3.3058 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2} \, dx\)

Optimal. Leaf size=93 \[ \frac{b \sqrt{d} \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{c^{3/2}}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{c} \]

[Out]

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

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Rubi [A]  time = 0.227829, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{b \sqrt{d} \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{c^{3/2}}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.6278, size = 75, normalized size = 0.81 \[ \frac{b \sqrt{d} \operatorname{atanh}{\left (\frac{b d + 2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{c^{\frac{3}{2}}} - \frac{2 \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.174704, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.045, size = 118, normalized size = 1.3 \[{1\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ( b\sqrt{{\frac{d}{x}}}x\ln \left ({1 \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ){\frac{1}{\sqrt{x}}}} \right ) c-2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{3/2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*(b*(d/x)^(1/2)*x*ln((2*c+b*(d/x)^(1/2)*x+2*c^(
1/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*c-2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c
^(3/2))/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^2), 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^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*sqrt(a + b*sqrt(d/x) + c/x)), x)

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GIAC/XCAS [A]  time = 0.457892, size = 123, normalized size = 1.32 \[ -\frac{{\left (\frac{b d{\rm ln}\left ({\left | -b d - 2 \, \sqrt{c}{\left (\sqrt{c} \sqrt{\frac{d}{x}} - \sqrt{b d \sqrt{\frac{d}{x}} + a d + \frac{c d}{x}}\right )} \right |}\right )}{c^{\frac{3}{2}}} + \frac{2 \, \sqrt{b d \sqrt{\frac{d}{x}} + a d + \frac{c d}{x}}}{c}\right )} \sqrt{d}}{{\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(d/x) + a + c/x)*x^2),x, algorithm="giac")

[Out]

-(b*d*ln(abs(-b*d - 2*sqrt(c)*(sqrt(c)*sqrt(d/x) - sqrt(b*d*sqrt(d/x) + a*d + c*
d/x))))/c^(3/2) + 2*sqrt(b*d*sqrt(d/x) + a*d + c*d/x)/c)*sqrt(d)/abs(d)