Integrand size = 22, antiderivative size = 134 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^9} \, dx=-\frac {c^3 (b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^5}+\frac {c^2 (4 b c-3 a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^5}-\frac {3 c (2 b c-a d) \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^5}+\frac {(4 b c-a d) \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^5}-\frac {b \left (c+\frac {d}{x^2}\right )^{11/2}}{11 d^5} \]
-1/3*c^3*(-a*d+b*c)*(c+d/x^2)^(3/2)/d^5+1/5*c^2*(-3*a*d+4*b*c)*(c+d/x^2)^( 5/2)/d^5-3/7*c*(-a*d+2*b*c)*(c+d/x^2)^(7/2)/d^5+1/9*(-a*d+4*b*c)*(c+d/x^2) ^(9/2)/d^5-1/11*b*(c+d/x^2)^(11/2)/d^5
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^9} \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (d+c x^2\right ) \left (11 a d x^2 \left (-35 d^3+30 c d^2 x^2-24 c^2 d x^4+16 c^3 x^6\right )+b \left (-315 d^4+280 c d^3 x^2-240 c^2 d^2 x^4+192 c^3 d x^6-128 c^4 x^8\right )\right )}{3465 d^5 x^{10}} \]
(Sqrt[c + d/x^2]*(d + c*x^2)*(11*a*d*x^2*(-35*d^3 + 30*c*d^2*x^2 - 24*c^2* d*x^4 + 16*c^3*x^6) + b*(-315*d^4 + 280*c*d^3*x^2 - 240*c^2*d^2*x^4 + 192* c^3*d*x^6 - 128*c^4*x^8)))/(3465*d^5*x^10)
Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {948, 86, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^9} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle -\frac {1}{2} \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^6}d\frac {1}{x^2}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {b \left (c+\frac {d}{x^2}\right )^{9/2}}{d^4}+\frac {(a d-4 b c) \left (c+\frac {d}{x^2}\right )^{7/2}}{d^4}+\frac {3 c (2 b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{d^4}-\frac {c^2 (4 b c-3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{d^4}+\frac {c^3 (b c-a d) \sqrt {c+\frac {d}{x^2}}}{d^4}\right )d\frac {1}{x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 c^3 \left (c+\frac {d}{x^2}\right )^{3/2} (b c-a d)}{3 d^5}+\frac {2 c^2 \left (c+\frac {d}{x^2}\right )^{5/2} (4 b c-3 a d)}{5 d^5}+\frac {2 \left (c+\frac {d}{x^2}\right )^{9/2} (4 b c-a d)}{9 d^5}-\frac {6 c \left (c+\frac {d}{x^2}\right )^{7/2} (2 b c-a d)}{7 d^5}-\frac {2 b \left (c+\frac {d}{x^2}\right )^{11/2}}{11 d^5}\right )\) |
((-2*c^3*(b*c - a*d)*(c + d/x^2)^(3/2))/(3*d^5) + (2*c^2*(4*b*c - 3*a*d)*( c + d/x^2)^(5/2))/(5*d^5) - (6*c*(2*b*c - a*d)*(c + d/x^2)^(7/2))/(7*d^5) + (2*(4*b*c - a*d)*(c + d/x^2)^(9/2))/(9*d^5) - (2*b*(c + d/x^2)^(11/2))/( 11*d^5))/2
3.10.36.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88
| method | result | size |
| gosper | \(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (176 a \,c^{3} d \,x^{8}-128 b \,c^{4} x^{8}-264 a \,c^{2} d^{2} x^{6}+192 b \,c^{3} d \,x^{6}+330 a c \,d^{3} x^{4}-240 b \,c^{2} d^{2} x^{4}-385 a \,d^{4} x^{2}+280 b c \,d^{3} x^{2}-315 b \,d^{4}\right ) \left (c \,x^{2}+d \right )}{3465 d^{5} x^{10}}\) | \(118\) |
| default | \(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (176 a \,c^{3} d \,x^{8}-128 b \,c^{4} x^{8}-264 a \,c^{2} d^{2} x^{6}+192 b \,c^{3} d \,x^{6}+330 a c \,d^{3} x^{4}-240 b \,c^{2} d^{2} x^{4}-385 a \,d^{4} x^{2}+280 b c \,d^{3} x^{2}-315 b \,d^{4}\right ) \left (c \,x^{2}+d \right )}{3465 d^{5} x^{10}}\) | \(118\) |
| risch | \(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (176 a \,c^{4} d \,x^{10}-128 b \,c^{5} x^{10}-88 a \,c^{3} d^{2} x^{8}+64 b \,c^{4} d \,x^{8}+66 a \,c^{2} d^{3} x^{6}-48 b \,c^{3} d^{2} x^{6}-55 a c \,d^{4} x^{4}+40 b \,c^{2} d^{3} x^{4}-385 a \,d^{5} x^{2}-35 b c \,d^{4} x^{2}-315 b \,d^{5}\right )}{3465 x^{10} d^{5}}\) | \(135\) |
| trager | \(\frac {\left (176 a \,c^{4} d \,x^{10}-128 b \,c^{5} x^{10}-88 a \,c^{3} d^{2} x^{8}+64 b \,c^{4} d \,x^{8}+66 a \,c^{2} d^{3} x^{6}-48 b \,c^{3} d^{2} x^{6}-55 a c \,d^{4} x^{4}+40 b \,c^{2} d^{3} x^{4}-385 a \,d^{5} x^{2}-35 b c \,d^{4} x^{2}-315 b \,d^{5}\right ) \sqrt {-\frac {-c \,x^{2}-d}{x^{2}}}}{3465 x^{10} d^{5}}\) | \(139\) |
1/3465*((c*x^2+d)/x^2)^(1/2)*(176*a*c^3*d*x^8-128*b*c^4*x^8-264*a*c^2*d^2* x^6+192*b*c^3*d*x^6+330*a*c*d^3*x^4-240*b*c^2*d^2*x^4-385*a*d^4*x^2+280*b* c*d^3*x^2-315*b*d^4)*(c*x^2+d)/d^5/x^10
Time = 0.36 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^9} \, dx=-\frac {{\left (16 \, {\left (8 \, b c^{5} - 11 \, a c^{4} d\right )} x^{10} - 8 \, {\left (8 \, b c^{4} d - 11 \, a c^{3} d^{2}\right )} x^{8} + 6 \, {\left (8 \, b c^{3} d^{2} - 11 \, a c^{2} d^{3}\right )} x^{6} + 315 \, b d^{5} - 5 \, {\left (8 \, b c^{2} d^{3} - 11 \, a c d^{4}\right )} x^{4} + 35 \, {\left (b c d^{4} + 11 \, a d^{5}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3465 \, d^{5} x^{10}} \]
-1/3465*(16*(8*b*c^5 - 11*a*c^4*d)*x^10 - 8*(8*b*c^4*d - 11*a*c^3*d^2)*x^8 + 6*(8*b*c^3*d^2 - 11*a*c^2*d^3)*x^6 + 315*b*d^5 - 5*(8*b*c^2*d^3 - 11*a* c*d^4)*x^4 + 35*(b*c*d^4 + 11*a*d^5)*x^2)*sqrt((c*x^2 + d)/x^2)/(d^5*x^10)
Time = 1.38 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^9} \, dx=- \frac {a \left (\begin {cases} \frac {2 \left (- \frac {c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{4 x^{8}} & \text {otherwise} \end {cases}\right )}{2} - \frac {b \left (\begin {cases} \frac {2 \left (\frac {c^{4} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {11}{2}}}{11}\right )}{d^{5}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{5 x^{10}} & \text {otherwise} \end {cases}\right )}{2} \]
-a*Piecewise((2*(-c**3*(c + d/x**2)**(3/2)/3 + 3*c**2*(c + d/x**2)**(5/2)/ 5 - 3*c*(c + d/x**2)**(7/2)/7 + (c + d/x**2)**(9/2)/9)/d**4, Ne(d, 0)), (s qrt(c)/(4*x**8), True))/2 - b*Piecewise((2*(c**4*(c + d/x**2)**(3/2)/3 - 4 *c**3*(c + d/x**2)**(5/2)/5 + 6*c**2*(c + d/x**2)**(7/2)/7 - 4*c*(c + d/x* *2)**(9/2)/9 + (c + d/x**2)**(11/2)/11)/d**5, Ne(d, 0)), (sqrt(c)/(5*x**10 ), True))/2
Time = 0.21 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^9} \, dx=-\frac {1}{3465} \, b {\left (\frac {315 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}}}{d^{5}} - \frac {1540 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} c}{d^{5}} + \frac {2970 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c^{2}}{d^{5}} - \frac {2772 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{3}}{d^{5}} + \frac {1155 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{4}}{d^{5}}\right )} - \frac {1}{315} \, a {\left (\frac {35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}}}{d^{4}} - \frac {135 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c}{d^{4}} + \frac {189 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{2}}{d^{4}} - \frac {105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{3}}{d^{4}}\right )} \]
-1/3465*b*(315*(c + d/x^2)^(11/2)/d^5 - 1540*(c + d/x^2)^(9/2)*c/d^5 + 297 0*(c + d/x^2)^(7/2)*c^2/d^5 - 2772*(c + d/x^2)^(5/2)*c^3/d^5 + 1155*(c + d /x^2)^(3/2)*c^4/d^5) - 1/315*a*(35*(c + d/x^2)^(9/2)/d^4 - 135*(c + d/x^2) ^(7/2)*c/d^4 + 189*(c + d/x^2)^(5/2)*c^2/d^4 - 105*(c + d/x^2)^(3/2)*c^3/d ^4)
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (114) = 228\).
Time = 1.49 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.21 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^9} \, dx=\frac {32 \, {\left (3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} a c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} b c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 4851 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {9}{2}} d \mathrm {sgn}\left (x\right ) + 7392 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {11}{2}} d \mathrm {sgn}\left (x\right ) + 231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {9}{2}} d^{2} \mathrm {sgn}\left (x\right ) + 2640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {11}{2}} d^{2} \mathrm {sgn}\left (x\right ) - 165 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {9}{2}} d^{3} \mathrm {sgn}\left (x\right ) - 1320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {11}{2}} d^{3} \mathrm {sgn}\left (x\right ) + 1815 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {9}{2}} d^{4} \mathrm {sgn}\left (x\right ) + 440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {11}{2}} d^{4} \mathrm {sgn}\left (x\right ) - 605 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {9}{2}} d^{5} \mathrm {sgn}\left (x\right ) - 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {11}{2}} d^{5} \mathrm {sgn}\left (x\right ) + 121 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {9}{2}} d^{6} \mathrm {sgn}\left (x\right ) + 8 \, b c^{\frac {11}{2}} d^{6} \mathrm {sgn}\left (x\right ) - 11 \, a c^{\frac {9}{2}} d^{7} \mathrm {sgn}\left (x\right )\right )}}{3465 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{11}} \]
32/3465*(3465*(sqrt(c)*x - sqrt(c*x^2 + d))^14*a*c^(9/2)*sgn(x) + 11088*(s qrt(c)*x - sqrt(c*x^2 + d))^12*b*c^(11/2)*sgn(x) - 4851*(sqrt(c)*x - sqrt( c*x^2 + d))^12*a*c^(9/2)*d*sgn(x) + 7392*(sqrt(c)*x - sqrt(c*x^2 + d))^10* b*c^(11/2)*d*sgn(x) + 231*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(9/2)*d^2*s gn(x) + 2640*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(11/2)*d^2*sgn(x) - 165*( sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(9/2)*d^3*sgn(x) - 1320*(sqrt(c)*x - sq rt(c*x^2 + d))^6*b*c^(11/2)*d^3*sgn(x) + 1815*(sqrt(c)*x - sqrt(c*x^2 + d) )^6*a*c^(9/2)*d^4*sgn(x) + 440*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(11/2)* d^4*sgn(x) - 605*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(9/2)*d^5*sgn(x) - 88 *(sqrt(c)*x - sqrt(c*x^2 + d))^2*b*c^(11/2)*d^5*sgn(x) + 121*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(9/2)*d^6*sgn(x) + 8*b*c^(11/2)*d^6*sgn(x) - 11*a*c ^(9/2)*d^7*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^11
Time = 10.11 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^9} \, dx=\frac {16\,a\,c^4\,\sqrt {c+\frac {d}{x^2}}}{315\,d^4}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{11\,x^{10}}-\frac {a\,\sqrt {c+\frac {d}{x^2}}}{9\,x^8}-\frac {128\,b\,c^5\,\sqrt {c+\frac {d}{x^2}}}{3465\,d^5}-\frac {a\,c\,\sqrt {c+\frac {d}{x^2}}}{63\,d\,x^6}-\frac {b\,c\,\sqrt {c+\frac {d}{x^2}}}{99\,d\,x^8}+\frac {2\,a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d^2\,x^4}-\frac {8\,a\,c^3\,\sqrt {c+\frac {d}{x^2}}}{315\,d^3\,x^2}+\frac {8\,b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{693\,d^2\,x^6}-\frac {16\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{1155\,d^3\,x^4}+\frac {64\,b\,c^4\,\sqrt {c+\frac {d}{x^2}}}{3465\,d^4\,x^2} \]
(16*a*c^4*(c + d/x^2)^(1/2))/(315*d^4) - (b*(c + d/x^2)^(1/2))/(11*x^10) - (a*(c + d/x^2)^(1/2))/(9*x^8) - (128*b*c^5*(c + d/x^2)^(1/2))/(3465*d^5) - (a*c*(c + d/x^2)^(1/2))/(63*d*x^6) - (b*c*(c + d/x^2)^(1/2))/(99*d*x^8) + (2*a*c^2*(c + d/x^2)^(1/2))/(105*d^2*x^4) - (8*a*c^3*(c + d/x^2)^(1/2))/ (315*d^3*x^2) + (8*b*c^2*(c + d/x^2)^(1/2))/(693*d^2*x^6) - (16*b*c^3*(c + d/x^2)^(1/2))/(1155*d^3*x^4) + (64*b*c^4*(c + d/x^2)^(1/2))/(3465*d^4*x^2 )