3.3057 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.203508, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.3472, size = 42, normalized size = 0.78 \[ \frac{2 \operatorname{atanh}{\left (\frac{2 a + b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.302518, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.044, size = 94, normalized size = 1.7 \[ 2\,{\frac{\sqrt{x}}{\sqrt{a}}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*ln(1/2
*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(
1/2))/a^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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