Integrand size = 15, antiderivative size = 99 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 a^4 \sqrt {a+\frac {b}{x}}}{b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^5}-\frac {12 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5} \] Output:
-2*a^4*(a+b/x)^(1/2)/b^5+8/3*a^3*(a+b/x)^(3/2)/b^5-12/5*a^2*(a+b/x)^(5/2)/ b^5+8/7*a*(a+b/x)^(7/2)/b^5-2/9*(a+b/x)^(9/2)/b^5
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 \sqrt {\frac {b+a x}{x}} \left (35 b^4-40 a b^3 x+48 a^2 b^2 x^2-64 a^3 b x^3+128 a^4 x^4\right )}{315 b^5 x^4} \] Input:
Integrate[1/(Sqrt[a + b/x]*x^6),x]
Output:
(-2*Sqrt[(b + a*x)/x]*(35*b^4 - 40*a*b^3*x + 48*a^2*b^2*x^2 - 64*a^3*b*x^3 + 128*a^4*x^4))/(315*b^5*x^4)
Time = 0.35 (sec) , antiderivative size = 99, 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.
\(\displaystyle \int \frac {1}{x^6 \sqrt {a+\frac {b}{x}}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {1}{\sqrt {a+\frac {b}{x}} x^4}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {a^4}{b^4 \sqrt {a+\frac {b}{x}}}-\frac {4 \sqrt {a+\frac {b}{x}} a^3}{b^4}+\frac {6 \left (a+\frac {b}{x}\right )^{3/2} a^2}{b^4}-\frac {4 \left (a+\frac {b}{x}\right )^{5/2} a}{b^4}+\frac {\left (a+\frac {b}{x}\right )^{7/2}}{b^4}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^4 \sqrt {a+\frac {b}{x}}}{b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^5}-\frac {12 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}\) |
Input:
Int[1/(Sqrt[a + b/x]*x^6),x]
Output:
(-2*a^4*Sqrt[a + b/x])/b^5 + (8*a^3*(a + b/x)^(3/2))/(3*b^5) - (12*a^2*(a + b/x)^(5/2))/(5*b^5) + (8*a*(a + b/x)^(7/2))/(7*b^5) - (2*(a + b/x)^(9/2) )/(9*b^5)
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 = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65
method | result | size |
orering | \(-\frac {2 \left (128 a^{4} x^{4}-64 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-40 a \,b^{3} x +35 b^{4}\right ) \left (a x +b \right )}{315 b^{5} x^{5} \sqrt {a +\frac {b}{x}}}\) | \(64\) |
trager | \(-\frac {2 \left (128 a^{4} x^{4}-64 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-40 a \,b^{3} x +35 b^{4}\right ) \sqrt {-\frac {-a x -b}{x}}}{315 x^{4} b^{5}}\) | \(65\) |
gosper | \(-\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}-64 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-40 a \,b^{3} x +35 b^{4}\right )}{315 x^{5} b^{5} \sqrt {\frac {a x +b}{x}}}\) | \(66\) |
risch | \(-\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}-64 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-40 a \,b^{3} x +35 b^{4}\right )}{315 x^{5} b^{5} \sqrt {\frac {a x +b}{x}}}\) | \(66\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, \left (630 \sqrt {x \left (a x +b \right )}\, a^{\frac {11}{2}} x^{6}+630 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} x^{6}+315 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b \,x^{6}-315 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b \,x^{6}-1260 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{4}+748 a^{\frac {7}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,x^{3}-492 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2} x^{2}+300 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} x -140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{4}\right )}{630 x^{5} \sqrt {x \left (a x +b \right )}\, b^{6} \sqrt {a}}\) | \(241\) |
Input:
int(1/(a+b/x)^(1/2)/x^6,x,method=_RETURNVERBOSE)
Output:
-2/315*(128*a^4*x^4-64*a^3*b*x^3+48*a^2*b^2*x^2-40*a*b^3*x+35*b^4)/b^5/x^5 *(a*x+b)/(a+b/x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 \, {\left (128 \, a^{4} x^{4} - 64 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 40 \, a b^{3} x + 35 \, b^{4}\right )} \sqrt {\frac {a x + b}{x}}}{315 \, b^{5} x^{4}} \] Input:
integrate(1/(a+b/x)^(1/2)/x^6,x, algorithm="fricas")
Output:
-2/315*(128*a^4*x^4 - 64*a^3*b*x^3 + 48*a^2*b^2*x^2 - 40*a*b^3*x + 35*b^4) *sqrt((a*x + b)/x)/(b^5*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 4901 vs. \(2 (85) = 170\).
Time = 3.20 (sec) , antiderivative size = 4901, normalized size of antiderivative = 49.51 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b/x)**(1/2)/x**6,x)
Output:
-256*a**(37/2)*b**(49/2)*x**14*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29 /2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 3 7800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a **(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/ 2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**3 8*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 2432*a**(35/2)*b**(51/2)*x**1 3*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x* *(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2 ) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66 150*a**(17/2)*b**35*x**(17/2) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a* *(13/2)*b**37*x**(13/2) + 3150*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b* *39*x**(9/2)) - 10336*a**(33/2)*b**(53/2)*x**12*sqrt(a*x/b + 1)/(315*a**(2 9/2)*b**29*x**(29/2) + 3150*a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b* *31*x**(25/2) + 37800*a**(23/2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x* *(21/2) + 79380*a**(19/2)*b**34*x**(19/2) + 66150*a**(17/2)*b**35*x**(17/2 ) + 37800*a**(15/2)*b**36*x**(15/2) + 14175*a**(13/2)*b**37*x**(13/2) + 31 50*a**(11/2)*b**38*x**(11/2) + 315*a**(9/2)*b**39*x**(9/2)) - 25840*a**(31 /2)*b**(55/2)*x**11*sqrt(a*x/b + 1)/(315*a**(29/2)*b**29*x**(29/2) + 3150* a**(27/2)*b**30*x**(27/2) + 14175*a**(25/2)*b**31*x**(25/2) + 37800*a**(23 /2)*b**32*x**(23/2) + 66150*a**(21/2)*b**33*x**(21/2) + 79380*a**(19/2)...
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}}}{9 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a}{7 \, b^{5}} - \frac {12 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3}}{3 \, b^{5}} - \frac {2 \, \sqrt {a + \frac {b}{x}} a^{4}}{b^{5}} \] Input:
integrate(1/(a+b/x)^(1/2)/x^6,x, algorithm="maxima")
Output:
-2/9*(a + b/x)^(9/2)/b^5 + 8/7*(a + b/x)^(7/2)*a/b^5 - 12/5*(a + b/x)^(5/2 )*a^2/b^5 + 8/3*(a + b/x)^(3/2)*a^3/b^5 - 2*sqrt(a + b/x)*a^4/b^5
Time = 0.14 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=\frac {2 \, {\left (1008 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} + 1680 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b + 1080 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} + 315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} + 35 \, b^{4}\right )}}{315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{9} \mathrm {sgn}\left (x\right )} \] Input:
integrate(1/(a+b/x)^(1/2)/x^6,x, algorithm="giac")
Output:
2/315*(1008*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2 + 1680*(sqrt(a)*x - sqrt (a*x^2 + b*x))^3*a^(3/2)*b + 1080*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2 + 315*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3 + 35*b^4)/((sqrt(a)*x - sqrt(a*x^2 + b*x))^9*sgn(x))
Time = 0.51 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=\frac {16\,a\,\sqrt {a+\frac {b}{x}}}{63\,b^2\,x^3}-\frac {2\,\sqrt {a+\frac {b}{x}}}{9\,b\,x^4}-\frac {256\,a^4\,\sqrt {a+\frac {b}{x}}}{315\,b^5}-\frac {32\,a^2\,\sqrt {a+\frac {b}{x}}}{105\,b^3\,x^2}+\frac {128\,a^3\,\sqrt {a+\frac {b}{x}}}{315\,b^4\,x} \] Input:
int(1/(x^6*(a + b/x)^(1/2)),x)
Output:
(16*a*(a + b/x)^(1/2))/(63*b^2*x^3) - (2*(a + b/x)^(1/2))/(9*b*x^4) - (256 *a^4*(a + b/x)^(1/2))/(315*b^5) - (32*a^2*(a + b/x)^(1/2))/(105*b^3*x^2) + (128*a^3*(a + b/x)^(1/2))/(315*b^4*x)
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^6} \, dx=\frac {-\frac {256 \sqrt {x}\, \sqrt {a x +b}\, a^{4} x^{4}}{315}+\frac {128 \sqrt {x}\, \sqrt {a x +b}\, a^{3} b \,x^{3}}{315}-\frac {32 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b^{2} x^{2}}{105}+\frac {16 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{3} x}{63}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, b^{4}}{9}+\frac {256 \sqrt {a}\, a^{4} x^{5}}{315}}{b^{5} x^{5}} \] Input:
int(1/(a+b/x)^(1/2)/x^6,x)
Output:
(2*( - 128*sqrt(x)*sqrt(a*x + b)*a**4*x**4 + 64*sqrt(x)*sqrt(a*x + b)*a**3 *b*x**3 - 48*sqrt(x)*sqrt(a*x + b)*a**2*b**2*x**2 + 40*sqrt(x)*sqrt(a*x + b)*a*b**3*x - 35*sqrt(x)*sqrt(a*x + b)*b**4 + 128*sqrt(a)*a**4*x**5))/(315 *b**5*x**5)