3.2976 \(\int \sqrt{a+b \sqrt{\frac{c}{x}}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+x \sqrt{a+b \sqrt{\frac{c}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.112581, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}}+x \sqrt{a+b \sqrt{\frac{c}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c/x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6816, size = 73, normalized size = 0.79 \[ x \sqrt{a + b \sqrt{\frac{c}{x}}} + \frac{b c \sqrt{a + b \sqrt{\frac{c}{x}}}}{2 a \sqrt{\frac{c}{x}}} - \frac{b^{2} c \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b*sqrt(c/x)) + b*c*sqrt(a + b*sqrt(c/x))/(2*a*sqrt(c/x)) - b**2*c*ata
nh(sqrt(a + b*sqrt(c/x))/sqrt(a))/(2*a**(3/2))

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Mathematica [A]  time = 0.105072, size = 79, normalized size = 0.86 \[ \frac{\sqrt{a} x \sqrt{a+b \sqrt{\frac{c}{x}}} \left (2 a+b \sqrt{\frac{c}{x}}\right )-b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.031, size = 147, normalized size = 1.6 \[{\frac{1}{4}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( 2\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{{\frac{c}{x}}}\sqrt{x}b-{b}^{2}c\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) a+4\,{a}^{5/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{x} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c/x)^(1/2))^(1/2),x)

[Out]

1/4*(a+b*(c/x)^(1/2))^(1/2)*x^(1/2)*(2*a^(3/2)*(a*x+b*(c/x)^(1/2)*x)^(1/2)*(c/x)
^(1/2)*x^(1/2)*b-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+4*a^(5/2)*(a*x+b*(c/x)^(1/2)*x)^(1/2)*x^(1/2)
)/(x*(a+b*(c/x)^(1/2)))^(1/2)/a^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291769, size = 1, normalized size = 0.01 \[ \left [\frac{b^{2} c \log \left (\frac{{\left (b \sqrt{\frac{c}{x}} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a}{\sqrt{\frac{c}{x}}}\right ) + 2 \,{\left (b x \sqrt{\frac{c}{x}} + 2 \, a x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a}}{4 \, a^{\frac{3}{2}}}, \frac{b^{2} c \arctan \left (\frac{a}{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}\right ) +{\left (\sqrt{-a} b x \sqrt{\frac{c}{x}} + 2 \, \sqrt{-a} a x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{2 \, \sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(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(sqrt(b*sqrt(c/x) + a),x, algorithm="giac")

[Out]

Exception raised: TypeError