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

Optimal. Leaf size=93 \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.304946, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a}}-\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.8151, size = 80, normalized size = 0.86 \[ - \frac{2 \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + \frac{d}{x}}}{\sqrt{d} \sqrt{a + \frac{b}{x}}} \right )}}{\sqrt{b}} + \frac{2 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + \frac{b}{x}}}{\sqrt{a} \sqrt{c + \frac{d}{x}}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.242667, size = 134, normalized size = 1.44 \[ -\frac{\sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+a d x+b c x+2 b d\right )}{\sqrt{b}}+\frac{\sqrt{c} \log \left (2 \sqrt{a} \sqrt{c} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+a (2 c x+d)+b c\right )}{\sqrt{a}}+\frac{\sqrt{d} \log (x)}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.048, size = 142, normalized size = 1.5 \[{x\sqrt{{\frac{ax+b}{x}}}\sqrt{{\frac{cx+d}{x}}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) c\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{bd}\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }+2\,bd \right ) } \right ) d\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.463091, 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(c + d/x)/(sqrt(a + b/x)*x),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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