Integrand size = 22, antiderivative size = 84 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx=-\frac {2 d (9 b c-4 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{315 c^3}+\frac {(9 b c-4 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{63 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{9 c} \] Output:
-2/315*d*(-4*a*d+9*b*c)*(c+d/x^2)^(5/2)*x^5/c^3+1/63*(-4*a*d+9*b*c)*(c+d/x ^2)^(5/2)*x^7/c^2+1/9*a*(c+d/x^2)^(5/2)*x^9/c
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.79 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (d+c x^2\right )^2 \left (9 b c \left (-2 d+5 c x^2\right )+a \left (8 d^2-20 c d x^2+35 c^2 x^4\right )\right )}{315 c^3} \] Input:
Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^8,x]
Output:
(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(9*b*c*(-2*d + 5*c*x^2) + a*(8*d^2 - 20*c *d*x^2 + 35*c^2*x^4)))/(315*c^3)
Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {955, 803, 796}
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.
\(\displaystyle \int x^8 \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle \frac {(9 b c-4 a d) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6dx}{9 c}+\frac {a x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {(9 b c-4 a d) \left (\frac {x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {2 d \int \left (c+\frac {d}{x^2}\right )^{3/2} x^4dx}{7 c}\right )}{9 c}+\frac {a x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle \frac {\left (\frac {x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {2 d x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{35 c^2}\right ) (9 b c-4 a d)}{9 c}+\frac {a x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}\) |
Input:
Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^8,x]
Output:
(a*(c + d/x^2)^(5/2)*x^9)/(9*c) + ((9*b*c - 4*a*d)*((-2*d*(c + d/x^2)^(5/2 )*x^5)/(35*c^2) + ((c + d/x^2)^(5/2)*x^7)/(7*c)))/(9*c)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (35 a \,c^{2} x^{4}-20 a d \,x^{2} c +45 b \,c^{2} x^{2}+8 a \,d^{2}-18 d b c \right ) \left (c \,x^{2}+d \right )}{315 c^{3}}\) | \(67\) |
default | \(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (35 a \,c^{2} x^{4}-20 a d \,x^{2} c +45 b \,c^{2} x^{2}+8 a \,d^{2}-18 d b c \right ) \left (c \,x^{2}+d \right )}{315 c^{3}}\) | \(67\) |
orering | \(\frac {\left (35 a \,c^{2} x^{4}-20 a d \,x^{2} c +45 b \,c^{2} x^{2}+8 a \,d^{2}-18 d b c \right ) \left (c \,x^{2}+d \right ) x^{5} \left (a +\frac {b}{x^{2}}\right ) \left (c +\frac {d}{x^{2}}\right )^{\frac {3}{2}}}{315 c^{3} \left (a \,x^{2}+b \right )}\) | \(79\) |
risch | \(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \left (35 a \,x^{8} c^{4}+50 a \,c^{3} d \,x^{6}+45 b \,c^{4} x^{6}+3 a \,c^{2} d^{2} x^{4}+72 b \,c^{3} d \,x^{4}-4 a c \,d^{3} x^{2}+9 b \,c^{2} d^{2} x^{2}+8 a \,d^{4}-18 b c \,d^{3}\right )}{315 c^{3}}\) | \(106\) |
trager | \(\frac {\left (35 a \,x^{8} c^{4}+50 a \,c^{3} d \,x^{6}+45 b \,c^{4} x^{6}+3 a \,c^{2} d^{2} x^{4}+72 b \,c^{3} d \,x^{4}-4 a c \,d^{3} x^{2}+9 b \,c^{2} d^{2} x^{2}+8 a \,d^{4}-18 b c \,d^{3}\right ) x \sqrt {-\frac {-c \,x^{2}-d}{x^{2}}}}{315 c^{3}}\) | \(110\) |
Input:
int((a+b/x^2)*(c+d/x^2)^(3/2)*x^8,x,method=_RETURNVERBOSE)
Output:
1/315*((c*x^2+d)/x^2)^(3/2)*x^3*(35*a*c^2*x^4-20*a*c*d*x^2+45*b*c^2*x^2+8* a*d^2-18*b*c*d)*(c*x^2+d)/c^3
Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.26 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx=\frac {{\left (35 \, a c^{4} x^{9} + 5 \, {\left (9 \, b c^{4} + 10 \, a c^{3} d\right )} x^{7} + 3 \, {\left (24 \, b c^{3} d + a c^{2} d^{2}\right )} x^{5} + {\left (9 \, b c^{2} d^{2} - 4 \, a c d^{3}\right )} x^{3} - 2 \, {\left (9 \, b c d^{3} - 4 \, a d^{4}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{315 \, c^{3}} \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^8,x, algorithm="fricas")
Output:
1/315*(35*a*c^4*x^9 + 5*(9*b*c^4 + 10*a*c^3*d)*x^7 + 3*(24*b*c^3*d + a*c^2 *d^2)*x^5 + (9*b*c^2*d^2 - 4*a*c*d^3)*x^3 - 2*(9*b*c*d^3 - 4*a*d^4)*x)*sqr t((c*x^2 + d)/x^2)/c^3
Leaf count of result is larger than twice the leaf count of optimal. 1340 vs. \(2 (78) = 156\).
Time = 4.18 (sec) , antiderivative size = 1340, normalized size of antiderivative = 15.95 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx=\text {Too large to display} \] Input:
integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**8,x)
Output:
35*a*c**8*d**(19/2)*x**14*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c** 6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 110*a*c**7*d**(21/2 )*x**12*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945 *c**5*d**11*x**2 + 315*c**4*d**12) + 114*a*c**6*d**(23/2)*x**10*sqrt(c*x** 2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 40*a*c**5*d**(25/2)*x**8*sqrt(c*x**2/d + 1)/(315*c**7*d **9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 1 5*a*c**5*d**(11/2)*x**10*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4 *d**5*x**2 + 105*c**3*d**6) - 5*a*c**4*d**(27/2)*x**6*sqrt(c*x**2/d + 1)/( 315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4* d**12) + 33*a*c**4*d**(13/2)*x**8*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) - 30*a*c**3*d**(29/2)*x**4*sqrt(c*x** 2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 17*a*c**3*d**(15/2)*x**6*sqrt(c*x**2/d + 1)/(105*c**5*d **4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) - 40*a*c**2*d**(31/2)*x**2* sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d* *11*x**2 + 315*c**4*d**12) + 3*a*c**2*d**(17/2)*x**4*sqrt(c*x**2/d + 1)/(1 05*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) - 16*a*c*d**(33/2) *sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d **11*x**2 + 315*c**4*d**12) + 12*a*c*d**(19/2)*x**2*sqrt(c*x**2/d + 1)/...
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.07 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx=\frac {{\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} x^{7} - 7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d x^{5}\right )} b}{35 \, c^{2}} + \frac {{\left (35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} x^{9} - 90 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d x^{7} + 63 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{2} x^{5}\right )} a}{315 \, c^{3}} \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^8,x, algorithm="maxima")
Output:
1/35*(5*(c + d/x^2)^(7/2)*x^7 - 7*(c + d/x^2)^(5/2)*d*x^5)*b/c^2 + 1/315*( 35*(c + d/x^2)^(9/2)*x^9 - 90*(c + d/x^2)^(7/2)*d*x^7 + 63*(c + d/x^2)^(5/ 2)*d^2*x^5)*a/c^3
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.25 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx=\frac {2 \, {\left (9 \, b c d^{\frac {7}{2}} - 4 \, a d^{\frac {9}{2}}\right )} \mathrm {sgn}\left (x\right )}{315 \, c^{3}} + \frac {35 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} a \mathrm {sgn}\left (x\right ) + 45 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c \mathrm {sgn}\left (x\right ) - 90 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a d \mathrm {sgn}\left (x\right ) - 63 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c d \mathrm {sgn}\left (x\right ) + 63 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d^{2} \mathrm {sgn}\left (x\right )}{315 \, c^{3}} \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^8,x, algorithm="giac")
Output:
2/315*(9*b*c*d^(7/2) - 4*a*d^(9/2))*sgn(x)/c^3 + 1/315*(35*(c*x^2 + d)^(9/ 2)*a*sgn(x) + 45*(c*x^2 + d)^(7/2)*b*c*sgn(x) - 90*(c*x^2 + d)^(7/2)*a*d*s gn(x) - 63*(c*x^2 + d)^(5/2)*b*c*d*sgn(x) + 63*(c*x^2 + d)^(5/2)*a*d^2*sgn (x))/c^3
Time = 3.86 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.15 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx=\sqrt {c+\frac {d}{x^2}}\,\left (\frac {x\,\left (8\,a\,d^4-18\,b\,c\,d^3\right )}{315\,c^3}+\frac {x^7\,\left (45\,b\,c^4+50\,a\,d\,c^3\right )}{315\,c^3}+\frac {a\,c\,x^9}{9}+\frac {d\,x^5\,\left (a\,d+24\,b\,c\right )}{105\,c}-\frac {d^2\,x^3\,\left (4\,a\,d-9\,b\,c\right )}{315\,c^2}\right ) \] Input:
int(x^8*(a + b/x^2)*(c + d/x^2)^(3/2),x)
Output:
(c + d/x^2)^(1/2)*((x*(8*a*d^4 - 18*b*c*d^3))/(315*c^3) + (x^7*(45*b*c^4 + 50*a*c^3*d))/(315*c^3) + (a*c*x^9)/9 + (d*x^5*(a*d + 24*b*c))/(105*c) - ( d^2*x^3*(4*a*d - 9*b*c))/(315*c^2))
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.18 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^8 \, dx=\frac {\sqrt {c \,x^{2}+d}\, \left (35 a \,c^{4} x^{8}+50 a \,c^{3} d \,x^{6}+45 b \,c^{4} x^{6}+3 a \,c^{2} d^{2} x^{4}+72 b \,c^{3} d \,x^{4}-4 a c \,d^{3} x^{2}+9 b \,c^{2} d^{2} x^{2}+8 a \,d^{4}-18 b c \,d^{3}\right )}{315 c^{3}} \] Input:
int((a+b/x^2)*(c+d/x^2)^(3/2)*x^8,x)
Output:
(sqrt(c*x**2 + d)*(35*a*c**4*x**8 + 50*a*c**3*d*x**6 + 3*a*c**2*d**2*x**4 - 4*a*c*d**3*x**2 + 8*a*d**4 + 45*b*c**4*x**6 + 72*b*c**3*d*x**4 + 9*b*c** 2*d**2*x**2 - 18*b*c*d**3))/(315*c**3)