3.2235 \(\int \frac{\sqrt{a+b \sqrt{x}}}{x^2} \, dx\)

Optimal. Leaf size=77 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{b \sqrt{a+b \sqrt{x}}}{2 a \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{x} \]

[Out]

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

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Rubi [A]  time = 0.101619, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{b \sqrt{a+b \sqrt{x}}}{2 a \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[x]]/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.0583, size = 63, normalized size = 0.82 \[ - \frac{\sqrt{a + b \sqrt{x}}}{x} - \frac{b \sqrt{a + b \sqrt{x}}}{2 a \sqrt{x}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0608848, size = 66, normalized size = 0.86 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{3/2}}+\left (-\frac{b}{2 a \sqrt{x}}-\frac{1}{x}\right ) \sqrt{a+b \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*Sqrt[x]]/x^2,x]

[Out]

(-x^(-1) - b/(2*a*Sqrt[x]))*Sqrt[a + b*Sqrt[x]] + (b^2*ArcTanh[Sqrt[a + b*Sqrt[x
]]/Sqrt[a]])/(2*a^(3/2))

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Maple [A]  time = 0.011, size = 59, normalized size = 0.8 \[ 4\,{b}^{2} \left ({\frac{1}{{b}^{2}x} \left ( -1/8\,{\frac{ \left ( a+b\sqrt{x} \right ) ^{3/2}}{a}}-1/8\,\sqrt{a+b\sqrt{x}} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^(1/2)/x^2,x)

[Out]

4*b^2*((-1/8/a*(a+b*x^(1/2))^(3/2)-1/8*(a+b*x^(1/2))^(1/2))/x/b^2+1/8/a^(3/2)*ar
ctanh((a+b*x^(1/2))^(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(x) + a)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2685, size = 1, normalized size = 0.01 \[ \left [\frac{b^{2} x \log \left (\frac{\sqrt{a} b \sqrt{x} + 2 \, \sqrt{b \sqrt{x} + a} a + 2 \, a^{\frac{3}{2}}}{\sqrt{x}}\right ) - 2 \,{\left (\sqrt{a} b \sqrt{x} + 2 \, a^{\frac{3}{2}}\right )} \sqrt{b \sqrt{x} + a}}{4 \, a^{\frac{3}{2}} x}, -\frac{b^{2} x \arctan \left (\frac{a}{\sqrt{b \sqrt{x} + a} \sqrt{-a}}\right ) +{\left (\sqrt{-a} b \sqrt{x} + 2 \, \sqrt{-a} a\right )} \sqrt{b \sqrt{x} + a}}{2 \, \sqrt{-a} a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*sqrt(x) + a)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 13.4252, size = 105, normalized size = 1.36 \[ - \frac{a}{\sqrt{b} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{3 \sqrt{b}}{2 x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{b^{\frac{3}{2}}}{2 a \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(sqrt(b)*x**(5/4)*sqrt(a/(b*sqrt(x)) + 1)) - 3*sqrt(b)/(2*x**(3/4)*sqrt(a/(b*
sqrt(x)) + 1)) - b**(3/2)/(2*a*x**(1/4)*sqrt(a/(b*sqrt(x)) + 1)) + b**2*asinh(sq
rt(a)/(sqrt(b)*x**(1/4)))/(2*a**(3/2))

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GIAC/XCAS [A]  time = 0.259851, size = 84, normalized size = 1.09 \[ -\frac{1}{2} \, b^{2}{\left (\frac{\arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} + \sqrt{b \sqrt{x} + a} a}{a b^{2} x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^2*(arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/(sqrt(-a)*a) + ((b*sqrt(x) + a)^(
3/2) + sqrt(b*sqrt(x) + a)*a)/(a*b^2*x))