3.2241 \(\int \frac{1}{\sqrt{a+b \sqrt{x}} x^2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.104909, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*Sqrt[x]]*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.3309, size = 70, normalized size = 0.88 \[ - \frac{\sqrt{a + b \sqrt{x}}}{a x} + \frac{3 b \sqrt{a + b \sqrt{x}}}{2 a^{2} \sqrt{x}} - \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0653856, size = 69, normalized size = 0.86 \[ \left (\frac{3 b}{2 a^2 \sqrt{x}}-\frac{1}{a x}\right ) \sqrt{a+b \sqrt{x}}-\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*Sqrt[x]]*x^2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 72, normalized size = 0.9 \[ 4\,{b}^{2} \left ( -1/4\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{2}x}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{a+b\sqrt{x}}}{ab\sqrt{x}}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*b^2*(-1/4*(a+b*x^(1/2))^(1/2)/a/x/b^2-3/4/a*(-1/2*(a+b*x^(1/2))^(1/2)/a/x^(1/2
)/b+1/2/a^(3/2)*arctanh((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(1/(sqrt(b*sqrt(x) + a)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25914, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} x \log \left (\frac{\sqrt{a} b \sqrt{x} - 2 \, \sqrt{b \sqrt{x} + a} a + 2 \, a^{\frac{3}{2}}}{\sqrt{x}}\right ) + 2 \,{\left (3 \, \sqrt{a} b \sqrt{x} - 2 \, a^{\frac{3}{2}}\right )} \sqrt{b \sqrt{x} + a}}{4 \, a^{\frac{5}{2}} x}, \frac{3 \, b^{2} x \arctan \left (\frac{a}{\sqrt{b \sqrt{x} + a} \sqrt{-a}}\right ) +{\left (3 \, \sqrt{-a} b \sqrt{x} - 2 \, \sqrt{-a} a\right )} \sqrt{b \sqrt{x} + a}}{2 \, \sqrt{-a} a^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(x) + a)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 16.7029, size = 110, normalized size = 1.38 \[ - \frac{1}{\sqrt{b} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{\sqrt{b}}{2 a x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{3 b^{\frac{3}{2}}}{2 a^{2} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(sqrt(b)*x**(5/4)*sqrt(a/(b*sqrt(x)) + 1)) + sqrt(b)/(2*a*x**(3/4)*sqrt(a/(b*
sqrt(x)) + 1)) + 3*b**(3/2)/(2*a**2*x**(1/4)*sqrt(a/(b*sqrt(x)) + 1)) - 3*b**2*a
sinh(sqrt(a)/(sqrt(b)*x**(1/4)))/(2*a**(5/2))

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GIAC/XCAS [A]  time = 0.256131, size = 89, normalized size = 1.11 \[ \frac{1}{2} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} - 5 \, \sqrt{b \sqrt{x} + a} a}{a^{2} b^{2} x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*sqrt(x) + a)*x^2),x, algorithm="giac")

[Out]

1/2*b^2*(3*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*sqrt(x) +
a)^(3/2) - 5*sqrt(b*sqrt(x) + a)*a)/(a^2*b^2*x))