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

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[(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]))

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Rubi [A]  time = 0.0585131, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{\tan ^{-1}\left (\frac{-a d+b c-2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (b c-a d)+a c-b d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]

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]))

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Rubi in Sympy [A]  time = 2.49379, size = 60, normalized size = 0.92 \[ - \frac{\operatorname{atan}{\left (\frac{- a d + b c - 2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{a c - b d x^{2} + x \left (- a d + b c\right )}} \right )}}{\sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((b*x+a)*(-d*x+c))**(1/2),x)

[Out]

-atan((-a*d + b*c - 2*b*d*x)/(2*sqrt(b)*sqrt(d)*sqrt(a*c - b*d*x**2 + x*(-a*d +
b*c))))/(sqrt(b)*sqrt(d))

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Mathematica [C]  time = 0.12729, size = 99, normalized size = 1.52 \[ \frac{i \sqrt{a+b x} \sqrt{c-d x} \log \left (2 \sqrt{a+b x} \sqrt{c-d x}-\frac{i (a d-b c+2 b d x)}{\sqrt{b} \sqrt{d}}\right )}{\sqrt{b} \sqrt{d} \sqrt{(a+b x) (c-d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 55, normalized size = 0.9 \[{1\arctan \left ({1\sqrt{bd} \left ( x-{\frac{-ad+bc}{2\,bd}} \right ){\frac{1}{\sqrt{ac+ \left ( -ad+bc \right ) x-bd{x}^{2}}}}} \right ){\frac{1}{\sqrt{bd}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(b*x + a)*(d*x - c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279448, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (4 \,{\left (2 \, b^{2} d^{2} x - b^{2} c d + a b d^{2}\right )} \sqrt{-b d x^{2} + a c +{\left (b c - a d\right )} x} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 8 \,{\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt{-b d}\right )}{2 \, \sqrt{-b d}}, \frac{\arctan \left (\frac{{\left (2 \, b d x - b c + a d\right )} \sqrt{b d}}{2 \, \sqrt{-b d x^{2} + a c +{\left (b c - a d\right )} x} b d}\right )}{\sqrt{b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(b*x + a)*(d*x - c)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29075, size = 80, normalized size = 1.23 \[ -\frac{{\rm ln}\left ({\left | b c - a d + 2 \, \sqrt{-b d}{\left (\sqrt{-b d} x - \sqrt{-b d x^{2} + b c x - a d x + a c}\right )} \right |}\right )}{\sqrt{-b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(b*x + a)*(d*x - c)),x, algorithm="giac")

[Out]

-ln(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)