3.2977 \(\int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x} \, dx\)

Optimal. Leaf size=51 \[ 4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-4 \sqrt{a+b \sqrt{\frac{c}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.09835, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ 4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-4 \sqrt{a+b \sqrt{\frac{c}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c/x]]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.12758, size = 41, normalized size = 0.8 \[ 4 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{\frac{c}{x}}}}{\sqrt{a}} \right )} - 4 \sqrt{a + b \sqrt{\frac{c}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0458404, size = 52, normalized size = 1.02 \[ 4 \left (\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )-\sqrt{a+b \sqrt{\frac{c}{x}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 150, normalized size = 2.9 \[ 2\,{\frac{1}{bx\sqrt{a}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( \ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) \sqrt{{\frac{c}{x}}}{x}^{3/2}ab+2\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}x-2\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{\frac{1}{\sqrt{{\frac{c}{x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(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))*(c/x)^(1/2)*x^(3/2)*a*b+2*a^(3/2)*(a*x+b*(c
/x)^(1/2)*x)^(1/2)*x-2*(a*x+b*(c/x)^(1/2)*x)^(3/2)*a^(1/2))/x/(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(sqrt(b*sqrt(c/x) + a)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266474, size = 1, normalized size = 0.02 \[ \left [2 \, \sqrt{a} \log \left (\frac{b \sqrt{\frac{c}{x}} + 2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a} + 2 \, a}{\sqrt{\frac{c}{x}}}\right ) - 4 \, \sqrt{b \sqrt{\frac{c}{x}} + a}, 4 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \sqrt{\frac{c}{x}} + a}}{\sqrt{-a}}\right ) - 4 \, \sqrt{b \sqrt{\frac{c}{x}} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError