3.1945 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^3} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{3 b \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

1/(3*b*(a + b/x^2)^(3/2))

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Rubi [A]  time = 0.0288461, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{1}{3 b \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(5/2)*x^3),x]

[Out]

1/(3*b*(a + b/x^2)^(3/2))

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Rubi in Sympy [A]  time = 2.11519, size = 14, normalized size = 0.78 \[ \frac{1}{3 b \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(5/2)/x**3,x)

[Out]

1/(3*b*(a + b/x**2)**(3/2))

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Mathematica [A]  time = 0.0310515, size = 30, normalized size = 1.67 \[ \frac{x^4 \sqrt{a+\frac{b}{x^2}}}{3 b \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(5/2)*x^3),x]

[Out]

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

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Maple [A]  time = 0.008, size = 29, normalized size = 1.6 \[{\frac{a{x}^{2}+b}{3\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(5/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.44066, size = 19, normalized size = 1.06 \[ \frac{1}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^3),x, algorithm="maxima")

[Out]

1/3/((a + b/x^2)^(3/2)*b)

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Fricas [A]  time = 0.241376, size = 55, normalized size = 3.06 \[ \frac{x^{4} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 13.2867, size = 48, normalized size = 2.67 \[ \begin{cases} \frac{1}{3 a b \sqrt{a + \frac{b}{x^{2}}} + \frac{3 b^{2} \sqrt{a + \frac{b}{x^{2}}}}{x^{2}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{5}{2}} x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(5/2)/x**3,x)

[Out]

Piecewise((1/(3*a*b*sqrt(a + b/x**2) + 3*b**2*sqrt(a + b/x**2)/x**2), Ne(b, 0)),
(-1/(2*a**(5/2)*x**2), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^3),x, algorithm="giac")

[Out]

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