3.1728 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0160528, 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} \[ \frac{2 a \sqrt{a+\frac{b}{x}}}{b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*x^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0171368, size = 29, normalized size = 0.81 \[ \frac{2 \sqrt{a+\frac{b}{x}} (2 a x-b)}{3 b^2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x]*x^3),x]

[Out]

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

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Maple [A]  time = 0.004, size = 33, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-b \right ) }{3\,{b}^{2}{x}^{2}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a+b/x)^(1/2),x)

[Out]

2/3*(a*x+b)*(2*a*x-b)/x^2/b^2/((a*x+b)/x)^(1/2)

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Maxima [A]  time = 1.01151, size = 41, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{2}} + \frac{2 \, \sqrt{a + \frac{b}{x}} a}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43524, size = 58, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (2 \, a x - b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.1837, size = 248, normalized size = 6.89 \begin{align*} \frac{4 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} + \frac{2 a^{\frac{5}{2}} b^{\frac{5}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{7}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{4} b x^{\frac{5}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{3} b^{2} x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b/x)**(1/2),x)

[Out]

4*a**(7/2)*b**(3/2)*x**2*sqrt(a*x/b + 1)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) + 2*a**(5/2)*b*
*(5/2)*x*sqrt(a*x/b + 1)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) - 2*a**(3/2)*b**(7/2)*sqrt(a*x/
b + 1)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) - 4*a**4*b*x**(5/2)/(3*a**(5/2)*b**3*x**(5/2) + 3
*a**(3/2)*b**4*x**(3/2)) - 4*a**3*b**2*x**(3/2)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2))

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Giac [A]  time = 1.14855, size = 55, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (3 \, a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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