3.1898 \(\int \left (a+\frac{b}{x^2}\right )^{3/2} x^2 \, dx\)

Optimal. Leaf size=61 \[ -b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )+b x \sqrt{a+\frac{b}{x^2}}+\frac{1}{3} x^3 \left (a+\frac{b}{x^2}\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.101419, antiderivative size = 61, 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 \[ -b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )+b x \sqrt{a+\frac{b}{x^2}}+\frac{1}{3} x^3 \left (a+\frac{b}{x^2}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(3/2)*x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.29897, size = 51, normalized size = 0.84 \[ - b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )} + b x \sqrt{a + \frac{b}{x^{2}}} + \frac{x^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.102408, size = 85, normalized size = 1.39 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (-3 b^{3/2} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+\sqrt{a x^2+b} \left (a x^2+4 b\right )+3 b^{3/2} \log (x)\right )}{3 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 78, normalized size = 1.3 \[ -{\frac{{x}^{3}}{3} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) - \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}-3\,\sqrt{a{x}^{2}+b}b \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(3/2)*x^2,x)

[Out]

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

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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 = 0.250717, size = 1, normalized size = 0.02 \[ \left [\frac{1}{2} \, b^{\frac{3}{2}} \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + \frac{1}{3} \,{\left (a x^{3} + 4 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}, -\sqrt{-b} b \arctan \left (\frac{b}{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) + \frac{1}{3} \,{\left (a x^{3} + 4 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.73216, size = 78, normalized size = 1.28 \[ \frac{a \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{b^{\frac{3}{2}} \log{\left (\frac{a x^{2}}{b} \right )}}{2} - b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a x^{2}}{b} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*sqrt(b)*x**2*sqrt(a*x**2/b + 1)/3 + 4*b**(3/2)*sqrt(a*x**2/b + 1)/3 + b**(3/2)
*log(a*x**2/b)/2 - b**(3/2)*log(sqrt(a*x**2/b + 1) + 1)

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GIAC/XCAS [A]  time = 0.238492, size = 119, normalized size = 1.95 \[ \frac{1}{3} \,{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} +{\left (a x^{2} + b\right )}^{\frac{3}{2}} + 3 \, \sqrt{a x^{2} + b} b\right )}{\rm sign}\left (x\right ) - \frac{{\left (3 \, b^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b} b^{\frac{3}{2}}\right )}{\rm sign}\left (x\right )}{3 \, \sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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