3.737 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=70 \[ \frac{\sqrt{a x^2+b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a x^2+b}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{d} x \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.163251, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\sqrt{a x^2+b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a x^2+b}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} \sqrt{d} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.69258, size = 63, normalized size = 0.9 \[ \frac{x \sqrt{a + \frac{b}{x^{2}}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a x^{2} + b}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{a} \sqrt{d} \sqrt{a x^{2} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b/x**2)*atanh(sqrt(d)*sqrt(a*x**2 + b)/(sqrt(a)*sqrt(c + d*x**2)))/(s
qrt(a)*sqrt(d)*sqrt(a*x**2 + b))

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Mathematica [A]  time = 0.0796566, size = 88, normalized size = 1.26 \[ \frac{\sqrt{a x^2+b} \log \left (2 \sqrt{a} \sqrt{d} \sqrt{a x^2+b} \sqrt{c+d x^2}+a c+2 a d x^2+b d\right )}{2 \sqrt{a} \sqrt{d} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.072, size = 117, normalized size = 1.7 \[{\frac{a{x}^{2}+b}{2\,x}\ln \left ({\frac{1}{2} \left ( 2\,ad{x}^{2}+2\,\sqrt{ad{x}^{4}+ac{x}^{2}+bd{x}^{2}+bc}\sqrt{ad}+ac+bd \right ){\frac{1}{\sqrt{ad}}}} \right ) \sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{\frac{1}{\sqrt{ad{x}^{4}+ac{x}^{2}+bd{x}^{2}+bc}}}{\frac{1}{\sqrt{ad}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/((a*x^2+b)/x^2)^(1/2)/x*(a*x^2+b)*ln(1/2*(2*a*d*x^2+2*(a*d*x^4+a*c*x^2+b*d*x
^2+b*c)^(1/2)*(a*d)^(1/2)+a*c+b*d)/(a*d)^(1/2))*(d*x^2+c)^(1/2)/(a*d*x^4+a*c*x^2
+b*d*x^2+b*c)^(1/2)/(a*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(d*x^2 + c)*sqrt(a + b/x^2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287848, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{a d} \log \left (4 \,{\left (2 \, a^{2} d^{2} x^{3} +{\left (a^{2} c d + a b d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (8 \, a^{2} d^{2} x^{4} + a^{2} c^{2} + 6 \, a b c d + b^{2} d^{2} + 8 \,{\left (a^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{a d}\right )}{4 \, a d}, -\frac{\sqrt{-a d} \arctan \left (\frac{{\left (2 \, a d x^{2} + a c + b d\right )} \sqrt{-a d}}{2 \, \sqrt{d x^{2} + c} a d x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )}{2 \, a d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x^2 + c)*sqrt(a + b/x^2)),x, algorithm="fricas")

[Out]

[1/4*sqrt(a*d)*log(4*(2*a^2*d^2*x^3 + (a^2*c*d + a*b*d^2)*x)*sqrt(d*x^2 + c)*sqr
t((a*x^2 + b)/x^2) + (8*a^2*d^2*x^4 + a^2*c^2 + 6*a*b*c*d + b^2*d^2 + 8*(a^2*c*d
+ a*b*d^2)*x^2)*sqrt(a*d))/(a*d), -1/2*sqrt(-a*d)*arctan(1/2*(2*a*d*x^2 + a*c +
b*d)*sqrt(-a*d)/(sqrt(d*x^2 + c)*a*d*x*sqrt((a*x^2 + b)/x^2)))/(a*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(a + b/x**2)*sqrt(c + d*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{a + \frac{b}{x^{2}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x^2 + c)*sqrt(a + b/x^2)),x, algorithm="giac")

[Out]

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