3.1899 \(\int \left (a+\frac{b}{x^2}\right )^{3/2} x \, dx\)

Optimal. Leaf size=63 \[ \frac{1}{2} x^2 \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{3}{2} b \sqrt{a+\frac{b}{x^2}}+\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0906803, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{1}{2} x^2 \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{3}{2} b \sqrt{a+\frac{b}{x^2}}+\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(3/2)*x,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.682, size = 56, normalized size = 0.89 \[ \frac{3 \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{2} - \frac{3 b \sqrt{a + \frac{b}{x^{2}}}}{2} + \frac{x^{2} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0623314, size = 79, normalized size = 1.25 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (\sqrt{a x^2+b} \left (a x^2-2 b\right )+3 \sqrt{a} b x \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )\right )}{2 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.011, size = 107, normalized size = 1.7 \[{\frac{{x}^{2}}{2\,b} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 2\,{a}^{3/2} \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{2}+3\,{a}^{3/2}\sqrt{a{x}^{2}+b}{x}^{2}b-2\, \left ( a{x}^{2}+b \right ) ^{5/2}\sqrt{a}+3\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) xa{b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(3/2)*x,x)

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.252853, size = 1, normalized size = 0.02 \[ \left [\frac{3}{4} \, \sqrt{a} b \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) + \frac{1}{2} \,{\left (a x^{2} - 2 \, b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}, \frac{3}{2} \, \sqrt{-a} b \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) + \frac{1}{2} \,{\left (a x^{2} - 2 \, b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 10.0842, size = 88, normalized size = 1.4 \[ \frac{3 \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2} + \frac{a^{2} x^{3}}{2 \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{a \sqrt{b} x}{2 \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{b^{\frac{3}{2}}}{x \sqrt{\frac{a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.274803, size = 107, normalized size = 1.7 \[ \frac{1}{2} \, \sqrt{a x^{2} + b} a x{\rm sign}\left (x\right ) - \frac{3}{4} \, \sqrt{a} b{\rm ln}\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \, \sqrt{a} b^{2}{\rm sign}\left (x\right )}{{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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