3.1902 \(\int \frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=71 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^2}}}{8 x}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{4 x} \]

[Out]

(-3*a*Sqrt[a + b/x^2])/(8*x) - (a + b/x^2)^(3/2)/(4*x) - (3*a^2*ArcTanh[Sqrt[b]/
(Sqrt[a + b/x^2]*x)])/(8*Sqrt[b])

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Rubi [A]  time = 0.0829662, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^2}}}{8 x}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(3/2)/x^2,x]

[Out]

(-3*a*Sqrt[a + b/x^2])/(8*x) - (a + b/x^2)^(3/2)/(4*x) - (3*a^2*ArcTanh[Sqrt[b]/
(Sqrt[a + b/x^2]*x)])/(8*Sqrt[b])

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Rubi in Sympy [A]  time = 6.22149, size = 61, normalized size = 0.86 \[ - \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{8 \sqrt{b}} - \frac{3 a \sqrt{a + \frac{b}{x^{2}}}}{8 x} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{4 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2*atanh(sqrt(b)/(x*sqrt(a + b/x**2)))/(8*sqrt(b)) - 3*a*sqrt(a + b/x**2)/(
8*x) - (a + b/x**2)**(3/2)/(4*x)

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Mathematica [A]  time = 0.0971036, size = 101, normalized size = 1.42 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (-3 a^2 x^4 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+3 a^2 x^4 \log (x)-\sqrt{b} \sqrt{a x^2+b} \left (5 a x^2+2 b\right )\right )}{8 \sqrt{b} x^3 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^2]*(-(Sqrt[b]*Sqrt[b + a*x^2]*(2*b + 5*a*x^2)) + 3*a^2*x^4*Log[x]
- 3*a^2*x^4*Log[b + Sqrt[b]*Sqrt[b + a*x^2]]))/(8*Sqrt[b]*x^3*Sqrt[b + a*x^2])

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Maple [B]  time = 0.012, size = 125, normalized size = 1.8 \[ -{\frac{1}{8\,{b}^{2}x} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{4}{a}^{2}- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{4}{a}^{2}+ \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{2}a-3\,\sqrt{a{x}^{2}+b}{x}^{4}{a}^{2}b+2\, \left ( a{x}^{2}+b \right ) ^{5/2}b \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(3/2)/x^2,x)

[Out]

-1/8*((a*x^2+b)/x^2)^(3/2)/x*(3*b^(3/2)*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*x^4*
a^2-(a*x^2+b)^(3/2)*x^4*a^2+(a*x^2+b)^(5/2)*x^2*a-3*(a*x^2+b)^(1/2)*x^4*a^2*b+2*
(a*x^2+b)^(5/2)*b)/(a*x^2+b)^(3/2)/b^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((a + b/x^2)^(3/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257626, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \sqrt{b} x^{3} \log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (5 \, a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, b x^{3}}, \frac{3 \, a^{2} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (5 \, a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*a^2*sqrt(b)*x^3*log((2*b*x*sqrt((a*x^2 + b)/x^2) - (a*x^2 + 2*b)*sqrt(b
))/x^2) - 2*(5*a*b*x^2 + 2*b^2)*sqrt((a*x^2 + b)/x^2))/(b*x^3), 1/8*(3*a^2*sqrt(
-b)*x^3*arctan(sqrt(-b)/(x*sqrt((a*x^2 + b)/x^2))) - (5*a*b*x^2 + 2*b^2)*sqrt((a
*x^2 + b)/x^2))/(b*x^3)]

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Sympy [A]  time = 11.4192, size = 71, normalized size = 1. \[ - \frac{5 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{8 x} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{2}}}}{4 x^{3}} - \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{8 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**(3/2)*sqrt(1 + b/(a*x**2))/(8*x) - sqrt(a)*b*sqrt(1 + b/(a*x**2))/(4*x**3)
- 3*a**2*asinh(sqrt(b)/(sqrt(a)*x))/(8*sqrt(b))

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GIAC/XCAS [A]  time = 0.249742, size = 85, normalized size = 1.2 \[ \frac{1}{8} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x^{2} + b} b}{a^{2} x^{4}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*a^2*(3*arctan(sqrt(a*x^2 + b)/sqrt(-b))/sqrt(-b) - (5*(a*x^2 + b)^(3/2) - 3*
sqrt(a*x^2 + b)*b)/(a^2*x^4))*sign(x)