\(\int \frac {1}{(a+\frac {b}{x})^{5/2} x^{3/2}} \, dx\) [1799]

Optimal result

Integrand size = 17, antiderivative size = 46 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {4 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}-\frac {2}{a \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \]

[Out]

-4/3*b/a^2/(a+b/x)^(3/2)/x^(3/2)-2/a/(a+b/x)^(3/2)/x^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {4 b}{3 a^2 x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}} \]

[In]

Int[1/((a + b/x)^(5/2)*x^(3/2)),x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 6.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} (2 b+3 a x)}{3 a^2 (b+a x)^2} \]

[In]

Integrate[1/((a + b/x)^(5/2)*x^(3/2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.72

method	result	size
gosper	\(-\frac {2 \left (3 a x +2 b \right ) \left (a x +b \right )}{3 a^{2} x^{\frac {5}{2}} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}\)	\(33\)
default	\(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (3 a x +2 b \right )}{3 \left (a x +b \right )^{2} a^{2}}\)	\(35\)

[In]

int(1/(a+b/x)^(5/2)/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {2 \, {\left (3 \, a x + 2 \, b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )}} \]

[In]

integrate(1/(a+b/x)^(5/2)/x^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (39) = 78\).

Time = 5.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.04 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=- \frac {6 a x}{3 a^{3} \sqrt {b} x \sqrt {\frac {a x}{b} + 1} + 3 a^{2} b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {4 b}{3 a^{3} \sqrt {b} x \sqrt {\frac {a x}{b} + 1} + 3 a^{2} b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}} \]

[In]

integrate(1/(a+b/x)**(5/2)/x**(3/2),x)

[Out]

-6*a*x/(3*a**3*sqrt(b)*x*sqrt(a*x/b + 1) + 3*a**2*b**(3/2)*sqrt(a*x/b + 1)) - 4*b/(3*a**3*sqrt(b)*x*sqrt(a*x/b
+ 1) + 3*a**2*b**(3/2)*sqrt(a*x/b + 1))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} x - b\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} x^{\frac {3}{2}}} \]

[In]

integrate(1/(a+b/x)^(5/2)/x^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {2 \, {\left (3 \, a x + 2 \, b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{2}} \]

[In]

integrate(1/(a+b/x)^(5/2)/x^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.36 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{3/2}}{a^3}+\frac {4\,b\,\sqrt {x}}{3\,a^4}\right )}{x^2+\frac {b^2}{a^2}+\frac {2\,b\,x}{a}} \]

[In]

int(1/(x^(3/2)*(a + b/x)^(5/2)),x)

[Out]

-((a + b/x)^(1/2)*((2*x^(3/2))/a^3 + (4*b*x^(1/2))/(3*a^4)))/(x^2 + b^2/a^2 + (2*b*x)/a)