3.1913 \(\int \frac{(a+\frac{b}{x^2})^{5/2}}{x^3} \, dx\)

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 = 0.0059163, 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} \[ -\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(5/2)/x^3,x]

[Out]

-(a + b/x^2)^(7/2)/(7*b)

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^2}\right )^{5/2}}{x^3} \, dx &=-\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0108239, size = 28, normalized size = 1.56 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{5/2} \left (a x^2+b\right )}{7 b x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(5/2)/x^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 29, normalized size = 1.6 \begin{align*} -{\frac{a{x}^{2}+b}{7\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^(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 = 1.00338, size = 19, normalized size = 1.06 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{7}{2}}}{7 \, b} \end{align*}

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 [B]  time = 1.52103, size = 108, normalized size = 6. \begin{align*} -\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}} \end{align*}

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 = 3.72643, size = 88, normalized size = 4.89 \begin{align*} \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} \end{align*}

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 [B]  time = 1.2684, size = 163, normalized size = 9.06 \begin{align*} \frac{2 \,{\left (7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{12} a^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{8} a^{\frac{7}{2}} b^{2} \mathrm{sgn}\left (x\right ) + 21 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{7}{2}} b^{4} \mathrm{sgn}\left (x\right ) + a^{\frac{7}{2}} b^{6} \mathrm{sgn}\left (x\right )\right )}}{7 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{7}} \end{align*}

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