3.1749 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} x^5} \, dx\)

Optimal. Leaf size=76 \[ -\frac{2 a^3}{3 b^4 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{6 a^2}{b^4 \sqrt{a+\frac{b}{x}}}+\frac{6 a \sqrt{a+\frac{b}{x}}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^4} \]

[Out]

(-2*a^3)/(3*b^4*(a + b/x)^(3/2)) + (6*a^2)/(b^4*Sqrt[a + b/x]) + (6*a*Sqrt[a + b/x])/b^4 - (2*(a + b/x)^(3/2))
/(3*b^4)

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Rubi [A]  time = 0.0319017, antiderivative size = 76, 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, Rules used = {266, 43} \[ -\frac{2 a^3}{3 b^4 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{6 a^2}{b^4 \sqrt{a+\frac{b}{x}}}+\frac{6 a \sqrt{a+\frac{b}{x}}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^5),x]

[Out]

(-2*a^3)/(3*b^4*(a + b/x)^(3/2)) + (6*a^2)/(b^4*Sqrt[a + b/x]) + (6*a*Sqrt[a + b/x])/b^4 - (2*(a + b/x)^(3/2))
/(3*b^4)

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^5} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{5/2}}+\frac{3 a^2}{b^3 (a+b x)^{3/2}}-\frac{3 a}{b^3 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^3}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^3}{3 b^4 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{6 a^2}{b^4 \sqrt{a+\frac{b}{x}}}+\frac{6 a \sqrt{a+\frac{b}{x}}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0304639, size = 58, normalized size = 0.76 \[ \frac{48 a^2 b x^2+32 a^3 x^3+12 a b^2 x-2 b^3}{3 b^4 x^2 \sqrt{a+\frac{b}{x}} (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^5),x]

[Out]

(-2*b^3 + 12*a*b^2*x + 48*a^2*b*x^2 + 32*a^3*x^3)/(3*b^4*Sqrt[a + b/x]*x^2*(b + a*x))

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Maple [A]  time = 0.004, size = 55, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}+24\,{a}^{2}b{x}^{2}+6\,xa{b}^{2}-{b}^{3} \right ) }{3\,{x}^{4}{b}^{4}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/x^5,x)

[Out]

2/3*(a*x+b)*(16*a^3*x^3+24*a^2*b*x^2+6*a*b^2*x-b^3)/x^4/b^4/((a*x+b)/x)^(5/2)

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Maxima [A]  time = 1.03179, size = 86, normalized size = 1.13 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{4}} + \frac{6 \, \sqrt{a + \frac{b}{x}} a}{b^{4}} + \frac{6 \, a^{2}}{\sqrt{a + \frac{b}{x}} b^{4}} - \frac{2 \, a^{3}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^5,x, algorithm="maxima")

[Out]

-2/3*(a + b/x)^(3/2)/b^4 + 6*sqrt(a + b/x)*a/b^4 + 6*a^2/(sqrt(a + b/x)*b^4) - 2/3*a^3/((a + b/x)^(3/2)*b^4)

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Fricas [A]  time = 1.61023, size = 142, normalized size = 1.87 \begin{align*} \frac{2 \,{\left (16 \, a^{3} x^{3} + 24 \, a^{2} b x^{2} + 6 \, a b^{2} x - b^{3}\right )} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{2} b^{4} x^{3} + 2 \, a b^{5} x^{2} + b^{6} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^5,x, algorithm="fricas")

[Out]

2/3*(16*a^3*x^3 + 24*a^2*b*x^2 + 6*a*b^2*x - b^3)*sqrt((a*x + b)/x)/(a^2*b^4*x^3 + 2*a*b^5*x^2 + b^6*x)

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Sympy [A]  time = 4.53977, size = 187, normalized size = 2.46 \begin{align*} \begin{cases} \frac{32 a^{3} x^{3}}{3 a b^{4} x^{3} \sqrt{a + \frac{b}{x}} + 3 b^{5} x^{2} \sqrt{a + \frac{b}{x}}} + \frac{48 a^{2} b x^{2}}{3 a b^{4} x^{3} \sqrt{a + \frac{b}{x}} + 3 b^{5} x^{2} \sqrt{a + \frac{b}{x}}} + \frac{12 a b^{2} x}{3 a b^{4} x^{3} \sqrt{a + \frac{b}{x}} + 3 b^{5} x^{2} \sqrt{a + \frac{b}{x}}} - \frac{2 b^{3}}{3 a b^{4} x^{3} \sqrt{a + \frac{b}{x}} + 3 b^{5} x^{2} \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{4 a^{\frac{5}{2}} x^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((32*a**3*x**3/(3*a*b**4*x**3*sqrt(a + b/x) + 3*b**5*x**2*sqrt(a + b/x)) + 48*a**2*b*x**2/(3*a*b**4*x
**3*sqrt(a + b/x) + 3*b**5*x**2*sqrt(a + b/x)) + 12*a*b**2*x/(3*a*b**4*x**3*sqrt(a + b/x) + 3*b**5*x**2*sqrt(a
+ b/x)) - 2*b**3/(3*a*b**4*x**3*sqrt(a + b/x) + 3*b**5*x**2*sqrt(a + b/x)), Ne(b, 0)), (-1/(4*a**(5/2)*x**4),
True))

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Giac [A]  time = 1.28539, size = 123, normalized size = 1.62 \begin{align*} -\frac{2}{3} \, b{\left (\frac{{\left (a^{3} - \frac{9 \,{\left (a x + b\right )} a^{2}}{x}\right )} x}{{\left (a x + b\right )} b^{5} \sqrt{\frac{a x + b}{x}}} - \frac{9 \, a b^{10} \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} b^{10} \sqrt{\frac{a x + b}{x}}}{x}}{b^{15}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^5,x, algorithm="giac")

[Out]

-2/3*b*((a^3 - 9*(a*x + b)*a^2/x)*x/((a*x + b)*b^5*sqrt((a*x + b)/x)) - (9*a*b^10*sqrt((a*x + b)/x) - (a*x + b
)*b^10*sqrt((a*x + b)/x)/x)/b^15)