3.1947 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^7} \, dx\)

Optimal. Leaf size=55 \[ \frac{a^2}{3 b^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{2 a}{b^3 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0925196, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2}{3 b^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{2 a}{b^3 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(5/2)*x^7),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.4993, size = 48, normalized size = 0.87 \[ \frac{a^{2}}{3 b^{3} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} - \frac{2 a}{b^{3} \sqrt{a + \frac{b}{x^{2}}}} - \frac{\sqrt{a + \frac{b}{x^{2}}}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(5/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0491817, size = 48, normalized size = 0.87 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (8 a^2 x^4+12 a b x^2+3 b^2\right )}{3 b^3 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(5/2)*x^7),x]

[Out]

-(Sqrt[a + b/x^2]*(3*b^2 + 12*a*b*x^2 + 8*a^2*x^4))/(3*b^3*(b + a*x^2)^2)

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Maple [A]  time = 0.008, size = 50, normalized size = 0.9 \[ -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 8\,{x}^{4}{a}^{2}+12\,ab{x}^{2}+3\,{b}^{2} \right ) }{3\,{b}^{3}{x}^{6}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(5/2)/x^7,x)

[Out]

-1/3*(a*x^2+b)*(8*a^2*x^4+12*a*b*x^2+3*b^2)/x^6/b^3/((a*x^2+b)/x^2)^(5/2)

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Maxima [A]  time = 1.44959, size = 63, normalized size = 1.15 \[ -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b^{3}} - \frac{2 \, a}{\sqrt{a + \frac{b}{x^{2}}} b^{3}} + \frac{a^{2}}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.242698, size = 82, normalized size = 1.49 \[ -\frac{{\left (8 \, a^{2} x^{4} + 12 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^7),x, algorithm="fricas")

[Out]

-1/3*(8*a^2*x^4 + 12*a*b*x^2 + 3*b^2)*sqrt((a*x^2 + b)/x^2)/(a^2*b^3*x^4 + 2*a*b
^4*x^2 + b^5)

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Sympy [A]  time = 31.2106, size = 153, normalized size = 2.78 \[ \begin{cases} - \frac{8 a^{2} x^{4}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{12 a b x^{2}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{3 b^{2}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{6 a^{\frac{5}{2}} x^{6}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(5/2)/x**7,x)

[Out]

Piecewise((-8*a**2*x**4/(3*a*b**3*x**4*sqrt(a + b/x**2) + 3*b**4*x**2*sqrt(a + b
/x**2)) - 12*a*b*x**2/(3*a*b**3*x**4*sqrt(a + b/x**2) + 3*b**4*x**2*sqrt(a + b/x
**2)) - 3*b**2/(3*a*b**3*x**4*sqrt(a + b/x**2) + 3*b**4*x**2*sqrt(a + b/x**2)),
Ne(b, 0)), (-1/(6*a**(5/2)*x**6), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(5/2)*x^7),x, algorithm="giac")

[Out]

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