3.1748 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} x^4} \, dx\)

Optimal. Leaf size=55 \[ \frac{2 a^2}{3 b^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 a}{b^3 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0248751, antiderivative size = 55, 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} \[ \frac{2 a^2}{3 b^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 a}{b^3 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.0310525, size = 44, normalized size = 0.8 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2+12 a b x+3 b^2\right )}{3 b^3 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^4),x]

[Out]

(-2*Sqrt[a + b/x]*(3*b^2 + 12*a*b*x + 8*a^2*x^2))/(3*b^3*(b + a*x)^2)

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Maple [A]  time = 0.003, size = 44, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}+12\,xab+3\,{b}^{2} \right ) }{3\,{b}^{3}{x}^{3}} \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^4,x)

[Out]

-2/3*(a*x+b)*(8*a^2*x^2+12*a*b*x+3*b^2)/x^3/b^3/((a*x+b)/x)^(5/2)

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Maxima [A]  time = 0.992823, size = 63, normalized size = 1.15 \begin{align*} -\frac{2 \, \sqrt{a + \frac{b}{x}}}{b^{3}} - \frac{4 \, a}{\sqrt{a + \frac{b}{x}} b^{3}} + \frac{2 \, a^{2}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.72509, size = 117, normalized size = 2.13 \begin{align*} -\frac{2 \,{\left (8 \, a^{2} x^{2} + 12 \, a b x + 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^4,x, algorithm="fricas")

[Out]

-2/3*(8*a^2*x^2 + 12*a*b*x + 3*b^2)*sqrt((a*x + b)/x)/(a^2*b^3*x^2 + 2*a*b^4*x + b^5)

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Sympy [A]  time = 3.6538, size = 136, normalized size = 2.47 \begin{align*} \begin{cases} - \frac{16 a^{2} x^{2}}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} - \frac{24 a b x}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} - \frac{6 b^{2}}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{3 a^{\frac{5}{2}} x^{3}} & \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**4,x)

[Out]

Piecewise((-16*a**2*x**2/(3*a*b**3*x**2*sqrt(a + b/x) + 3*b**4*x*sqrt(a + b/x)) - 24*a*b*x/(3*a*b**3*x**2*sqrt
(a + b/x) + 3*b**4*x*sqrt(a + b/x)) - 6*b**2/(3*a*b**3*x**2*sqrt(a + b/x) + 3*b**4*x*sqrt(a + b/x)), Ne(b, 0))
, (-1/(3*a**(5/2)*x**3), True))

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Giac [A]  time = 1.30114, size = 78, normalized size = 1.42 \begin{align*} \frac{2}{3} \, b{\left (\frac{{\left (a^{2} - \frac{6 \,{\left (a x + b\right )} a}{x}\right )} x}{{\left (a x + b\right )} b^{4} \sqrt{\frac{a x + b}{x}}} - \frac{3 \, \sqrt{\frac{a x + b}{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^4,x, algorithm="giac")

[Out]

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