3.2039 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x^7} \, dx\)

Optimal. Leaf size=38 \[ -\frac{2 a}{3 b^2 \sqrt{a+\frac{b}{x^3}}}-\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0699912, antiderivative size = 38, 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 \[ -\frac{2 a}{3 b^2 \sqrt{a+\frac{b}{x^3}}}-\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)^(3/2)*x^7),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.00207, size = 36, normalized size = 0.95 \[ - \frac{2 a}{3 b^{2} \sqrt{a + \frac{b}{x^{3}}}} - \frac{2 \sqrt{a + \frac{b}{x^{3}}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**3)**(3/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0275304, size = 29, normalized size = 0.76 \[ -\frac{2 \left (2 a x^3+b\right )}{3 b^2 x^3 \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^3)^(3/2)*x^7),x]

[Out]

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

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Maple [A]  time = 0.007, size = 37, normalized size = 1. \[ -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 2\,a{x}^{3}+b \right ) }{3\,{b}^{2}{x}^{6}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)^(3/2)/x^7,x)

[Out]

-2/3*(a*x^3+b)*(2*a*x^3+b)/x^6/b^2/((a*x^3+b)/x^3)^(3/2)

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Maxima [A]  time = 1.44626, size = 41, normalized size = 1.08 \[ -\frac{2 \, \sqrt{a + \frac{b}{x^{3}}}}{3 \, b^{2}} - \frac{2 \, a}{3 \, \sqrt{a + \frac{b}{x^{3}}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.241147, size = 50, normalized size = 1.32 \[ -\frac{2 \,{\left (2 \, a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{3 \,{\left (a b^{2} x^{3} + b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^7),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 21.1159, size = 51, normalized size = 1.34 \[ \begin{cases} - \frac{4 a}{3 b^{2} \sqrt{a + \frac{b}{x^{3}}}} - \frac{2}{3 b x^{3} \sqrt{a + \frac{b}{x^{3}}}} & \text{for}\: b \neq 0 \\- \frac{1}{6 a^{\frac{3}{2}} x^{6}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)**(3/2)/x**7,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^7),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^3)^(3/2)*x^7), x)