\(\int \frac {(a+\frac {b}{x})^{5/2}}{x^2} \, dx\) [1717]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2} \, dx=-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b} \]

[Out]

-2/7*(a+b/x)^(7/2)/b

Rubi [A] (verified)

Time = 0.00 (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} \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2} \, dx=-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b} \]

[In]

Int[(a + b/x)^(5/2)/x^2,x]

[Out]

(-2*(a + b/x)^(7/2))/(7*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}\right )^{7/2}}{7 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2} \, dx=-\frac {2 \left (\frac {b+a x}{x}\right )^{7/2}}{7 b} \]

[In]

Integrate[(a + b/x)^(5/2)/x^2,x]

[Out]

(-2*((b + a*x)/x)^(7/2))/(7*b)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-\frac {2 \left (a +\frac {b}{x}\right )^{\frac {7}{2}}}{7 b}\)	\(15\)
gosper	\(-\frac {2 \left (a x +b \right ) \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}{7 x b}\)	\(25\)
risch	\(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a^{3} x^{3}+3 a^{2} b \,x^{2}+3 a \,b^{2} x +b^{3}\right )}{7 x^{3} b}\)	\(47\)
trager	\(-\frac {2 \left (a^{3} x^{3}+3 a^{2} b \,x^{2}+3 a \,b^{2} x +b^{3}\right ) \sqrt {-\frac {-a x -b}{x}}}{7 x^{3} b}\)	\(51\)
default	\(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (a^{2} x^{2}+2 a b x +b^{2}\right )}{7 x^{4} b \sqrt {x \left (a x +b \right )}}\)	\(56\)

[In]

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

[Out]

-2/7*(a+b/x)^(7/2)/b

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (14) = 28\).

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2} \, dx=-\frac {2 \, {\left (a^{3} x^{3} + 3 \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3}\right )} \sqrt {\frac {a x + b}{x}}}{7 \, b x^{3}} \]

[In]

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

[Out]

-2/7*(a^3*x^3 + 3*a^2*b*x^2 + 3*a*b^2*x + b^3)*sqrt((a*x + b)/x)/(b*x^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (14) = 28\).

Time = 0.39 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.44 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2} \, dx=\begin {cases} - \frac {2 a^{3} \sqrt {a + \frac {b}{x}}}{7 b} - \frac {6 a^{2} \sqrt {a + \frac {b}{x}}}{7 x} - \frac {6 a b \sqrt {a + \frac {b}{x}}}{7 x^{2}} - \frac {2 b^{2} \sqrt {a + \frac {b}{x}}}{7 x^{3}} & \text {for}\: b \neq 0 \\- \frac {a^{\frac {5}{2}}}{x} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*a**3*sqrt(a + b/x)/(7*b) - 6*a**2*sqrt(a + b/x)/(7*x) - 6*a*b*sqrt(a + b/x)/(7*x**2) - 2*b**2*sq
rt(a + b/x)/(7*x**3), Ne(b, 0)), (-a**(5/2)/x, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{7 \, b} \]

[In]

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

[Out]

-2/7*(a + b/x)^(7/2)/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (14) = 28\).

Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 11.50 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2} \, dx=\frac {2 \, {\left (7 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} \mathrm {sgn}\left (x\right ) + 21 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b \mathrm {sgn}\left (x\right ) + 35 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{2} \mathrm {sgn}\left (x\right ) + 35 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{3} \mathrm {sgn}\left (x\right ) + 21 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{4} \mathrm {sgn}\left (x\right ) + 7 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{5} \mathrm {sgn}\left (x\right ) + b^{6} \mathrm {sgn}\left (x\right )\right )}}{7 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7}} \]

[In]

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

[Out]

2/7*(7*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*sgn(x) + 21*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b*sgn(x) +
35*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^2*sgn(x) + 35*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^3*sgn(x)
+ 21*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^4*sgn(x) + 7*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^5*sgn(x) + b
^6*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^7

Mupad [B] (verification not implemented)

Time = 6.68 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.78 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2} \, dx=-\frac {2\,a^3\,\sqrt {a+\frac {b}{x}}}{7\,b}-\frac {6\,a^2\,\sqrt {a+\frac {b}{x}}}{7\,x}-\frac {2\,b^2\,\sqrt {a+\frac {b}{x}}}{7\,x^3}-\frac {6\,a\,b\,\sqrt {a+\frac {b}{x}}}{7\,x^2} \]

[In]

int((a + b/x)^(5/2)/x^2,x)

[Out]

- (2*a^3*(a + b/x)^(1/2))/(7*b) - (6*a^2*(a + b/x)^(1/2))/(7*x) - (2*b^2*(a + b/x)^(1/2))/(7*x^3) - (6*a*b*(a
+ b/x)^(1/2))/(7*x^2)