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

Optimal. Leaf size=123 \[ x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} \sqrt{c}}-2 \sqrt{b} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.353574, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} \sqrt{c}}-2 \sqrt{b} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right ) \]

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

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Rubi in Sympy [A]  time = 32.876, size = 104, normalized size = 0.85 \[ - 2 \sqrt{b} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + \frac{d}{x}}}{\sqrt{d} \sqrt{a + \frac{b}{x}}} \right )} + x \sqrt{a + \frac{b}{x}} \sqrt{c + \frac{d}{x}} + \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + \frac{b}{x}}}{\sqrt{a} \sqrt{c + \frac{d}{x}}} \right )}}{\sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.355393, size = 168, normalized size = 1.37 \[ x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}-\sqrt{b} \sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+a d x+b c x+2 b d\right )+\frac{(a d+b c) \log \left (2 \sqrt{a} \sqrt{c} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+a (2 c x+d)+b c\right )}{2 \sqrt{a} \sqrt{c}}+\sqrt{b} \sqrt{d} \log (x) \]

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

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Maple [B]  time = 0.033, size = 253, normalized size = 2.1 \[{\frac{x}{2}\sqrt{{\frac{cx+d}{x}}}\sqrt{{\frac{ax+b}{x}}} \left ( \sqrt{bd}\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ac{x}^{2}+adx+bcx+bd}\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) ad+\sqrt{bd}\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ac{x}^{2}+adx+bcx+bd}\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) bc-2\,bd\ln \left ({\frac{adx+bcx+2\,\sqrt{bd}\sqrt{ac{x}^{2}+adx+bcx+bd}+2\,bd}{x}} \right ) \sqrt{ac}+2\,\sqrt{ac{x}^{2}+adx+bcx+bd}\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{ac{x}^{2}+adx+bcx+bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.728648, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d/x)**(1/2)*(a+b/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. \[ \int \sqrt{a + \frac{b}{x}} \sqrt{c + \frac{d}{x}}\,{d x} \]

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)