Integrand size = 15, antiderivative size = 101 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^6} \, dx=-\frac {2 a^4 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5}-\frac {12 a^2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{13/2}}{13 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{15/2}}{15 b^5} \]
-2/7*a^4*(a+b/x)^(7/2)/b^5+8/9*a^3*(a+b/x)^(9/2)/b^5-12/11*a^2*(a+b/x)^(11 /2)/b^5+8/13*a*(a+b/x)^(13/2)/b^5-2/15*(a+b/x)^(15/2)/b^5
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^6} \, dx=-\frac {2 (b+a x)^3 \sqrt {\frac {b+a x}{x}} \left (3003 b^4-1848 a b^3 x+1008 a^2 b^2 x^2-448 a^3 b x^3+128 a^4 x^4\right )}{45045 b^5 x^7} \]
(-2*(b + a*x)^3*Sqrt[(b + a*x)/x]*(3003*b^4 - 1848*a*b^3*x + 1008*a^2*b^2* x^2 - 448*a^3*b*x^3 + 128*a^4*x^4))/(45045*b^5*x^7)
Time = 0.21 (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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^6} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {\left (a+\frac {b}{x}\right )^{13/2}}{b^4}-\frac {4 a \left (a+\frac {b}{x}\right )^{11/2}}{b^4}+\frac {6 a^2 \left (a+\frac {b}{x}\right )^{9/2}}{b^4}-\frac {4 a^3 \left (a+\frac {b}{x}\right )^{7/2}}{b^4}+\frac {a^4 \left (a+\frac {b}{x}\right )^{5/2}}{b^4}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^4 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^5}-\frac {12 a^2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{15/2}}{15 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{13/2}}{13 b^5}\) |
(-2*a^4*(a + b/x)^(7/2))/(7*b^5) + (8*a^3*(a + b/x)^(9/2))/(9*b^5) - (12*a ^2*(a + b/x)^(11/2))/(11*b^5) + (8*a*(a + b/x)^(13/2))/(13*b^5) - (2*(a + b/x)^(15/2))/(15*b^5)
3.18.21.3.1 Defintions of rubi rules used
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 = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(-\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}-448 a^{3} b \,x^{3}+1008 a^{2} b^{2} x^{2}-1848 a \,b^{3} x +3003 b^{4}\right ) \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}{45045 b^{5} x^{5}}\) | \(66\) |
risch | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (128 a^{7} x^{7}-64 a^{6} b \,x^{6}+48 a^{5} b^{2} x^{5}-40 a^{4} b^{3} x^{4}+35 a^{3} b^{4} x^{3}+4473 b^{5} x^{2} a^{2}+7161 b^{6} x a +3003 b^{7}\right )}{45045 x^{7} b^{5}}\) | \(94\) |
trager | \(-\frac {2 \left (128 a^{7} x^{7}-64 a^{6} b \,x^{6}+48 a^{5} b^{2} x^{5}-40 a^{4} b^{3} x^{4}+35 a^{3} b^{4} x^{3}+4473 b^{5} x^{2} a^{2}+7161 b^{6} x a +3003 b^{7}\right ) \sqrt {-\frac {-a x -b}{x}}}{45045 x^{7} b^{5}}\) | \(98\) |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (128 a^{6} x^{6}-192 a^{5} b \,x^{5}+240 a^{4} b^{2} x^{4}-280 a^{3} x^{3} b^{3}+315 b^{4} x^{2} a^{2}+4158 a x \,b^{5}+3003 b^{6}\right )}{45045 x^{8} b^{5} \sqrt {x \left (a x +b \right )}}\) | \(103\) |
-2/45045*(a*x+b)*(128*a^4*x^4-448*a^3*b*x^3+1008*a^2*b^2*x^2-1848*a*b^3*x+ 3003*b^4)*((a*x+b)/x)^(5/2)/b^5/x^5
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^6} \, dx=-\frac {2 \, {\left (128 \, a^{7} x^{7} - 64 \, a^{6} b x^{6} + 48 \, a^{5} b^{2} x^{5} - 40 \, a^{4} b^{3} x^{4} + 35 \, a^{3} b^{4} x^{3} + 4473 \, a^{2} b^{5} x^{2} + 7161 \, a b^{6} x + 3003 \, b^{7}\right )} \sqrt {\frac {a x + b}{x}}}{45045 \, b^{5} x^{7}} \]
-2/45045*(128*a^7*x^7 - 64*a^6*b*x^6 + 48*a^5*b^2*x^5 - 40*a^4*b^3*x^4 + 3 5*a^3*b^4*x^3 + 4473*a^2*b^5*x^2 + 7161*a*b^6*x + 3003*b^7)*sqrt((a*x + b) /x)/(b^5*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 5482 vs. \(2 (87) = 174\).
Time = 2.81 (sec) , antiderivative size = 5482, normalized size of antiderivative = 54.28 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^6} \, dx=\text {Too large to display} \]
-256*a**(49/2)*b**(49/2)*x**17*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**( 35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/ 2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 2432 *a**(47/2)*b**(51/2)*x**16*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2 ) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 1 1351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 54 05400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 4504 50*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 10336*a* *(45/2)*b**(53/2)*x**15*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 54 05400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 1135 1340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 54054 00*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450* a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 25840*a**(4 3/2)*b**(55/2)*x**14*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 45 0450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 54...
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^6} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {15}{2}}}{15 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{2}} a}{13 \, b^{5}} - \frac {12 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} a^{2}}{11 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a^{3}}{9 \, b^{5}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{4}}{7 \, b^{5}} \]
-2/15*(a + b/x)^(15/2)/b^5 + 8/13*(a + b/x)^(13/2)*a/b^5 - 12/11*(a + b/x) ^(11/2)*a^2/b^5 + 8/9*(a + b/x)^(9/2)*a^3/b^5 - 2/7*(a + b/x)^(7/2)*a^4/b^ 5
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (81) = 162\).
Time = 0.31 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.29 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^6} \, dx=\frac {2 \, {\left (144144 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{10} a^{5} \mathrm {sgn}\left (x\right ) + 960960 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{9} a^{\frac {9}{2}} b \mathrm {sgn}\left (x\right ) + 2934360 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{8} a^{4} b^{2} \mathrm {sgn}\left (x\right ) + 5360355 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7} a^{\frac {7}{2}} b^{3} \mathrm {sgn}\left (x\right ) + 6451445 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} b^{4} \mathrm {sgn}\left (x\right ) + 5324319 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b^{5} \mathrm {sgn}\left (x\right ) + 3042585 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{6} \mathrm {sgn}\left (x\right ) + 1186185 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{7} \mathrm {sgn}\left (x\right ) + 301455 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{8} \mathrm {sgn}\left (x\right ) + 45045 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{9} \mathrm {sgn}\left (x\right ) + 3003 \, b^{10} \mathrm {sgn}\left (x\right )\right )}}{45045 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{15}} \]
2/45045*(144144*(sqrt(a)*x - sqrt(a*x^2 + b*x))^10*a^5*sgn(x) + 960960*(sq rt(a)*x - sqrt(a*x^2 + b*x))^9*a^(9/2)*b*sgn(x) + 2934360*(sqrt(a)*x - sqr t(a*x^2 + b*x))^8*a^4*b^2*sgn(x) + 5360355*(sqrt(a)*x - sqrt(a*x^2 + b*x)) ^7*a^(7/2)*b^3*sgn(x) + 6451445*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*b^4* sgn(x) + 5324319*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b^5*sgn(x) + 30 42585*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^6*sgn(x) + 1186185*(sqrt(a)* x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^7*sgn(x) + 301455*(sqrt(a)*x - sqrt(a*x ^2 + b*x))^2*a*b^8*sgn(x) + 45045*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)* b^9*sgn(x) + 3003*b^10*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^15
Time = 7.84 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^6} \, dx=\frac {16\,a^4\,\sqrt {a+\frac {b}{x}}}{9009\,b^2\,x^3}-\frac {142\,a^2\,\sqrt {a+\frac {b}{x}}}{715\,x^5}-\frac {2\,b^2\,\sqrt {a+\frac {b}{x}}}{15\,x^7}-\frac {2\,a^3\,\sqrt {a+\frac {b}{x}}}{1287\,b\,x^4}-\frac {256\,a^7\,\sqrt {a+\frac {b}{x}}}{45045\,b^5}-\frac {32\,a^5\,\sqrt {a+\frac {b}{x}}}{15015\,b^3\,x^2}+\frac {128\,a^6\,\sqrt {a+\frac {b}{x}}}{45045\,b^4\,x}-\frac {62\,a\,b\,\sqrt {a+\frac {b}{x}}}{195\,x^6} \]
(16*a^4*(a + b/x)^(1/2))/(9009*b^2*x^3) - (142*a^2*(a + b/x)^(1/2))/(715*x ^5) - (2*b^2*(a + b/x)^(1/2))/(15*x^7) - (2*a^3*(a + b/x)^(1/2))/(1287*b*x ^4) - (256*a^7*(a + b/x)^(1/2))/(45045*b^5) - (32*a^5*(a + b/x)^(1/2))/(15 015*b^3*x^2) + (128*a^6*(a + b/x)^(1/2))/(45045*b^4*x) - (62*a*b*(a + b/x) ^(1/2))/(195*x^6)