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

Optimal. Leaf size=61 \[ \frac{\tanh ^{-1}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (a d+b c)+a c+b d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

ArcTanh[(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 = 0.0583457, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{x (a d+b c)+a c+b d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(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 = 2.43263, size = 58, normalized size = 0.95 \[ \frac{\operatorname{atanh}{\left (\frac{a d + b c + 2 b d x}{2 \sqrt{b} \sqrt{d} \sqrt{a c + b d x^{2} + x \left (a d + b c\right )}} \right )}}{\sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh((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 [A]  time = 0.0489312, size = 87, normalized size = 1.43 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 49, normalized size = 0.8 \[{1\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{ac+ \left ( ad+bc \right ) x+bd{x}^{2}} \right ){\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(a*c+(a*d+b*c)*x+b*d*x^2)^(1/2))/(b*d)^(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 = 0.2842, 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)*sqr
t(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 = 0.295161, size = 92, normalized size = 1.51 \[ -\frac{\sqrt{b d}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b d} x - \sqrt{b d x^{2} + b c x + a d x + a c}\right )} b d - \sqrt{b d} b c - \sqrt{b d} a d \right |}\right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b*d)*ln(abs(-2*(sqrt(b*d)*x - sqrt(b*d*x^2 + b*c*x + a*d*x + a*c))*b*d - s
qrt(b*d)*b*c - sqrt(b*d)*a*d))/(b*d)