3.2013 \(\int \frac {(a+\frac {b}{x^3})^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=18 \[ -\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b} \]

[Out]

-2/15*(a+b/x^3)^(5/2)/b

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {261} \[ -\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^3)^(3/2)/x^4,x]

[Out]

(-2*(a + b/x^3)^(5/2))/(15*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+\frac {b}{x^3}\right )^{3/2}}{x^4} \, dx &=-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.56 \[ -\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2} \left (a x^3+b\right )}{15 b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^3)^(3/2)/x^4,x]

[Out]

(-2*(a + b/x^3)^(3/2)*(b + a*x^3))/(15*b*x^3)

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fricas [B]  time = 1.05, size = 39, normalized size = 2.17 \[ -\frac {2 \, {\left (a^{2} x^{6} + 2 \, a b x^{3} + b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{15 \, b x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(3/2)/x^4,x, algorithm="fricas")

[Out]

-2/15*(a^2*x^6 + 2*a*b*x^3 + b^2)*sqrt((a*x^3 + b)/x^3)/(b*x^6)

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giac [A]  time = 0.18, size = 14, normalized size = 0.78 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(3/2)/x^4,x, algorithm="giac")

[Out]

-2/15*(a + b/x^3)^(5/2)/b

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maple [A]  time = 0.00, size = 29, normalized size = 1.61 \[ -\frac {2 \left (a \,x^{3}+b \right ) \left (\frac {a \,x^{3}+b}{x^{3}}\right )^{\frac {3}{2}}}{15 b \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^3)^(3/2)/x^4,x)

[Out]

-2/15*(a*x^3+b)/x^3/b*((a*x^3+b)/x^3)^(3/2)

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maxima [A]  time = 0.85, size = 14, normalized size = 0.78 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(3/2)/x^4,x, algorithm="maxima")

[Out]

-2/15*(a + b/x^3)^(5/2)/b

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mupad [B]  time = 1.89, size = 26, normalized size = 1.44 \[ -\frac {2\,\sqrt {a+\frac {b}{x^3}}\,{\left (a\,x^3+b\right )}^2}{15\,b\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^3)^(3/2)/x^4,x)

[Out]

-(2*(a + b/x^3)^(1/2)*(b + a*x^3)^2)/(15*b*x^6)

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sympy [B]  time = 1.36, size = 71, normalized size = 3.94 \[ - \frac {2 a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x^{3}}}}{15 b} - \frac {4 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}}{15 x^{3}} - \frac {2 \sqrt {a} b \sqrt {1 + \frac {b}{a x^{3}}}}{15 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**3)**(3/2)/x**4,x)

[Out]

-2*a**(5/2)*sqrt(1 + b/(a*x**3))/(15*b) - 4*a**(3/2)*sqrt(1 + b/(a*x**3))/(15*x**3) - 2*sqrt(a)*b*sqrt(1 + b/(
a*x**3))/(15*x**6)

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