\(\int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx\) [1699]

Optimal result

Integrand size = 15, antiderivative size = 80 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx=\frac {2 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^4}-\frac {6 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4}+\frac {6 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^4} \]

[Out]

2/3*a^3*(a+b/x)^(3/2)/b^4-6/5*a^2*(a+b/x)^(5/2)/b^4+6/7*a*(a+b/x)^(7/2)/b^4-2/9*(a+b/x)^(9/2)/b^4

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 80, 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} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx=\frac {2 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^4}-\frac {6 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^4}+\frac {6 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4} \]

[In]

Int[Sqrt[a + b/x]/x^5,x]

[Out]

(2*a^3*(a + b/x)^(3/2))/(3*b^4) - (6*a^2*(a + b/x)^(5/2))/(5*b^4) + (6*a*(a + b/x)^(7/2))/(7*b^4) - (2*(a + b/
x)^(9/2))/(9*b^4)

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 x^3 \sqrt {a+b x} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^4}-\frac {6 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4}+\frac {6 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx=\frac {2 \sqrt {\frac {b+a x}{x}} \left (-35 b^4-5 a b^3 x+6 a^2 b^2 x^2-8 a^3 b x^3+16 a^4 x^4\right )}{315 b^4 x^4} \]

[In]

Integrate[Sqrt[a + b/x]/x^5,x]

[Out]

(2*Sqrt[(b + a*x)/x]*(-35*b^4 - 5*a*b^3*x + 6*a^2*b^2*x^2 - 8*a^3*b*x^3 + 16*a^4*x^4))/(315*b^4*x^4)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.69

method	result	size
gosper	\(\frac {2 \left (a x +b \right ) \left (16 a^{3} x^{3}-24 a^{2} b \,x^{2}+30 a \,b^{2} x -35 b^{3}\right ) \sqrt {\frac {a x +b}{x}}}{315 b^{4} x^{4}}\)	\(55\)
risch	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (16 a^{4} x^{4}-8 a^{3} b \,x^{3}+6 a^{2} b^{2} x^{2}-5 a \,b^{3} x -35 b^{4}\right )}{315 x^{4} b^{4}}\)	\(61\)
trager	\(\frac {2 \left (16 a^{4} x^{4}-8 a^{3} b \,x^{3}+6 a^{2} b^{2} x^{2}-5 a \,b^{3} x -35 b^{4}\right ) \sqrt {-\frac {-a x -b}{x}}}{315 x^{4} b^{4}}\)	\(65\)
default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (16 a^{3} x^{3}-24 a^{2} b \,x^{2}+30 a \,b^{2} x -35 b^{3}\right )}{315 x^{5} \sqrt {x \left (a x +b \right )}\, b^{4}}\)	\(70\)

[In]

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

[Out]

2/315*(a*x+b)*(16*a^3*x^3-24*a^2*b*x^2+30*a*b^2*x-35*b^3)*((a*x+b)/x)^(1/2)/b^4/x^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx=\frac {2 \, {\left (16 \, a^{4} x^{4} - 8 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 5 \, a b^{3} x - 35 \, b^{4}\right )} \sqrt {\frac {a x + b}{x}}}{315 \, b^{4} x^{4}} \]

[In]

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

[Out]

2/315*(16*a^4*x^4 - 8*a^3*b*x^3 + 6*a^2*b^2*x^2 - 5*a*b^3*x - 35*b^4)*sqrt((a*x + b)/x)/(b^4*x^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2297 vs. \(2 (68) = 136\).

Time = 1.55 (sec) , antiderivative size = 2297, normalized size of antiderivative = 28.71 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

32*a**(29/2)*b**(23/2)*x**10*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) +
4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(1
1/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 176*a**(27/2)*b**(25/2)*x**9*sqrt(a*x/b + 1)/(315*a**(21
/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x
**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 39
6*a**(25/2)*b**(27/2)*x**8*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4
725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/
2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 462*a**(23/2)*b**(29/2)*x**7*sqrt(a*x/b + 1)/(315*a**(21/2
)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**
(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) + 210*
a**(21/2)*b**(31/2)*x**6*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 472
5*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)
*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 378*a**(19/2)*b**(33/2)*x**5*sqrt(a*x/b + 1)/(315*a**(21/2)*
b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(1
5/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 1134*a
**(17/2)*b**(35/2)*x**4*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725
*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*
b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 1494*a**(15/2)*b**(37/2)*x**3*sqrt(a*x/b + 1)/(315*a**(21/2)*
b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(1
5/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 1098*a
**(13/2)*b**(39/2)*x**2*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725
*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*
b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 430*a**(11/2)*b**(41/2)*x*sqrt(a*x/b + 1)/(315*a**(21/2)*b**1
5*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2)
+ 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 70*a**(9/2
)*b**(43/2)*sqrt(a*x/b + 1)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b
**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11
/2) + 315*a**(9/2)*b**21*x**(9/2)) - 32*a**15*b**11*x**(21/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*
b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(1
3/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 192*a**14*b**12*x**(19/2)/(315*a**(21/2
)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**
(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 480*
a**13*b**13*x**(17/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x
**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) +
315*a**(9/2)*b**21*x**(9/2)) - 640*a**12*b**14*x**(15/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16
*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2)
+ 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 480*a**11*b**15*x**(13/2)/(315*a**(21/2)*b**
15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2
) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2)) - 192*a**10
*b**16*x**(11/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/2) + 4725*a**(17/2)*b**17*x**(17
/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a**(11/2)*b**20*x**(11/2) + 315*a
**(9/2)*b**21*x**(9/2)) - 32*a**9*b**17*x**(9/2)/(315*a**(21/2)*b**15*x**(21/2) + 1890*a**(19/2)*b**16*x**(19/
2) + 4725*a**(17/2)*b**17*x**(17/2) + 6300*a**(15/2)*b**18*x**(15/2) + 4725*a**(13/2)*b**19*x**(13/2) + 1890*a
**(11/2)*b**20*x**(11/2) + 315*a**(9/2)*b**21*x**(9/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}}}{9 \, b^{4}} + \frac {6 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a}{7 \, b^{4}} - \frac {6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3}}{3 \, b^{4}} \]

[In]

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

[Out]

-2/9*(a + b/x)^(9/2)/b^4 + 6/7*(a + b/x)^(7/2)*a/b^4 - 6/5*(a + b/x)^(5/2)*a^2/b^4 + 2/3*(a + b/x)^(3/2)*a^3/b
^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (64) = 128\).

Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx=\frac {2 \, {\left (630 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 1764 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b \mathrm {sgn}\left (x\right ) + 1995 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (x\right ) + 1125 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{3} \mathrm {sgn}\left (x\right ) + 315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{4} \mathrm {sgn}\left (x\right ) + 35 \, b^{5} \mathrm {sgn}\left (x\right )\right )}}{315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{9}} \]

[In]

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

[Out]

2/315*(630*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*sgn(x) + 1764*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b*sgn
(x) + 1995*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^2*sgn(x) + 1125*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^3
*sgn(x) + 315*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^4*sgn(x) + 35*b^5*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*
x))^9

Mupad [B] (verification not implemented)

Time = 6.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^5} \, dx=\frac {32\,a^4\,\sqrt {a+\frac {b}{x}}}{315\,b^4}-\frac {2\,\sqrt {a+\frac {b}{x}}}{9\,x^4}-\frac {2\,a\,\sqrt {a+\frac {b}{x}}}{63\,b\,x^3}+\frac {4\,a^2\,\sqrt {a+\frac {b}{x}}}{105\,b^2\,x^2}-\frac {16\,a^3\,\sqrt {a+\frac {b}{x}}}{315\,b^3\,x} \]

[In]

int((a + b/x)^(1/2)/x^5,x)

[Out]

(32*a^4*(a + b/x)^(1/2))/(315*b^4) - (2*(a + b/x)^(1/2))/(9*x^4) - (2*a*(a + b/x)^(1/2))/(63*b*x^3) + (4*a^2*(
a + b/x)^(1/2))/(105*b^2*x^2) - (16*a^3*(a + b/x)^(1/2))/(315*b^3*x)