3.1939 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^4} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{b x \sqrt{a+\frac{b}{x^2}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0761371, antiderivative size = 47, 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 \[ \frac{1}{b x \sqrt{a+\frac{b}{x^2}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(3/2)*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.37041, size = 37, normalized size = 0.79 \[ \frac{1}{b x \sqrt{a + \frac{b}{x^{2}}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(3/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0604653, size = 73, normalized size = 1.55 \[ \frac{\log (x) \sqrt{a x^2+b}-\sqrt{a x^2+b} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+\sqrt{b}}{b^{3/2} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(3/2)*x^4),x]

[Out]

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

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Maple [A]  time = 0.012, size = 65, normalized size = 1.4 \[{\frac{a{x}^{2}+b}{{x}^{3}} \left ({b}^{{\frac{3}{2}}}-\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) b\sqrt{a{x}^{2}+b} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2487, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + b\right )} \sqrt{b} \log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right )}{2 \,{\left (a b^{2} x^{2} + b^{3}\right )}}, \frac{b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )}{a b^{2} x^{2} + b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 9.50262, size = 184, normalized size = 3.91 \[ \frac{a b^{2} x^{2} \log{\left (\frac{a x^{2}}{b} \right )}}{2 a b^{\frac{7}{2}} x^{2} + 2 b^{\frac{9}{2}}} - \frac{2 a b^{2} x^{2} \log{\left (\sqrt{\frac{a x^{2}}{b} + 1} + 1 \right )}}{2 a b^{\frac{7}{2}} x^{2} + 2 b^{\frac{9}{2}}} + \frac{2 b^{3} \sqrt{\frac{a x^{2}}{b} + 1}}{2 a b^{\frac{7}{2}} x^{2} + 2 b^{\frac{9}{2}}} + \frac{b^{3} \log{\left (\frac{a x^{2}}{b} \right )}}{2 a b^{\frac{7}{2}} x^{2} + 2 b^{\frac{9}{2}}} - \frac{2 b^{3} \log{\left (\sqrt{\frac{a x^{2}}{b} + 1} + 1 \right )}}{2 a b^{\frac{7}{2}} x^{2} + 2 b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(3/2)/x**4,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(3/2)*x^4), x)