3.2982 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}}} \, dx\)

Optimal. Leaf size=95 \[ \frac{3 b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a^2 \sqrt{\frac{c}{x}}}+\frac{x \sqrt{a+b \sqrt{\frac{c}{x}}}}{a} \]

[Out]

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

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Rubi [A]  time = 0.10965, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{3 b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a^2 \sqrt{\frac{c}{x}}}+\frac{x \sqrt{a+b \sqrt{\frac{c}{x}}}}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a + b*Sqrt[c/x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 12.088, size = 80, normalized size = 0.84 \[ \frac{x \sqrt{a + b \sqrt{\frac{c}{x}}}}{a} - \frac{3 b c \sqrt{a + b \sqrt{\frac{c}{x}}}}{2 a^{2} \sqrt{\frac{c}{x}}} + \frac{3 b^{2} c \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.167126, size = 89, normalized size = 0.94 \[ \frac{3 b^2 c \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b \sqrt{\frac{c}{x}}}}\right )}{2 a^{5/2}}+\frac{2 a^2 x-a b x \sqrt{\frac{c}{x}}-3 b^2 c}{2 a^2 \sqrt{a+b \sqrt{\frac{c}{x}}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.054, size = 233, normalized size = 2.5 \[ -{\frac{1}{4}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( 8\,b\sqrt{{\frac{c}{x}}}\sqrt{x}\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{a}^{5/2}-4\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{x}{a}^{7/2}-2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}b\sqrt{{\frac{c}{x}}}\sqrt{x}{a}^{5/2}+{b}^{2}c\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{2}-4\,{b}^{2}c\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{2} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(a+b*(c/x)^(1/2))^(1/2)*x^(1/2)*(8*b*(c/x)^(1/2)*x^(1/2)*(x*(a+b*(c/x)^(1/2
)))^(1/2)*a^(5/2)-4*(a*x+b*(c/x)^(1/2)*x)^(1/2)*x^(1/2)*a^(7/2)-2*(a*x+b*(c/x)^(
1/2)*x)^(1/2)*b*(c/x)^(1/2)*x^(1/2)*a^(5/2)+b^2*c*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+
2*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^2-4*b^2*c*ln(1/2*(
b*(c/x)^(1/2)*x^(1/2)+2*(x*(a+b*(c/x)^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2)
)*a^2)/(x*(a+b*(c/x)^(1/2)))^(1/2)/a^(9/2)

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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(1/sqrt(b*sqrt(c/x) + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260911, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c \log \left (\frac{{\left (b \sqrt{\frac{c}{x}} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a}{\sqrt{\frac{c}{x}}}\right ) - 2 \,{\left (3 \, b x \sqrt{\frac{c}{x}} - 2 \, a x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a}}{4 \, a^{\frac{5}{2}}}, -\frac{3 \, b^{2} c \arctan \left (\frac{a}{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}\right ) +{\left (3 \, \sqrt{-a} b x \sqrt{\frac{c}{x}} - 2 \, \sqrt{-a} a x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{2 \, \sqrt{-a} a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(3*b^2*c*log(((b*sqrt(c/x) + 2*a)*sqrt(a) + 2*sqrt(b*sqrt(c/x) + a)*a)/sqrt
(c/x)) - 2*(3*b*x*sqrt(c/x) - 2*a*x)*sqrt(b*sqrt(c/x) + a)*sqrt(a))/a^(5/2), -1/
2*(3*b^2*c*arctan(a/(sqrt(b*sqrt(c/x) + a)*sqrt(-a))) + (3*sqrt(-a)*b*x*sqrt(c/x
) - 2*sqrt(-a)*a*x)*sqrt(b*sqrt(c/x) + a))/(sqrt(-a)*a^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(a + b*sqrt(c/x)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError