3.3056 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx\)

Optimal. Leaf size=135 \[ -\frac{\left (4 a c-3 b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{5/2}}-\frac{3 b d \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a^2 \sqrt{\frac{d}{x}}}+\frac{x \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{a} \]

[Out]

(-3*b*d*Sqrt[a + b*Sqrt[d/x] + c/x])/(2*a^2*Sqrt[d/x]) + (Sqrt[a + b*Sqrt[d/x] +
c/x]*x)/a - ((4*a*c - 3*b^2*d)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a +
b*Sqrt[d/x] + c/x])])/(4*a^(5/2))

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Rubi [A]  time = 0.485129, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (4 a c-3 b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{4 a^{5/2}}-\frac{3 b d \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{2 a^2 \sqrt{\frac{d}{x}}}+\frac{x \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{a} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*d*Sqrt[a + b*Sqrt[d/x] + c/x])/(2*a^2*Sqrt[d/x]) + (Sqrt[a + b*Sqrt[d/x] +
c/x]*x)/a - ((4*a*c - 3*b^2*d)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a +
b*Sqrt[d/x] + c/x])])/(4*a^(5/2))

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Rubi in Sympy [A]  time = 34.4416, size = 109, normalized size = 0.81 \[ \frac{x \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{a} - \frac{3 b d \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{2 a^{2} \sqrt{\frac{d}{x}}} - \frac{\left (4 a c - 3 b^{2} d\right ) \operatorname{atanh}{\left (\frac{2 a + b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b*sqrt(d/x) + c/x)/a - 3*b*d*sqrt(a + b*sqrt(d/x) + c/x)/(2*a**2*sqrt
(d/x)) - (4*a*c - 3*b**2*d)*atanh((2*a + b*sqrt(d/x))/(2*sqrt(a)*sqrt(a + b*sqrt
(d/x) + c/x)))/(4*a**(5/2))

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Mathematica [A]  time = 0.355117, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.043, size = 213, normalized size = 1.6 \[{\frac{1}{4}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 4\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}-6\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}b+3\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) da{b}^{2}-4\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{2}c \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(4*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)
^(1/2)*x^(1/2)-6*a^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*b+3*l
n(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))*d*a*b^2-4*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^2*c)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/a^(7/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(b*sqrt(d/x) + a + c/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, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.505475, size = 232, normalized size = 1.72 \[ \frac{{\left (3 \, b^{2} d{\rm ln}\left ({\left | -b d + 2 \, \sqrt{c d} \sqrt{a} \right |}\right ) - 4 \, a c{\rm ln}\left ({\left | -b d + 2 \, \sqrt{c d} \sqrt{a} \right |}\right ) + 6 \, \sqrt{c d} \sqrt{a} b\right )}{\rm sign}\left (x\right )}{4 \, a^{\frac{5}{2}}} - \frac{2 \, \sqrt{a d x + \sqrt{d x} b d + c d}{\left (\frac{3 \, b d}{a^{2}} - \frac{2 \, \sqrt{d x}}{a}\right )} + \frac{{\left (3 \, b^{2} d^{2} - 4 \, a c d\right )}{\rm ln}\left ({\left | -b d - 2 \,{\left (\sqrt{d x} \sqrt{a} - \sqrt{a d x + \sqrt{d x} b d + c d}\right )} \sqrt{a} \right |}\right )}{a^{\frac{5}{2}}}}{4 \, d{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(3*b^2*d*ln(abs(-b*d + 2*sqrt(c*d)*sqrt(a))) - 4*a*c*ln(abs(-b*d + 2*sqrt(c*
d)*sqrt(a))) + 6*sqrt(c*d)*sqrt(a)*b)*sign(x)/a^(5/2) - 1/4*(2*sqrt(a*d*x + sqrt
(d*x)*b*d + c*d)*(3*b*d/a^2 - 2*sqrt(d*x)/a) + (3*b^2*d^2 - 4*a*c*d)*ln(abs(-b*d
- 2*(sqrt(d*x)*sqrt(a) - sqrt(a*d*x + sqrt(d*x)*b*d + c*d))*sqrt(a)))/a^(5/2))/
(d*sign(x))