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

Optimal. Leaf size=81 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c} \]

[Out]

(Sqrt[a + b/x]*Sqrt[c + d/x]*x)/c + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])
/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*c^(3/2))

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Rubi [A]  time = 0.192844, antiderivative size = 81, 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 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/Sqrt[c + d/x],x]

[Out]

(Sqrt[a + b/x]*Sqrt[c + d/x]*x)/c + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])
/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*c^(3/2))

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Rubi in Sympy [A]  time = 14.8743, size = 65, normalized size = 0.8 \[ \frac{x \sqrt{a + \frac{b}{x}} \sqrt{c + \frac{d}{x}}}{c} - \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + \frac{b}{x}}}{\sqrt{a} \sqrt{c + \frac{d}{x}}} \right )}}{\sqrt{a} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b/x)*sqrt(c + d/x)/c - (a*d - b*c)*atanh(sqrt(c)*sqrt(a + b/x)/(sqrt(
a)*sqrt(c + d/x)))/(sqrt(a)*c**(3/2))

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Mathematica [A]  time = 0.19538, size = 98, normalized size = 1.21 \[ \frac{(b c-a d) \log \left (2 \sqrt{a} \sqrt{c} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+2 a c x+a d+b c\right )}{2 \sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]/Sqrt[c + d/x],x]

[Out]

(Sqrt[a + b/x]*Sqrt[c + d/x]*x)/c + ((b*c - a*d)*Log[b*c + a*d + 2*a*c*x + 2*Sqr
t[a]*Sqrt[c]*Sqrt[a + b/x]*Sqrt[c + d/x]*x])/(2*Sqrt[a]*c^(3/2))

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Maple [B]  time = 0.034, size = 155, normalized size = 1.9 \[ -{\frac{x}{2\,c}\sqrt{{\frac{ax+b}{x}}}\sqrt{{\frac{cx+d}{x}}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) ad-\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) bc-2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.327013, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} -{\left (b c - a d\right )} \log \left (4 \,{\left (2 \, a^{2} c^{2} x^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} -{\left (8 \, a^{2} c^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}\right )}{4 \, \sqrt{a c} c}, \frac{2 \, \sqrt{-a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} +{\left (b c - a d\right )} \arctan \left (\frac{2 \, \sqrt{-a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}}}{2 \, a c x + b c + a d}\right )}{2 \, \sqrt{-a c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x)/sqrt(c + d/x), x)