Out]

1/(b*d)^(1/2)*arctan((b*d)^(1/2)*(x-1/2*(-a*d+b*c)/b/d)/(a*c+(-a*d+b*c)*x-b*d*x^2)^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\left (a+b\,x\right )\,\left (c-d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)*(c - d*x))^(1/2),x)

[Out]

int(1/((a + b*x)*(c - d*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (a + b x\right ) \left (c - d x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((b*x+a)*(-d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt((a + b*x)*(c - d*x)), x)

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