3.1929 \(\int \frac{x}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{3 x^2 \sqrt{a+\frac{b}{x^2}}}{2 a^2}-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.103591, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{3 x^2 \sqrt{a+\frac{b}{x^2}}}{2 a^2}-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.61652, size = 61, normalized size = 0.88 \[ - \frac{x^{2}}{a \sqrt{a + \frac{b}{x^{2}}}} + \frac{3 x^{2} \sqrt{a + \frac{b}{x^{2}}}}{2 a^{2}} - \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0513118, size = 76, normalized size = 1.1 \[ \frac{\sqrt{a} x \left (a x^2+3 b\right )-3 b \sqrt{a x^2+b} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{2 a^{5/2} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 73, normalized size = 1.1 \[{\frac{a{x}^{2}+b}{2\,{x}^{3}} \left ({x}^{3}{a}^{{\frac{5}{2}}}+3\,{a}^{3/2}xb-3\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) \sqrt{a{x}^{2}+b}ab \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^2)^(3/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245536, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a b x^{2} + b^{2}\right )} \sqrt{a} \log \left (2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \,{\left (a^{4} x^{2} + a^{3} b\right )}}, \frac{3 \,{\left (a b x^{2} + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.6551, size = 71, normalized size = 1.03 \[ \frac{x^{3}}{2 a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{3 \sqrt{b} x}{2 a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268809, size = 131, normalized size = 1.9 \[ \frac{1}{2} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, a - \frac{3 \,{\left (a x^{2} + b\right )}}{x^{2}}}{{\left (a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}\right )} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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