Integrand size = 15, antiderivative size = 101 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx=-\frac {2 a^4 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}-\frac {12 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^5} \] Output:
-2/3*a^4*(a+b/x)^(3/2)/b^5+8/5*a^3*(a+b/x)^(5/2)/b^5-12/7*a^2*(a+b/x)^(7/2 )/b^5+8/9*a*(a+b/x)^(9/2)/b^5-2/11*(a+b/x)^(11/2)/b^5
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx=-\frac {2 \sqrt {\frac {b+a x}{x}} \left (315 b^5+35 a b^4 x-40 a^2 b^3 x^2+48 a^3 b^2 x^3-64 a^4 b x^4+128 a^5 x^5\right )}{3465 b^5 x^5} \] Input:
Integrate[Sqrt[a + b/x]/x^6,x]
Output:
(-2*Sqrt[(b + a*x)/x]*(315*b^5 + 35*a*b^4*x - 40*a^2*b^3*x^2 + 48*a^3*b^2* x^3 - 64*a^4*b*x^4 + 128*a^5*x^5))/(3465*b^5*x^5)
Time = 0.35 (sec) , antiderivative size = 101, 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.
\(\displaystyle \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {\sqrt {a+\frac {b}{x}}}{x^4}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {\left (a+\frac {b}{x}\right )^{9/2}}{b^4}-\frac {4 a \left (a+\frac {b}{x}\right )^{7/2}}{b^4}+\frac {6 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{b^4}-\frac {4 a^3 \left (a+\frac {b}{x}\right )^{3/2}}{b^4}+\frac {a^4 \sqrt {a+\frac {b}{x}}}{b^4}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^4 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}-\frac {12 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5}\) |
Input:
Int[Sqrt[a + b/x]/x^6,x]
Output:
(-2*a^4*(a + b/x)^(3/2))/(3*b^5) + (8*a^3*(a + b/x)^(5/2))/(5*b^5) - (12*a ^2*(a + b/x)^(7/2))/(7*b^5) + (8*a*(a + b/x)^(9/2))/(9*b^5) - (2*(a + b/x) ^(11/2))/(11*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 = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.63
method | result | size |
orering | \(-\frac {2 \left (128 a^{4} x^{4}-192 a^{3} b \,x^{3}+240 a^{2} b^{2} x^{2}-280 a \,b^{3} x +315 b^{4}\right ) \left (a x +b \right ) \sqrt {a +\frac {b}{x}}}{3465 b^{5} x^{5}}\) | \(64\) |
gosper | \(-\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}-192 a^{3} b \,x^{3}+240 a^{2} b^{2} x^{2}-280 a \,b^{3} x +315 b^{4}\right ) \sqrt {\frac {a x +b}{x}}}{3465 b^{5} x^{5}}\) | \(66\) |
risch | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (128 a^{5} x^{5}-64 a^{4} b \,x^{4}+48 a^{3} b^{2} x^{3}-40 a^{2} b^{3} x^{2}+35 b^{4} x a +315 b^{5}\right )}{3465 x^{5} b^{5}}\) | \(72\) |
trager | \(-\frac {2 \left (128 a^{5} x^{5}-64 a^{4} b \,x^{4}+48 a^{3} b^{2} x^{3}-40 a^{2} b^{3} x^{2}+35 b^{4} x a +315 b^{5}\right ) \sqrt {-\frac {-a x -b}{x}}}{3465 x^{5} b^{5}}\) | \(76\) |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (128 a^{4} x^{4}-192 a^{3} b \,x^{3}+240 a^{2} b^{2} x^{2}-280 a \,b^{3} x +315 b^{4}\right )}{3465 x^{6} \sqrt {x \left (a x +b \right )}\, b^{5}}\) | \(81\) |
Input:
int((a+b/x)^(1/2)/x^6,x,method=_RETURNVERBOSE)
Output:
-2/3465*(128*a^4*x^4-192*a^3*b*x^3+240*a^2*b^2*x^2-280*a*b^3*x+315*b^4)/b^ 5/x^5*(a*x+b)*(a+b/x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx=-\frac {2 \, {\left (128 \, a^{5} x^{5} - 64 \, a^{4} b x^{4} + 48 \, a^{3} b^{2} x^{3} - 40 \, a^{2} b^{3} x^{2} + 35 \, a b^{4} x + 315 \, b^{5}\right )} \sqrt {\frac {a x + b}{x}}}{3465 \, b^{5} x^{5}} \] Input:
integrate((a+b/x)^(1/2)/x^6,x, algorithm="fricas")
Output:
-2/3465*(128*a^5*x^5 - 64*a^4*b*x^4 + 48*a^3*b^2*x^3 - 40*a^2*b^3*x^2 + 35 *a*b^4*x + 315*b^5)*sqrt((a*x + b)/x)/(b^5*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 5095 vs. \(2 (87) = 174\).
Time = 3.20 (sec) , antiderivative size = 5095, normalized size of antiderivative = 50.45 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx=\text {Too large to display} \] Input:
integrate((a+b/x)**(1/2)/x**6,x)
Output:
-256*a**(41/2)*b**(49/2)*x**15*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(3 1/2) + 34650*a**(29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 87 3180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800 *a**(17/2)*b**36*x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**( 13/2)*b**38*x**(13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 2432*a**(39/2)*b **(51/2)*x**14*sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a** (29/2)*b**30*x**(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/ 2)*b**32*x**(25/2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b **34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36 *x**(17/2) + 155925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**( 13/2) + 3465*a**(11/2)*b**39*x**(11/2)) - 10336*a**(37/2)*b**(53/2)*x**13* sqrt(a*x/b + 1)/(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x* *(29/2) + 155925*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25 /2) + 727650*a**(23/2)*b**33*x**(23/2) + 873180*a**(21/2)*b**34*x**(21/2) + 727650*a**(19/2)*b**35*x**(19/2) + 415800*a**(17/2)*b**36*x**(17/2) + 15 5925*a**(15/2)*b**37*x**(15/2) + 34650*a**(13/2)*b**38*x**(13/2) + 3465*a* *(11/2)*b**39*x**(11/2)) - 25840*a**(35/2)*b**(55/2)*x**12*sqrt(a*x/b + 1) /(3465*a**(31/2)*b**29*x**(31/2) + 34650*a**(29/2)*b**30*x**(29/2) + 15592 5*a**(27/2)*b**31*x**(27/2) + 415800*a**(25/2)*b**32*x**(25/2) + 727650...
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}}}{11 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a}{9 \, b^{5}} - \frac {12 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3}}{5 \, b^{5}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4}}{3 \, b^{5}} \] Input:
integrate((a+b/x)^(1/2)/x^6,x, algorithm="maxima")
Output:
-2/11*(a + b/x)^(11/2)/b^5 + 8/9*(a + b/x)^(9/2)*a/b^5 - 12/7*(a + b/x)^(7 /2)*a^2/b^5 + 8/5*(a + b/x)^(5/2)*a^3/b^5 - 2/3*(a + b/x)^(3/2)*a^4/b^5
Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (81) = 162\).
Time = 0.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx=\frac {2 \, {\left (11088 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} \mathrm {sgn}\left (x\right ) + 36960 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b \mathrm {sgn}\left (x\right ) + 51480 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{2} \mathrm {sgn}\left (x\right ) + 38115 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{3} \mathrm {sgn}\left (x\right ) + 15785 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{4} \mathrm {sgn}\left (x\right ) + 3465 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{5} \mathrm {sgn}\left (x\right ) + 315 \, b^{6} \mathrm {sgn}\left (x\right )\right )}}{3465 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{11}} \] Input:
integrate((a+b/x)^(1/2)/x^6,x, algorithm="giac")
Output:
2/3465*(11088*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*sgn(x) + 36960*(sqrt(a )*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b*sgn(x) + 51480*(sqrt(a)*x - sqrt(a*x^ 2 + b*x))^4*a^2*b^2*sgn(x) + 38115*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/ 2)*b^3*sgn(x) + 15785*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^4*sgn(x) + 346 5*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^5*sgn(x) + 315*b^6*sgn(x))/(sq rt(a)*x - sqrt(a*x^2 + b*x))^11
Time = 0.99 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx=\frac {16\,a^2\,\sqrt {a+\frac {b}{x}}}{693\,b^2\,x^3}-\frac {256\,a^5\,\sqrt {a+\frac {b}{x}}}{3465\,b^5}-\frac {2\,a\,\sqrt {a+\frac {b}{x}}}{99\,b\,x^4}-\frac {2\,\sqrt {a+\frac {b}{x}}}{11\,x^5}-\frac {32\,a^3\,\sqrt {a+\frac {b}{x}}}{1155\,b^3\,x^2}+\frac {128\,a^4\,\sqrt {a+\frac {b}{x}}}{3465\,b^4\,x} \] Input:
int((a + b/x)^(1/2)/x^6,x)
Output:
(16*a^2*(a + b/x)^(1/2))/(693*b^2*x^3) - (256*a^5*(a + b/x)^(1/2))/(3465*b ^5) - (2*a*(a + b/x)^(1/2))/(99*b*x^4) - (2*(a + b/x)^(1/2))/(11*x^5) - (3 2*a^3*(a + b/x)^(1/2))/(1155*b^3*x^2) + (128*a^4*(a + b/x)^(1/2))/(3465*b^ 4*x)
Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^6} \, dx=\frac {-\frac {256 \sqrt {x}\, \sqrt {a x +b}\, a^{5} x^{5}}{3465}+\frac {128 \sqrt {x}\, \sqrt {a x +b}\, a^{4} b \,x^{4}}{3465}-\frac {32 \sqrt {x}\, \sqrt {a x +b}\, a^{3} b^{2} x^{3}}{1155}+\frac {16 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b^{3} x^{2}}{693}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{4} x}{99}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, b^{5}}{11}+\frac {256 \sqrt {a}\, a^{5} x^{6}}{3465}}{b^{5} x^{6}} \] Input:
int((a+b/x)^(1/2)/x^6,x)
Output:
(2*( - 128*sqrt(x)*sqrt(a*x + b)*a**5*x**5 + 64*sqrt(x)*sqrt(a*x + b)*a**4 *b*x**4 - 48*sqrt(x)*sqrt(a*x + b)*a**3*b**2*x**3 + 40*sqrt(x)*sqrt(a*x + b)*a**2*b**3*x**2 - 35*sqrt(x)*sqrt(a*x + b)*a*b**4*x - 315*sqrt(x)*sqrt(a *x + b)*b**5 + 128*sqrt(a)*a**5*x**6))/(3465*b**5*x**6)