3.1930 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x} \, dx\)

Optimal. Leaf size=41 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.0808789, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(3/2)*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.22218, size = 34, normalized size = 0.83 \[ - \frac{1}{a \sqrt{a + \frac{b}{x^{2}}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(3/2)/x,x)

[Out]

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

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Mathematica [A]  time = 0.0398808, size = 63, normalized size = 1.54 \[ \frac{\sqrt{a x^2+b} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )-\sqrt{a} x}{a^{3/2} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(3/2)*x),x]

[Out]

(-(Sqrt[a]*x) + Sqrt[b + a*x^2]*Log[a*x + Sqrt[a]*Sqrt[b + a*x^2]])/(a^(3/2)*Sqr
t[a + b/x^2]*x)

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Maple [A]  time = 0.012, size = 63, normalized size = 1.5 \[ -{\frac{a{x}^{2}+b}{{x}^{3}} \left ( x{a}^{{\frac{3}{2}}}-\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) a\sqrt{a{x}^{2}+b} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244282, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + b\right )} \sqrt{a} \log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right )}{2 \,{\left (a^{3} x^{2} + a^{2} b\right )}}, -\frac{a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )}{a^{3} x^{2} + a^{2} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.9315, size = 187, normalized size = 4.56 \[ - \frac{2 a^{3} x^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} - \frac{a^{3} x^{2} \log{\left (\frac{b}{a x^{2}} \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} + \frac{2 a^{3} x^{2} \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x^{2}} \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(3/2)/x,x)

[Out]

-2*a**3*x**2*sqrt(1 + b/(a*x**2))/(2*a**(9/2)*x**2 + 2*a**(7/2)*b) - a**3*x**2*l
og(b/(a*x**2))/(2*a**(9/2)*x**2 + 2*a**(7/2)*b) + 2*a**3*x**2*log(sqrt(1 + b/(a*
x**2)) + 1)/(2*a**(9/2)*x**2 + 2*a**(7/2)*b) - a**2*b*log(b/(a*x**2))/(2*a**(9/2
)*x**2 + 2*a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x**2)) + 1)/(2*a**(9/2)*x**2
+ 2*a**(7/2)*b)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(3/2)*x), x)