\(\int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx\) [1994]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx=-\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b} \]

[Out]

-2/9*(a+b/x^3)^(3/2)/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx=-\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b} \]

[In]

Int[Sqrt[a + b/x^3]/x^4,x]

[Out]

(-2*(a + b/x^3)^(3/2))/(9*b)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx=-\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b} \]

[In]

Integrate[Sqrt[a + b/x^3]/x^4,x]

[Out]

(-2*(a + b/x^3)^(3/2))/(9*b)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-\frac {2 \left (a +\frac {b}{x^{3}}\right )^{\frac {3}{2}}}{9 b}\)	\(15\)
gosper	\(-\frac {2 \left (a \,x^{3}+b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{9 x^{3} b}\)	\(29\)
risch	\(-\frac {2 \left (a \,x^{3}+b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{9 x^{3} b}\)	\(29\)
trager	\(-\frac {2 \left (a \,x^{3}+b \right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{9 x^{3} b}\)	\(33\)
default	\(-\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {a \,x^{4}+b x}\, \left (a \,x^{3}+b \right )}{9 x^{3} \sqrt {x \left (a \,x^{3}+b \right )}\, b}\)	\(51\)

[In]

int((a+b/x^3)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-2/9*(a+b/x^3)^(3/2)/b

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx=-\frac {2 \, {\left (a x^{3} + b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{9 \, b x^{3}} \]

[In]

integrate((a+b/x^3)^(1/2)/x^4,x, algorithm="fricas")

[Out]

-2/9*(a*x^3 + b)*sqrt((a*x^3 + b)/x^3)/(b*x^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).

Time = 0.58 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx=- \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}}{9 b} - \frac {2 \sqrt {a} \sqrt {1 + \frac {b}{a x^{3}}}}{9 x^{3}} \]

[In]

integrate((a+b/x**3)**(1/2)/x**4,x)

[Out]

-2*a**(3/2)*sqrt(1 + b/(a*x**3))/(9*b) - 2*sqrt(a)*sqrt(1 + b/(a*x**3))/(9*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, b} \]

[In]

integrate((a+b/x^3)^(1/2)/x^4,x, algorithm="maxima")

[Out]

-2/9*(a + b/x^3)^(3/2)/b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, b} \]

[In]

integrate((a+b/x^3)^(1/2)/x^4,x, algorithm="giac")

[Out]

-2/9*(a + b/x^3)^(3/2)/b

Mupad [B] (verification not implemented)

Time = 6.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx=-\frac {2\,\sqrt {a+\frac {b}{x^3}}\,\left (a\,x^3+b\right )}{9\,b\,x^3} \]

[In]

int((a + b/x^3)^(1/2)/x^4,x)

[Out]

-(2*(a + b/x^3)^(1/2)*(b + a*x^3))/(9*b*x^3)