\(\int (a+\frac {b}{x^2}) (c+\frac {d}{x^2})^{3/2} x^{12} \, dx\) [161]

Optimal result

Integrand size = 22, antiderivative size = 150 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{12} \, dx=-\frac {16 d^3 (13 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{15015 c^5}+\frac {8 d^2 (13 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{3003 c^4}-\frac {2 d (13 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^9}{429 c^3}+\frac {(13 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^{11}}{143 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^{13}}{13 c} \] Output:

-16/15015*d^3*(-8*a*d+13*b*c)*(c+d/x^2)^(5/2)*x^5/c^5+8/3003*d^2*(-8*a*d+1 
3*b*c)*(c+d/x^2)^(5/2)*x^7/c^4-2/429*d*(-8*a*d+13*b*c)*(c+d/x^2)^(5/2)*x^9 
/c^3+1/143*(-8*a*d+13*b*c)*(c+d/x^2)^(5/2)*x^11/c^2+1/13*a*(c+d/x^2)^(5/2) 
*x^13/c
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.73 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{12} \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (d+c x^2\right )^2 \left (13 b c \left (-16 d^3+40 c d^2 x^2-70 c^2 d x^4+105 c^3 x^6\right )+a \left (128 d^4-320 c d^3 x^2+560 c^2 d^2 x^4-840 c^3 d x^6+1155 c^4 x^8\right )\right )}{15015 c^5} \] Input:

Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x]
 

Output:

(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(13*b*c*(-16*d^3 + 40*c*d^2*x^2 - 70*c^2* 
d*x^4 + 105*c^3*x^6) + a*(128*d^4 - 320*c*d^3*x^2 + 560*c^2*d^2*x^4 - 840* 
c^3*d*x^6 + 1155*c^4*x^8)))/(15015*c^5)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {955, 803, 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.

\(\displaystyle \int x^{12} \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} \, dx\)

\(\Big \downarrow \) 955

\(\displaystyle \frac {(13 b c-8 a d) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^{10}dx}{13 c}+\frac {a x^{13} \left (c+\frac {d}{x^2}\right )^{5/2}}{13 c}\)

\(\Big \downarrow \) 803

\(\displaystyle \frac {(13 b c-8 a d) \left (\frac {x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c}-\frac {6 d \int \left (c+\frac {d}{x^2}\right )^{3/2} x^8dx}{11 c}\right )}{13 c}+\frac {a x^{13} \left (c+\frac {d}{x^2}\right )^{5/2}}{13 c}\)

\(\Big \downarrow \) 803

\(\displaystyle \frac {(13 b c-8 a d) \left (\frac {x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c}-\frac {6 d \left (\frac {x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}-\frac {4 d \int \left (c+\frac {d}{x^2}\right )^{3/2} x^6dx}{9 c}\right )}{11 c}\right )}{13 c}+\frac {a x^{13} \left (c+\frac {d}{x^2}\right )^{5/2}}{13 c}\)

\(\Big \downarrow \) 803

\(\displaystyle \frac {(13 b c-8 a d) \left (\frac {x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c}-\frac {6 d \left (\frac {x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}-\frac {4 d \left (\frac {x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {2 d \int \left (c+\frac {d}{x^2}\right )^{3/2} x^4dx}{7 c}\right )}{9 c}\right )}{11 c}\right )}{13 c}+\frac {a x^{13} \left (c+\frac {d}{x^2}\right )^{5/2}}{13 c}\)

\(\Big \downarrow \) 796

\(\displaystyle \frac {\left (\frac {x^{11} \left (c+\frac {d}{x^2}\right )^{5/2}}{11 c}-\frac {6 d \left (\frac {x^9 \left (c+\frac {d}{x^2}\right )^{5/2}}{9 c}-\frac {4 d \left (\frac {x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c}-\frac {2 d x^5 \left (c+\frac {d}{x^2}\right )^{5/2}}{35 c^2}\right )}{9 c}\right )}{11 c}\right ) (13 b c-8 a d)}{13 c}+\frac {a x^{13} \left (c+\frac {d}{x^2}\right )^{5/2}}{13 c}\)

Input:

Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x]
 

Output:

(a*(c + d/x^2)^(5/2)*x^13)/(13*c) + ((13*b*c - 8*a*d)*(((c + d/x^2)^(5/2)* 
x^11)/(11*c) - (6*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)))/ 
(13*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 = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.77

method	result	size
gosper	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (1155 a \,x^{8} c^{4}-840 a \,c^{3} d \,x^{6}+1365 b \,c^{4} x^{6}+560 a \,c^{2} d^{2} x^{4}-910 b \,c^{3} d \,x^{4}-320 a c \,d^{3} x^{2}+520 b \,c^{2} d^{2} x^{2}+128 a \,d^{4}-208 b c \,d^{3}\right ) \left (c \,x^{2}+d \right )}{15015 c^{5}}\)	\(115\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{3} \left (1155 a \,x^{8} c^{4}-840 a \,c^{3} d \,x^{6}+1365 b \,c^{4} x^{6}+560 a \,c^{2} d^{2} x^{4}-910 b \,c^{3} d \,x^{4}-320 a c \,d^{3} x^{2}+520 b \,c^{2} d^{2} x^{2}+128 a \,d^{4}-208 b c \,d^{3}\right ) \left (c \,x^{2}+d \right )}{15015 c^{5}}\)	\(115\)
orering	\(\frac {\left (1155 a \,x^{8} c^{4}-840 a \,c^{3} d \,x^{6}+1365 b \,c^{4} x^{6}+560 a \,c^{2} d^{2} x^{4}-910 b \,c^{3} d \,x^{4}-320 a c \,d^{3} x^{2}+520 b \,c^{2} d^{2} x^{2}+128 a \,d^{4}-208 b c \,d^{3}\right ) \left (c \,x^{2}+d \right ) x^{5} \left (a +\frac {b}{x^{2}}\right ) \left (c +\frac {d}{x^{2}}\right )^{\frac {3}{2}}}{15015 c^{5} \left (a \,x^{2}+b \right )}\)	\(127\)
risch	\(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \left (1155 a \,c^{6} x^{12}+1470 a \,c^{5} d \,x^{10}+1365 b \,c^{6} x^{10}+35 a \,c^{4} d^{2} x^{8}+1820 b \,c^{5} d \,x^{8}-40 a \,c^{3} d^{3} x^{6}+65 b \,c^{4} d^{2} x^{6}+48 a \,c^{2} d^{4} x^{4}-78 b \,c^{3} d^{3} x^{4}-64 a c \,d^{5} x^{2}+104 b \,c^{2} d^{4} x^{2}+128 a \,d^{6}-208 b c \,d^{5}\right )}{15015 c^{5}}\)	\(154\)
trager	\(\frac {\left (1155 a \,c^{6} x^{12}+1470 a \,c^{5} d \,x^{10}+1365 b \,c^{6} x^{10}+35 a \,c^{4} d^{2} x^{8}+1820 b \,c^{5} d \,x^{8}-40 a \,c^{3} d^{3} x^{6}+65 b \,c^{4} d^{2} x^{6}+48 a \,c^{2} d^{4} x^{4}-78 b \,c^{3} d^{3} x^{4}-64 a c \,d^{5} x^{2}+104 b \,c^{2} d^{4} x^{2}+128 a \,d^{6}-208 b c \,d^{5}\right ) x \sqrt {-\frac {-c \,x^{2}-d}{x^{2}}}}{15015 c^{5}}\)	\(158\)

Input:

int((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x,method=_RETURNVERBOSE)
 

Output:

1/15015*((c*x^2+d)/x^2)^(3/2)*x^3*(1155*a*c^4*x^8-840*a*c^3*d*x^6+1365*b*c 
^4*x^6+560*a*c^2*d^2*x^4-910*b*c^3*d*x^4-320*a*c*d^3*x^2+520*b*c^2*d^2*x^2 
+128*a*d^4-208*b*c*d^3)*(c*x^2+d)/c^5
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{12} \, dx=\frac {{\left (1155 \, a c^{6} x^{13} + 105 \, {\left (13 \, b c^{6} + 14 \, a c^{5} d\right )} x^{11} + 35 \, {\left (52 \, b c^{5} d + a c^{4} d^{2}\right )} x^{9} + 5 \, {\left (13 \, b c^{4} d^{2} - 8 \, a c^{3} d^{3}\right )} x^{7} - 6 \, {\left (13 \, b c^{3} d^{3} - 8 \, a c^{2} d^{4}\right )} x^{5} + 8 \, {\left (13 \, b c^{2} d^{4} - 8 \, a c d^{5}\right )} x^{3} - 16 \, {\left (13 \, b c d^{5} - 8 \, a d^{6}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15015 \, c^{5}} \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x, algorithm="fricas")
 

Output:

1/15015*(1155*a*c^6*x^13 + 105*(13*b*c^6 + 14*a*c^5*d)*x^11 + 35*(52*b*c^5 
*d + a*c^4*d^2)*x^9 + 5*(13*b*c^4*d^2 - 8*a*c^3*d^3)*x^7 - 6*(13*b*c^3*d^3 
 - 8*a*c^2*d^4)*x^5 + 8*(13*b*c^2*d^4 - 8*a*c*d^5)*x^3 - 16*(13*b*c*d^5 - 
8*a*d^6)*x)*sqrt((c*x^2 + d)/x^2)/c^5
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3351 vs. \(2 (146) = 292\).

Time = 7.23 (sec) , antiderivative size = 3351, normalized size of antiderivative = 22.34 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{12} \, dx=\text {Too large to display} \] Input:

integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**12,x)
 

Output:

693*a*c**12*d**(51/2)*x**22*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 4 
5045*c**10*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45 
045*c**7*d**29*x**2 + 9009*c**6*d**30) + 3528*a*c**11*d**(53/2)*x**20*sqrt 
(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26*x**8 + 90090*c* 
*9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6* 
d**30) + 7175*a*c**10*d**(55/2)*x**18*sqrt(c*x**2/d + 1)/(9009*c**11*d**25 
*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**28 
*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 7290*a*c**9*d**(57/2)*x 
**16*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26*x**8 + 
 90090*c**9*d**27*x**6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9 
009*c**6*d**30) + 315*a*c**9*d**(35/2)*x**18*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) + 3699*a*c**8*d**(59/2)*x**14*sqrt(c*x**2/d + 
 1)/(9009*c**11*d**25*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*d**27*x* 
*6 + 90090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 12 
95*a*c**8*d**(37/2)*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**19*x**2 + 3465*c* 
*5*d**20) + 756*a*c**7*d**(61/2)*x**12*sqrt(c*x**2/d + 1)/(9009*c**11*d**2 
5*x**10 + 45045*c**10*d**26*x**8 + 90090*c**9*d**27*x**6 + 90090*c**8*d**2 
8*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 1990*a*c**7*d**(39/...
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{12} \, dx=\frac {{\left (105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}} x^{11} - 385 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} d x^{9} + 495 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d^{2} x^{7} - 231 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} x^{5}\right )} b}{1155 \, c^{4}} + \frac {{\left (1155 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {13}{2}} x^{13} - 5460 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}} d x^{11} + 10010 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} d^{2} x^{9} - 8580 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d^{3} x^{7} + 3003 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{4} x^{5}\right )} a}{15015 \, c^{5}} \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x, algorithm="maxima")
 

Output:

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)*b/c^4 + 1/15015*( 
1155*(c + d/x^2)^(13/2)*x^13 - 5460*(c + d/x^2)^(11/2)*d*x^11 + 10010*(c + 
 d/x^2)^(9/2)*d^2*x^9 - 8580*(c + d/x^2)^(7/2)*d^3*x^7 + 3003*(c + d/x^2)^ 
(5/2)*d^4*x^5)*a/c^5
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.17 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{12} \, dx=\frac {16 \, {\left (13 \, b c d^{\frac {11}{2}} - 8 \, a d^{\frac {13}{2}}\right )} \mathrm {sgn}\left (x\right )}{15015 \, c^{5}} + \frac {1155 \, {\left (c x^{2} + d\right )}^{\frac {13}{2}} a \mathrm {sgn}\left (x\right ) + 1365 \, {\left (c x^{2} + d\right )}^{\frac {11}{2}} b c \mathrm {sgn}\left (x\right ) - 5460 \, {\left (c x^{2} + d\right )}^{\frac {11}{2}} a d \mathrm {sgn}\left (x\right ) - 5005 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} b c d \mathrm {sgn}\left (x\right ) + 10010 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} a d^{2} \mathrm {sgn}\left (x\right ) + 6435 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c d^{2} \mathrm {sgn}\left (x\right ) - 8580 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a d^{3} \mathrm {sgn}\left (x\right ) - 3003 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c d^{3} \mathrm {sgn}\left (x\right ) + 3003 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d^{4} \mathrm {sgn}\left (x\right )}{15015 \, c^{5}} \] Input:

integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x, algorithm="giac")
 

Output:

16/15015*(13*b*c*d^(11/2) - 8*a*d^(13/2))*sgn(x)/c^5 + 1/15015*(1155*(c*x^ 
2 + d)^(13/2)*a*sgn(x) + 1365*(c*x^2 + d)^(11/2)*b*c*sgn(x) - 5460*(c*x^2 
+ d)^(11/2)*a*d*sgn(x) - 5005*(c*x^2 + d)^(9/2)*b*c*d*sgn(x) + 10010*(c*x^ 
2 + d)^(9/2)*a*d^2*sgn(x) + 6435*(c*x^2 + d)^(7/2)*b*c*d^2*sgn(x) - 8580*( 
c*x^2 + d)^(7/2)*a*d^3*sgn(x) - 3003*(c*x^2 + d)^(5/2)*b*c*d^3*sgn(x) + 30 
03*(c*x^2 + d)^(5/2)*a*d^4*sgn(x))/c^5
 

Mupad [B] (verification not implemented)

Time = 4.02 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{12} \, dx=\sqrt {c+\frac {d}{x^2}}\,\left (\frac {x\,\left (128\,a\,d^6-208\,b\,c\,d^5\right )}{15015\,c^5}+\frac {x^{11}\,\left (1365\,b\,c^6+1470\,a\,d\,c^5\right )}{15015\,c^5}+\frac {a\,c\,x^{13}}{13}+\frac {d\,x^9\,\left (a\,d+52\,b\,c\right )}{429\,c}-\frac {d^2\,x^7\,\left (8\,a\,d-13\,b\,c\right )}{3003\,c^2}+\frac {2\,d^3\,x^5\,\left (8\,a\,d-13\,b\,c\right )}{5005\,c^3}-\frac {8\,d^4\,x^3\,\left (8\,a\,d-13\,b\,c\right )}{15015\,c^4}\right ) \] Input:

int(x^12*(a + b/x^2)*(c + d/x^2)^(3/2),x)
 

Output:

(c + d/x^2)^(1/2)*((x*(128*a*d^6 - 208*b*c*d^5))/(15015*c^5) + (x^11*(1365 
*b*c^6 + 1470*a*c^5*d))/(15015*c^5) + (a*c*x^13)/13 + (d*x^9*(a*d + 52*b*c 
))/(429*c) - (d^2*x^7*(8*a*d - 13*b*c))/(3003*c^2) + (2*d^3*x^5*(8*a*d - 1 
3*b*c))/(5005*c^3) - (8*d^4*x^3*(8*a*d - 13*b*c))/(15015*c^4))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.98 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^{12} \, dx=\frac {\sqrt {c \,x^{2}+d}\, \left (1155 a \,c^{6} x^{12}+1470 a \,c^{5} d \,x^{10}+1365 b \,c^{6} x^{10}+35 a \,c^{4} d^{2} x^{8}+1820 b \,c^{5} d \,x^{8}-40 a \,c^{3} d^{3} x^{6}+65 b \,c^{4} d^{2} x^{6}+48 a \,c^{2} d^{4} x^{4}-78 b \,c^{3} d^{3} x^{4}-64 a c \,d^{5} x^{2}+104 b \,c^{2} d^{4} x^{2}+128 a \,d^{6}-208 b c \,d^{5}\right )}{15015 c^{5}} \] Input:

int((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x)
 

Output:

(sqrt(c*x**2 + d)*(1155*a*c**6*x**12 + 1470*a*c**5*d*x**10 + 35*a*c**4*d** 
2*x**8 - 40*a*c**3*d**3*x**6 + 48*a*c**2*d**4*x**4 - 64*a*c*d**5*x**2 + 12 
8*a*d**6 + 1365*b*c**6*x**10 + 1820*b*c**5*d*x**8 + 65*b*c**4*d**2*x**6 - 
78*b*c**3*d**3*x**4 + 104*b*c**2*d**4*x**2 - 208*b*c*d**5))/(15015*c**5)