\(\int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx\) [1728]

Optimal result

Integrand size = 15, antiderivative size = 36 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx=\frac {2 a \sqrt {a+\frac {b}{x}}}{b^2}-\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx=\frac {2 a \sqrt {a+\frac {b}{x}}}{b^2}-\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^2} \]

[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 45

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 272

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*} \text {integral}& = -\text {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {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] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx=\frac {2 \sqrt {\frac {b+a x}{x}} (-b+2 a x)}{3 b^2 x} \]

[In]

Integrate[1/(Sqrt[a + b/x]*x^3),x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89

method	result	size
trager	\(\frac {2 \left (2 a x -b \right ) \sqrt {-\frac {-a x -b}{x}}}{3 x \,b^{2}}\)	\(32\)
gosper	\(\frac {2 \left (a x +b \right ) \left (2 a x -b \right )}{3 x^{2} b^{2} \sqrt {\frac {a x +b}{x}}}\)	\(33\)
risch	\(\frac {2 \left (a x +b \right ) \left (2 a x -b \right )}{3 x^{2} b^{2} \sqrt {\frac {a x +b}{x}}}\)	\(33\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (6 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} x^{3}+6 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} x^{3}+3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b \,x^{3}-3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b \,x^{3}-12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} x +4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \sqrt {a}\right )}{6 x^{2} \sqrt {x \left (a x +b \right )}\, b^{3} \sqrt {a}}\)	\(175\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx=\frac {2 \, {\left (2 \, a x - b\right )} \sqrt {\frac {a x + b}{x}}}{3 \, b^{2} x} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (29) = 58\).

Time = 0.69 (sec) , antiderivative size = 248, normalized size of antiderivative = 6.89 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx=\frac {4 a^{\frac {7}{2}} b^{\frac {3}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} + \frac {2 a^{\frac {5}{2}} b^{\frac {5}{2}} x \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {2 a^{\frac {3}{2}} b^{\frac {7}{2}} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {4 a^{4} b x^{\frac {5}{2}}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} - \frac {4 a^{3} b^{2} x^{\frac {3}{2}}}{3 a^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{3 \, b^{2}} + \frac {2 \, \sqrt {a + \frac {b}{x}} a}{b^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}}{3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^3} \, dx=-\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (b-2\,a\,x\right )}{3\,b^2\,x} \]

[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)