\(\int \frac {1}{(a+\frac {b}{x})^{3/2} x^5} \, dx\) [1739]

Optimal result

Integrand size = 15, antiderivative size = 74 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^5} \, dx=-\frac {2 a^3}{b^4 \sqrt {a+\frac {b}{x}}}-\frac {6 a^2 \sqrt {a+\frac {b}{x}}}{b^4}+\frac {2 a \left (a+\frac {b}{x}\right )^{3/2}}{b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4} \]

[Out]

2*a*(a+b/x)^(3/2)/b^4-2/5*(a+b/x)^(5/2)/b^4-2*a^3/b^4/(a+b/x)^(1/2)-6*a^2*(a+b/x)^(1/2)/b^4

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, 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} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^5} \, dx=-\frac {2 a^3}{b^4 \sqrt {a+\frac {b}{x}}}-\frac {6 a^2 \sqrt {a+\frac {b}{x}}}{b^4}+\frac {2 a \left (a+\frac {b}{x}\right )^{3/2}}{b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4} \]

[In]

Int[1/((a + b/x)^(3/2)*x^5),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^5} \, dx=-\frac {2 \sqrt {\frac {b+a x}{x}} \left (b^3-2 a b^2 x+8 a^2 b x^2+16 a^3 x^3\right )}{5 b^4 x^2 (b+a x)} \]

[In]

Integrate[1/((a + b/x)^(3/2)*x^5),x]

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.72

method	result	size
gosper	\(-\frac {2 \left (a x +b \right ) \left (16 a^{3} x^{3}+8 a^{2} b \,x^{2}-2 a \,b^{2} x +b^{3}\right )}{5 x^{4} b^{4} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\)	\(53\)
trager	\(-\frac {2 \left (16 a^{3} x^{3}+8 a^{2} b \,x^{2}-2 a \,b^{2} x +b^{3}\right ) \sqrt {-\frac {-a x -b}{x}}}{5 x^{2} b^{4} \left (a x +b \right )}\)	\(59\)
risch	\(-\frac {2 \left (a x +b \right ) \left (11 a^{2} x^{2}-3 a b x +b^{2}\right )}{5 b^{4} x^{3} \sqrt {\frac {a x +b}{x}}}-\frac {2 a^{3}}{b^{4} \sqrt {\frac {a x +b}{x}}}\)	\(62\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-10 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} x^{6}-5 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b \,x^{6}-10 a^{\frac {11}{2}} \sqrt {x \left (a x +b \right )}\, x^{6}+5 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b \,x^{6}+30 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{4}-20 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b \,x^{5}-10 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{2} x^{5}-10 a^{\frac {9}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} x^{4}-20 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}\, b \,x^{5}+10 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{2} x^{5}+52 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b \,x^{3}-10 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{2} x^{4}-5 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{3} x^{4}-10 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} x^{4}+5 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{3} x^{4}+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2} x^{2}-4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} x +2 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{4}\right )}{5 x^{3} \sqrt {x \left (a x +b \right )}\, b^{5} \sqrt {a}\, \left (a x +b \right )^{2}}\)	\(497\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^5} \, dx=-\frac {2 \, {\left (16 \, a^{3} x^{3} + 8 \, a^{2} b x^{2} - 2 \, a b^{2} x + b^{3}\right )} \sqrt {\frac {a x + b}{x}}}{5 \, {\left (a b^{4} x^{3} + b^{5} x^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2032 vs. \(2 (63) = 126\).

Time = 1.65 (sec) , antiderivative size = 2032, normalized size of antiderivative = 27.46 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^5} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-32*a**(21/2)*b**(23/2)*x**8*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*
a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**
(7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 176*a**(19/2)*b**(25/2)*x**7*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2)
+ 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b
**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 396*a**(17/2)*b**(27/2)*x**6*sqrt(a*
x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(
11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) -
462*a**(15/2)*b**(29/2)*x**5*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75
*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x*
*(7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 290*a**(13/2)*b**(31/2)*x**4*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2
) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*
b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 92*a**(11/2)*b**(33/2)*x**3*sqrt(a*
x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(
11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) -
16*a**(9/2)*b**(35/2)*x**2*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a
**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(
7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 6*a**(7/2)*b**(37/2)*x*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a
**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x*
*(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 2*a**(5/2)*b**(39/2)*sqrt(a*x/b + 1)/(5*a**
(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**
(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 32*a**11*b**11
*x**(17/2)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a*
*(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2))
+ 192*a**10*b**12*x**(15/2)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*
x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/
2)*b**21*x**(5/2)) + 480*a**9*b**13*x**(13/2)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75
*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x*
*(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 640*a**8*b**14*x**(11/2)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**
16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*
a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 480*a**7*b**15*x**(9/2)/(5*a**(17/2)*b**15*x**(17/2) +
30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**1
9*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 192*a**6*b**16*x**(7/2)/(5*a**(17/2)*b*
*15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) +
75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 32*a**5*b**17*x**(5/2)/
(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**
18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^5} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{5 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a}{b^{4}} - \frac {6 \, \sqrt {a + \frac {b}{x}} a^{2}}{b^{4}} - \frac {2 \, a^{3}}{\sqrt {a + \frac {b}{x}} b^{4}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(1/((a + b/x)^(3/2)*x^5), x)

Mupad [B] (verification not implemented)

Time = 5.92 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^5} \, dx=-\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (16\,a^3\,x^3+8\,a^2\,b\,x^2-2\,a\,b^2\,x+b^3\right )}{5\,b^4\,x^2\,\left (b+a\,x\right )} \]

[In]

int(1/(x^5*(a + b/x)^(3/2)),x)

[Out]

-(2*(a + b/x)^(1/2)*(b^3 + 16*a^3*x^3 + 8*a^2*b*x^2 - 2*a*b^2*x))/(5*b^4*x^2*(b + a*x))