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]

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

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.86, size = 27, normalized size = 0.75 \[ \frac {2 \, {\left (2 \, a x - b\right )} \sqrt {\frac {a x + b}{x}}}{3 \, b^{2} x} \]

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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giac [A]  time = 0.19, size = 41, normalized size = 1.14 \[ \frac {2 \, {\left (3 \, a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )}}{3 \, b^{2}} \]

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

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maple [A]  time = 0.00, size = 33, normalized size = 0.92 \[ \frac {2 \left (a x +b \right ) \left (2 a x -b \right )}{3 \sqrt {\frac {a x +b}{x}}\, b^{2} x^{2}} \]

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 = 0.90, size = 30, normalized size = 0.83 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{3 \, b^{2}} + \frac {2 \, \sqrt {a + \frac {b}{x}} a}{b^{2}} \]

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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mupad [B]  time = 1.30, size = 23, normalized size = 0.64 \[ -\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (b-2\,a\,x\right )}{3\,b^2\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x)^(1/2)),x)

[Out]

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

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sympy [B]  time = 1.45, size = 248, normalized size = 6.89 \[ \frac {4 a^{\frac {7}{2}} b^{\frac {3}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} + \frac {2 a^{\frac {5}{2}} b^{\frac {5}{2}} x \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {2 a^{\frac {3}{2}} b^{\frac {7}{2}} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {4 a^{4} b x^{\frac {5}{2}}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {4 a^{3} b^{2} x^{\frac {3}{2}}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} \]

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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