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*(a + b/x^3)^(5/2))/(15*b)

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Rubi [A]  time = 0.0064478, 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, 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*}

Mathematica [A]  time = 0.0098076, 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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Maple [A]  time = 0.004, size = 29, normalized size = 1.6 \begin{align*} -{\frac{2\,a{x}^{3}+2\,b}{15\,b{x}^{3}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.969006, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b} \end{align*}

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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Fricas [B]  time = 1.51056, size = 88, normalized size = 4.89 \begin{align*} -\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}} \end{align*}

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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Sympy [B]  time = 1.39658, size = 71, normalized size = 3.94 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 1.17954, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b} \end{align*}

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