3.1940 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^6} \, dx\)

Optimal. Leaf size=71 \[ \frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{5/2}}-\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{1}{b x^3 \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.110937, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{5/2}}-\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{1}{b x^3 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(3/2)*x^6),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.8548, size = 63, normalized size = 0.89 \[ \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2 b^{\frac{5}{2}}} + \frac{1}{b x^{3} \sqrt{a + \frac{b}{x^{2}}}} - \frac{3 \sqrt{a + \frac{b}{x^{2}}}}{2 b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(3/2)/x**6,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0753506, size = 95, normalized size = 1.34 \[ \frac{-\sqrt{b} \left (3 a x^2+b\right )-3 a x^2 \log (x) \sqrt{a x^2+b}+3 a x^2 \sqrt{a x^2+b} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )}{2 b^{5/2} x^3 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(3/2)*x^6),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 79, normalized size = 1.1 \[ -{\frac{a{x}^{2}+b}{2\,{x}^{5}} \left ( 3\,{b}^{3/2}{x}^{2}a-3\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) \sqrt{a{x}^{2}+b}{x}^{2}ab+{b}^{{\frac{5}{2}}} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x^6,x)

[Out]

-1/2*(a*x^2+b)*(3*b^(3/2)*x^2*a-3*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*(a*x^2+b)^
(1/2)*x^2*a*b+b^(5/2))/((a*x^2+b)/x^2)^(3/2)/x^5/b^(7/2)

_______________________________________________________________________________________

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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.249167, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a^{2} x^{3} + a b x\right )} \sqrt{b} \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (3 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \,{\left (a b^{3} x^{3} + b^{4} x\right )}}, -\frac{3 \,{\left (a^{2} x^{3} + a b x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (3 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \,{\left (a b^{3} x^{3} + b^{4} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^6),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 16.6653, size = 73, normalized size = 1.03 \[ - \frac{3 \sqrt{a}}{2 b^{2} x \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2 b^{\frac{5}{2}}} - \frac{1}{2 \sqrt{a} b x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(3/2)/x**6,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^6),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(3/2)*x^6), x)