3.1911 \(\int \frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b} \]

[Out]

-(a + b/x^2)^(7/2)/(7*b)

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Rubi [A]  time = 0.0272453, 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 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(5/2)/x^3,x]

[Out]

-(a + b/x^2)^(7/2)/(7*b)

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Rubi in Sympy [A]  time = 2.13972, size = 14, normalized size = 0.78 \[ - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b/x**2)**(7/2)/(7*b)

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Mathematica [A]  time = 0.0329176, size = 30, normalized size = 1.67 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )^3}{7 b x^6} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b/x^2]*(b + a*x^2)^3)/(7*b*x^6)

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Maple [A]  time = 0.008, size = 29, normalized size = 1.6 \[ -{\frac{a{x}^{2}+b}{7\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(5/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.44165, size = 19, normalized size = 1.06 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{7}{2}}}{7 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*(a + b/x^2)^(7/2)/b

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Fricas [A]  time = 0.251446, size = 68, normalized size = 3.78 \[ -\frac{{\left (a^{3} x^{6} + 3 \, a^{2} b x^{4} + 3 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{7 \, b x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*(a^3*x^6 + 3*a^2*b*x^4 + 3*a*b^2*x^2 + b^3)*sqrt((a*x^2 + b)/x^2)/(b*x^6)

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Sympy [A]  time = 12.6234, size = 88, normalized size = 4.89 \[ \begin{cases} - \frac{a^{3} \sqrt{a + \frac{b}{x^{2}}}}{7 b} - \frac{3 a^{2} \sqrt{a + \frac{b}{x^{2}}}}{7 x^{2}} - \frac{3 a b \sqrt{a + \frac{b}{x^{2}}}}{7 x^{4}} - \frac{b^{2} \sqrt{a + \frac{b}{x^{2}}}}{7 x^{6}} & \text{for}\: b \neq 0 \\- \frac{a^{\frac{5}{2}}}{2 x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a**3*sqrt(a + b/x**2)/(7*b) - 3*a**2*sqrt(a + b/x**2)/(7*x**2) - 3*a
*b*sqrt(a + b/x**2)/(7*x**4) - b**2*sqrt(a + b/x**2)/(7*x**6), Ne(b, 0)), (-a**(
5/2)/(2*x**2), True))

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GIAC/XCAS [A]  time = 0.253855, size = 163, normalized size = 9.06 \[ \frac{2 \,{\left (7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{12} a^{\frac{7}{2}}{\rm sign}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{8} a^{\frac{7}{2}} b^{2}{\rm sign}\left (x\right ) + 21 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{7}{2}} b^{4}{\rm sign}\left (x\right ) + a^{\frac{7}{2}} b^{6}{\rm sign}\left (x\right )\right )}}{7 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(7*(sqrt(a)*x - sqrt(a*x^2 + b))^12*a^(7/2)*sign(x) + 35*(sqrt(a)*x - sqrt(a
*x^2 + b))^8*a^(7/2)*b^2*sign(x) + 21*(sqrt(a)*x - sqrt(a*x^2 + b))^4*a^(7/2)*b^
4*sign(x) + a^(7/2)*b^6*sign(x))/((sqrt(a)*x - sqrt(a*x^2 + b))^2 - b)^7