3.1747 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} x^3} \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0172852, antiderivative size = 36, 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} \[ \frac{2}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0217715, size = 33, normalized size = 0.92 \[ \frac{4 a x+6 b}{3 b^2 \sqrt{a+\frac{b}{x}} (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^3),x]

[Out]

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

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Maple [A]  time = 0.003, size = 33, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax+3\,b \right ) }{3\,{b}^{2}{x}^{2}} \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^3,x)

[Out]

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

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Maxima [A]  time = 1.28351, size = 41, normalized size = 1.14 \begin{align*} \frac{2}{\sqrt{a + \frac{b}{x}} b^{2}} - \frac{2 \, a}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.592, size = 99, normalized size = 2.75 \begin{align*} \frac{2 \,{\left (2 \, a x^{2} + 3 \, b x\right )} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.98322, size = 82, normalized size = 2.28 \begin{align*} \begin{cases} \frac{4 a x}{3 a b^{2} x \sqrt{a + \frac{b}{x}} + 3 b^{3} \sqrt{a + \frac{b}{x}}} + \frac{6 b}{3 a b^{2} x \sqrt{a + \frac{b}{x}} + 3 b^{3} \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{5}{2}} x^{2}} & \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**3,x)

[Out]

Piecewise((4*a*x/(3*a*b**2*x*sqrt(a + b/x) + 3*b**3*sqrt(a + b/x)) + 6*b/(3*a*b**2*x*sqrt(a + b/x) + 3*b**3*sq
rt(a + b/x)), Ne(b, 0)), (-1/(2*a**(5/2)*x**2), True))

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Giac [A]  time = 1.28229, size = 49, normalized size = 1.36 \begin{align*} -\frac{2 \,{\left (a - \frac{3 \,{\left (a x + b\right )}}{x}\right )} x}{3 \,{\left (a x + b\right )} b^{2} \sqrt{\frac{a x + b}{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^3,x, algorithm="giac")

[Out]

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