3.2983 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x} \, dx\)

Optimal. Leaf size=31 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0778016, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.68235, size = 26, normalized size = 0.84 \[ \frac{4 \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*atanh(sqrt(a + b*sqrt(c/x))/sqrt(a))/sqrt(a)

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Mathematica [A]  time = 0.0604787, size = 31, normalized size = 1. \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b \sqrt{\frac{c}{x}}}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.059, size = 200, normalized size = 6.5 \[{\frac{1}{b}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( \ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) b\sqrt{{\frac{c}{x}}}\sqrt{x}+\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}-2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{\frac{1}{\sqrt{{\frac{c}{x}}}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262137, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \log \left (\frac{{\left (b \sqrt{\frac{c}{x}} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a}{\sqrt{\frac{c}{x}}}\right )}{\sqrt{a}}, -\frac{4 \, \arctan \left (\frac{a}{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*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),x, algorithm="giac")

[Out]

Exception raised: TypeError