\(\int (a+\frac {b}{x^2}) (c+\frac {d}{x^2})^{3/2} x^{10} \, dx\) [162]

Optimal result

Integrand size = 22, antiderivative size = 117 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx=\frac {8 d^2 (11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{3465 c^4}-\frac {4 d (11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{693 c^3}+\frac {(11 b c-6 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{99 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^{11}}{11 c} \] Output:

8/3465*d^2*(-6*a*d+11*b*c)*(c+d/x^2)^(5/2)*x^5/c^4-4/693*d*(-6*a*d+11*b*c) 
*(c+d/x^2)^(5/2)*x^7/c^3+1/99*(-6*a*d+11*b*c)*(c+d/x^2)^(5/2)*x^9/c^2+1/11 
*a*(c+d/x^2)^(5/2)*x^11/c
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.76 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (d+c x^2\right )^2 \left (11 b c \left (8 d^2-20 c d x^2+35 c^2 x^4\right )+3 a \left (-16 d^3+40 c d^2 x^2-70 c^2 d x^4+105 c^3 x^6\right )\right )}{3465 c^4} \] Input:

Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^10,x]
 

Output:

(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(11*b*c*(8*d^2 - 20*c*d*x^2 + 35*c^2*x^4) 
 + 3*a*(-16*d^3 + 40*c*d^2*x^2 - 70*c^2*d*x^4 + 105*c^3*x^6)))/(3465*c^4)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {955, 803, 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.

\(\displaystyle \int x^{10} \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} \, dx\)

\(\Big \downarrow \) 955

\(\displaystyle \frac {(11 b c-6 a d) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^8dx}{11 c}+\frac {a x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c}\)

\(\Big \downarrow \) 803

\(\displaystyle \frac {(11 b c-6 a d) \left (\frac {x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}-\frac {4 d \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6dx}{9 c}\right )}{11 c}+\frac {a x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c}\)

\(\Big \downarrow \) 803

\(\displaystyle \frac {(11 b c-6 a d) \left (\frac {x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}-\frac {4 d \left (\frac {x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {2 d \int \left (c+\frac {d}{x^2}\right )^{3/2} x^4dx}{7 c}\right )}{9 c}\right )}{11 c}+\frac {a x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c}\)

\(\Big \downarrow \) 796

\(\displaystyle \frac {\left (\frac {x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}-\frac {4 d \left (\frac {x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {2 d x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{35 c^2}\right )}{9 c}\right ) (11 b c-6 a d)}{11 c}+\frac {a x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c}\)

Input:

Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^10,x]
 

Output:

(a*(c + d/x^2)^(5/2)*x^11)/(11*c) + ((11*b*c - 6*a*d)*(((c + d/x^2)^(5/2)* 
x^9)/(9*c) - (4*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)))/(11*c)
 

Defintions of rubi rules used

rule 796

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]
 

rule 803

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]
 

rule 955

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]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78

method	result	size
gosper	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (315 a \,x^{6} c^{3}-210 a \,c^{2} d \,x^{4}+385 b \,c^{3} x^{4}+120 a c \,d^{2} x^{2}-220 b \,c^{2} d \,x^{2}-48 a \,d^{3}+88 b c \,d^{2}\right ) \left (c \,x^{2}+d \right )}{3465 c^{4}}\)	\(91\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (315 a \,x^{6} c^{3}-210 a \,c^{2} d \,x^{4}+385 b \,c^{3} x^{4}+120 a c \,d^{2} x^{2}-220 b \,c^{2} d \,x^{2}-48 a \,d^{3}+88 b c \,d^{2}\right ) \left (c \,x^{2}+d \right )}{3465 c^{4}}\)	\(91\)
orering	\(\frac {\left (315 a \,x^{6} c^{3}-210 a \,c^{2} d \,x^{4}+385 b \,c^{3} x^{4}+120 a c \,d^{2} x^{2}-220 b \,c^{2} d \,x^{2}-48 a \,d^{3}+88 b c \,d^{2}\right ) \left (c \,x^{2}+d \right ) x^{5} \left (a +\frac {b}{x^{2}}\right ) \left (c +\frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3465 c^{4} \left (a \,x^{2}+b \right )}\)	\(103\)
risch	\(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \left (315 a \,c^{5} x^{10}+420 a \,c^{4} d \,x^{8}+385 b \,c^{5} x^{8}+15 a \,c^{3} d^{2} x^{6}+550 b \,c^{4} d \,x^{6}-18 a \,c^{2} d^{3} x^{4}+33 b \,c^{3} d^{2} x^{4}+24 a c \,d^{4} x^{2}-44 b \,c^{2} d^{3} x^{2}-48 a \,d^{5}+88 b c \,d^{4}\right )}{3465 c^{4}}\)	\(130\)
trager	\(\frac {\left (315 a \,c^{5} x^{10}+420 a \,c^{4} d \,x^{8}+385 b \,c^{5} x^{8}+15 a \,c^{3} d^{2} x^{6}+550 b \,c^{4} d \,x^{6}-18 a \,c^{2} d^{3} x^{4}+33 b \,c^{3} d^{2} x^{4}+24 a c \,d^{4} x^{2}-44 b \,c^{2} d^{3} x^{2}-48 a \,d^{5}+88 b c \,d^{4}\right ) x \sqrt {-\frac {-c \,x^{2}-d}{x^{2}}}}{3465 c^{4}}\)	\(134\)

Input:

int((a+b/x^2)*(c+d/x^2)^(3/2)*x^10,x,method=_RETURNVERBOSE)
 

Output:

1/3465*((c*x^2+d)/x^2)^(3/2)*x^3*(315*a*c^3*x^6-210*a*c^2*d*x^4+385*b*c^3* 
x^4+120*a*c*d^2*x^2-220*b*c^2*d*x^2-48*a*d^3+88*b*c*d^2)*(c*x^2+d)/c^4
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.13 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx=\frac {{\left (315 \, a c^{5} x^{11} + 35 \, {\left (11 \, b c^{5} + 12 \, a c^{4} d\right )} x^{9} + 5 \, {\left (110 \, b c^{4} d + 3 \, a c^{3} d^{2}\right )} x^{7} + 3 \, {\left (11 \, b c^{3} d^{2} - 6 \, a c^{2} d^{3}\right )} x^{5} - 4 \, {\left (11 \, b c^{2} d^{3} - 6 \, a c d^{4}\right )} x^{3} + 8 \, {\left (11 \, b c d^{4} - 6 \, a d^{5}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3465 \, c^{4}} \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^10,x, algorithm="fricas")
 

Output:

1/3465*(315*a*c^5*x^11 + 35*(11*b*c^5 + 12*a*c^4*d)*x^9 + 5*(110*b*c^4*d + 
 3*a*c^3*d^2)*x^7 + 3*(11*b*c^3*d^2 - 6*a*c^2*d^3)*x^5 - 4*(11*b*c^2*d^3 - 
 6*a*c*d^4)*x^3 + 8*(11*b*c*d^4 - 6*a*d^5)*x)*sqrt((c*x^2 + d)/x^2)/c^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2304 vs. \(2 (112) = 224\).

Time = 5.27 (sec) , antiderivative size = 2304, normalized size of antiderivative = 19.69 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx=\text {Too large to display} \] Input:

integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**10,x)
 

Output:

315*a*c**10*d**(33/2)*x**18*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 138 
60*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465* 
c**5*d**20) + 1295*a*c**9*d**(35/2)*x**16*sqrt(c*x**2/d + 1)/(3465*c**9*d* 
*16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 13860*c**6*d**1 
9*x**2 + 3465*c**5*d**20) + 1990*a*c**8*d**(37/2)*x**14*sqrt(c*x**2/d + 1) 
/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 1 
3860*c**6*d**19*x**2 + 3465*c**5*d**20) + 1358*a*c**7*d**(39/2)*x**12*sqrt 
(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7* 
d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 35*a*c**7*d**(21/2 
)*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) + 343*a*c**6*d**(41/2)*x**10*sqrt(c*x** 
2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18* 
x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 110*a*c**6*d**(23/2)*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) + 35*a*c**5*d**(43/2)*x**8*sqrt(c*x**2/d + 1 
)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 
13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 114*a*c**5*d**(25/2)*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) + 280*a*c**4*d**(45/2)*x**6*sqrt(c*x**2/d + 1)/(346 
5*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 20790*c**7*d**18*x**4 + 138...
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx=\frac {{\left (35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} x^{9} - 90 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d x^{7} + 63 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{2} x^{5}\right )} b}{315 \, c^{3}} + \frac {{\left (105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}} x^{11} - 385 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} d x^{9} + 495 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d^{2} x^{7} - 231 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} x^{5}\right )} a}{1155 \, c^{4}} \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^10,x, algorithm="maxima")
 

Output:

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)*b/c^3 + 1/1155*(105*(c + d/x^2)^(11/2)*x^11 - 385*(c + 
d/x^2)^(9/2)*d*x^9 + 495*(c + d/x^2)^(7/2)*d^2*x^7 - 231*(c + d/x^2)^(5/2) 
*d^3*x^5)*a/c^4
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.20 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx=-\frac {8 \, {\left (11 \, b c d^{\frac {9}{2}} - 6 \, a d^{\frac {11}{2}}\right )} \mathrm {sgn}\left (x\right )}{3465 \, c^{4}} + \frac {315 \, {\left (c x^{2} + d\right )}^{\frac {11}{2}} a \mathrm {sgn}\left (x\right ) + 385 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} b c \mathrm {sgn}\left (x\right ) - 1155 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} a d \mathrm {sgn}\left (x\right ) - 990 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c d \mathrm {sgn}\left (x\right ) + 1485 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a d^{2} \mathrm {sgn}\left (x\right ) + 693 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c d^{2} \mathrm {sgn}\left (x\right ) - 693 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d^{3} \mathrm {sgn}\left (x\right )}{3465 \, c^{4}} \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^10,x, algorithm="giac")
 

Output:

-8/3465*(11*b*c*d^(9/2) - 6*a*d^(11/2))*sgn(x)/c^4 + 1/3465*(315*(c*x^2 + 
d)^(11/2)*a*sgn(x) + 385*(c*x^2 + d)^(9/2)*b*c*sgn(x) - 1155*(c*x^2 + d)^( 
9/2)*a*d*sgn(x) - 990*(c*x^2 + d)^(7/2)*b*c*d*sgn(x) + 1485*(c*x^2 + d)^(7 
/2)*a*d^2*sgn(x) + 693*(c*x^2 + d)^(5/2)*b*c*d^2*sgn(x) - 693*(c*x^2 + d)^ 
(5/2)*a*d^3*sgn(x))/c^4
 

Mupad [B] (verification not implemented)

Time = 3.85 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx=\sqrt {c+\frac {d}{x^2}}\,\left (\frac {x^9\,\left (385\,b\,c^5+420\,a\,d\,c^4\right )}{3465\,c^4}-\frac {x\,\left (48\,a\,d^5-88\,b\,c\,d^4\right )}{3465\,c^4}+\frac {a\,c\,x^{11}}{11}+\frac {d\,x^7\,\left (3\,a\,d+110\,b\,c\right )}{693\,c}-\frac {d^2\,x^5\,\left (6\,a\,d-11\,b\,c\right )}{1155\,c^2}+\frac {4\,d^3\,x^3\,\left (6\,a\,d-11\,b\,c\right )}{3465\,c^3}\right ) \] Input:

int(x^10*(a + b/x^2)*(c + d/x^2)^(3/2),x)
 

Output:

(c + d/x^2)^(1/2)*((x^9*(385*b*c^5 + 420*a*c^4*d))/(3465*c^4) - (x*(48*a*d 
^5 - 88*b*c*d^4))/(3465*c^4) + (a*c*x^11)/11 + (d*x^7*(3*a*d + 110*b*c))/( 
693*c) - (d^2*x^5*(6*a*d - 11*b*c))/(1155*c^2) + (4*d^3*x^3*(6*a*d - 11*b* 
c))/(3465*c^3))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{10} \, dx=\frac {\sqrt {c \,x^{2}+d}\, \left (315 a \,c^{5} x^{10}+420 a \,c^{4} d \,x^{8}+385 b \,c^{5} x^{8}+15 a \,c^{3} d^{2} x^{6}+550 b \,c^{4} d \,x^{6}-18 a \,c^{2} d^{3} x^{4}+33 b \,c^{3} d^{2} x^{4}+24 a c \,d^{4} x^{2}-44 b \,c^{2} d^{3} x^{2}-48 a \,d^{5}+88 b c \,d^{4}\right )}{3465 c^{4}} \] Input:

int((a+b/x^2)*(c+d/x^2)^(3/2)*x^10,x)
 

Output:

(sqrt(c*x**2 + d)*(315*a*c**5*x**10 + 420*a*c**4*d*x**8 + 15*a*c**3*d**2*x 
**6 - 18*a*c**2*d**3*x**4 + 24*a*c*d**4*x**2 - 48*a*d**5 + 385*b*c**5*x**8 
 + 550*b*c**4*d*x**6 + 33*b*c**3*d**2*x**4 - 44*b*c**2*d**3*x**2 + 88*b*c* 
d**4))/(3465*c**4)