Integrand size = 15, antiderivative size = 76 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^5} \, dx=\frac {2 a^3 \sqrt {a+\frac {b}{x}}}{b^4}-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{b^4}+\frac {6 a \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4} \] Output:
2*a^3*(a+b/x)^(1/2)/b^4-2*a^2*(a+b/x)^(3/2)/b^4+6/5*a*(a+b/x)^(5/2)/b^4-2/ 7*(a+b/x)^(7/2)/b^4
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^5} \, dx=\frac {2 \sqrt {\frac {b+a x}{x}} \left (-5 b^3+6 a b^2 x-8 a^2 b x^2+16 a^3 x^3\right )}{35 b^4 x^3} \] Input:
Integrate[1/(Sqrt[a + b/x]*x^5),x]
Output:
(2*Sqrt[(b + a*x)/x]*(-5*b^3 + 6*a*b^2*x - 8*a^2*b*x^2 + 16*a^3*x^3))/(35* b^4*x^3)
Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^5 \sqrt {a+\frac {b}{x}}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {1}{\sqrt {a+\frac {b}{x}} x^3}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (-\frac {a^3}{b^3 \sqrt {a+\frac {b}{x}}}+\frac {3 \sqrt {a+\frac {b}{x}} a^2}{b^3}-\frac {3 \left (a+\frac {b}{x}\right )^{3/2} a}{b^3}+\frac {\left (a+\frac {b}{x}\right )^{5/2}}{b^3}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^3 \sqrt {a+\frac {b}{x}}}{b^4}-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4}+\frac {6 a \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4}\) |
Input:
Int[1/(Sqrt[a + b/x]*x^5),x]
Output:
(2*a^3*Sqrt[a + b/x])/b^4 - (2*a^2*(a + b/x)^(3/2))/b^4 + (6*a*(a + b/x)^( 5/2))/(5*b^4) - (2*(a + b/x)^(7/2))/(7*b^4)
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] && 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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70
method | result | size |
orering | \(\frac {2 \left (16 a^{3} x^{3}-8 a^{2} b \,x^{2}+6 a \,b^{2} x -5 b^{3}\right ) \left (a x +b \right )}{35 b^{4} x^{4} \sqrt {a +\frac {b}{x}}}\) | \(53\) |
trager | \(\frac {2 \left (16 a^{3} x^{3}-8 a^{2} b \,x^{2}+6 a \,b^{2} x -5 b^{3}\right ) \sqrt {-\frac {-a x -b}{x}}}{35 x^{3} b^{4}}\) | \(54\) |
gosper | \(\frac {2 \left (a x +b \right ) \left (16 a^{3} x^{3}-8 a^{2} b \,x^{2}+6 a \,b^{2} x -5 b^{3}\right )}{35 x^{4} b^{4} \sqrt {\frac {a x +b}{x}}}\) | \(55\) |
risch | \(\frac {2 \left (a x +b \right ) \left (16 a^{3} x^{3}-8 a^{2} b \,x^{2}+6 a \,b^{2} x -5 b^{3}\right )}{35 x^{4} b^{4} \sqrt {\frac {a x +b}{x}}}\) | \(55\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (70 \sqrt {x \left (a x +b \right )}\, a^{\frac {9}{2}} x^{5}+70 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} x^{5}+35 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b \,x^{5}-35 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b \,x^{5}-140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{3}+76 a^{\frac {5}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,x^{2}-44 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} x +20 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{3}\right )}{70 x^{4} \sqrt {x \left (a x +b \right )}\, b^{5} \sqrt {a}}\) | \(219\) |
Input:
int(1/(a+b/x)^(1/2)/x^5,x,method=_RETURNVERBOSE)
Output:
2/35*(16*a^3*x^3-8*a^2*b*x^2+6*a*b^2*x-5*b^3)/b^4/x^4*(a*x+b)/(a+b/x)^(1/2 )
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^5} \, dx=\frac {2 \, {\left (16 \, a^{3} x^{3} - 8 \, a^{2} b x^{2} + 6 \, a b^{2} x - 5 \, b^{3}\right )} \sqrt {\frac {a x + b}{x}}}{35 \, b^{4} x^{3}} \] Input:
integrate(1/(a+b/x)^(1/2)/x^5,x, algorithm="fricas")
Output:
2/35*(16*a^3*x^3 - 8*a^2*b*x^2 + 6*a*b^2*x - 5*b^3)*sqrt((a*x + b)/x)/(b^4 *x^3)
Leaf count of result is larger than twice the leaf count of optimal. 2164 vs. \(2 (65) = 130\).
Time = 1.90 (sec) , antiderivative size = 2164, normalized size of antiderivative = 28.47 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b/x)**(1/2)/x**5,x)
Output:
32*a**(25/2)*b**(23/2)*x**9*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**( 13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20 *x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) + 176*a**(23/2)*b**(25/2)*x**8*sqr t(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a** (11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x **(7/2)) + 396*a**(21/2)*b**(27/2)*x**7*sqrt(a*x/b + 1)/(35*a**(19/2)*b**1 5*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2 ) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a* *(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) + 462*a**(19/2)*b**(29 /2)*x**6*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b** 16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/ 2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**( 7/2)*b**21*x**(7/2)) + 280*a**(17/2)*b**(31/2)*x**5*sqrt(a*x/b + 1)/(35*a* *(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b* *17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11 /2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) + 42*a**(1 5/2)*b**(33/2)*x**4*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a* *(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)...
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^5} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{7 \, b^{4}} + \frac {6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a}{5 \, b^{4}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}{b^{4}} + \frac {2 \, \sqrt {a + \frac {b}{x}} a^{3}}{b^{4}} \] Input:
integrate(1/(a+b/x)^(1/2)/x^5,x, algorithm="maxima")
Output:
-2/7*(a + b/x)^(7/2)/b^4 + 6/5*(a + b/x)^(5/2)*a/b^4 - 2*(a + b/x)^(3/2)*a ^2/b^4 + 2*sqrt(a + b/x)*a^3/b^4
Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^5} \, dx=\frac {2 \, {\left (70 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} + 84 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b + 35 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{2} + 5 \, b^{3}\right )}}{35 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7} \mathrm {sgn}\left (x\right )} \] Input:
integrate(1/(a+b/x)^(1/2)/x^5,x, algorithm="giac")
Output:
2/35*(70*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2) + 84*(sqrt(a)*x - sqrt( a*x^2 + b*x))^2*a*b + 35*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^2 + 5*b ^3)/((sqrt(a)*x - sqrt(a*x^2 + b*x))^7*sgn(x))
Time = 0.49 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^5} \, dx=\frac {32\,a^3\,\sqrt {a+\frac {b}{x}}}{35\,b^4}-\frac {2\,\sqrt {a+\frac {b}{x}}}{7\,b\,x^3}+\frac {12\,a\,\sqrt {a+\frac {b}{x}}}{35\,b^2\,x^2}-\frac {16\,a^2\,\sqrt {a+\frac {b}{x}}}{35\,b^3\,x} \] Input:
int(1/(x^5*(a + b/x)^(1/2)),x)
Output:
(32*a^3*(a + b/x)^(1/2))/(35*b^4) - (2*(a + b/x)^(1/2))/(7*b*x^3) + (12*a* (a + b/x)^(1/2))/(35*b^2*x^2) - (16*a^2*(a + b/x)^(1/2))/(35*b^3*x)
Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^5} \, dx=\frac {\frac {32 \sqrt {x}\, \sqrt {a x +b}\, a^{3} x^{3}}{35}-\frac {16 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b \,x^{2}}{35}+\frac {12 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{2} x}{35}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, b^{3}}{7}-\frac {32 \sqrt {a}\, a^{3} x^{4}}{35}}{b^{4} x^{4}} \] Input:
int(1/(a+b/x)^(1/2)/x^5,x)
Output:
(2*(16*sqrt(x)*sqrt(a*x + b)*a**3*x**3 - 8*sqrt(x)*sqrt(a*x + b)*a**2*b*x* *2 + 6*sqrt(x)*sqrt(a*x + b)*a*b**2*x - 5*sqrt(x)*sqrt(a*x + b)*b**3 - 16* sqrt(a)*a**3*x**4))/(35*b**4*x**4)