3.1900 \(\int \left (a+\frac{b}{x^2}\right )^{3/2} \, dx\)

Optimal. Leaf size=64 \[ x \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0864575, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ x \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.27497, size = 56, normalized size = 0.88 \[ - \frac{3 a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2} - \frac{3 b \sqrt{a + \frac{b}{x^{2}}}}{2 x} + x \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0810075, size = 95, normalized size = 1.48 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (-\left (b-2 a x^2\right ) \sqrt{a x^2+b}+3 a \sqrt{b} x^2 \log (x)-3 a \sqrt{b} x^2 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )\right )}{2 x \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 100, normalized size = 1.6 \[ -{\frac{x}{2\,b} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}a- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{2}a+ \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}-3\,\sqrt{a{x}^{2}+b}{x}^{2}ab \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(3/2),x)

[Out]

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

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253555, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{b} x \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \,{\left (2 \, a x^{2} - b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \, x}, -\frac{3 \, a \sqrt{-b} x \arctan \left (\frac{b}{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (2 \, a x^{2} - b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.64989, size = 88, normalized size = 1.38 \[ \frac{a^{\frac{3}{2}} x}{\sqrt{1 + \frac{b}{a x^{2}}}} + \frac{\sqrt{a} b}{2 x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2} - \frac{b^{2}}{2 \sqrt{a} x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.24969, size = 80, normalized size = 1.25 \[ \frac{1}{2} \,{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{a x^{2} + b} - \frac{\sqrt{a x^{2} + b} b}{a x^{2}}\right )} a{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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