\(\int \frac {(a+b x^2)^{3/2} (A+B x^2)}{x^{12}} \, dx\) [221]

Optimal result

Integrand size = 22, antiderivative size = 117 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{12}} \, dx=-\frac {A \left (a+b x^2\right )^{5/2}}{11 a x^{11}}+\frac {(6 A b-11 a B) \left (a+b x^2\right )^{5/2}}{99 a^2 x^9}-\frac {4 b (6 A b-11 a B) \left (a+b x^2\right )^{5/2}}{693 a^3 x^7}+\frac {8 b^2 (6 A b-11 a B) \left (a+b x^2\right )^{5/2}}{3465 a^4 x^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{12}} \, dx=\frac {\left (a+b x^2\right )^{5/2} \left (-315 a^3 A+210 a^2 A b x^2-385 a^3 B x^2-120 a A b^2 x^4+220 a^2 b B x^4+48 A b^3 x^6-88 a b^2 B x^6\right )}{3465 a^4 x^{11}} \] Input:

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

Output:

((a + b*x^2)^(5/2)*(-315*a^3*A + 210*a^2*A*b*x^2 - 385*a^3*B*x^2 - 120*a*A 
*b^2*x^4 + 220*a^2*b*B*x^4 + 48*A*b^3*x^6 - 88*a*b^2*B*x^6))/(3465*a^4*x^1 
1)
 

Rubi [A] (verified)

Time = 0.21 (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 = {359, 245, 245, 242}

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.

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

\(\Big \downarrow \) 359

\(\displaystyle -\frac {(6 A b-11 a B) \int \frac {\left (b x^2+a\right )^{3/2}}{x^{10}}dx}{11 a}-\frac {A \left (a+b x^2\right )^{5/2}}{11 a x^{11}}\)

\(\Big \downarrow \) 245

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

\(\Big \downarrow \) 245

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

\(\Big \downarrow \) 242

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

Input:

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

Output:

-1/11*(A*(a + b*x^2)^(5/2))/(a*x^11) - ((6*A*b - 11*a*B)*(-1/9*(a + b*x^2) 
^(5/2)/(a*x^9) - (4*b*(-1/7*(a + b*x^2)^(5/2)/(a*x^7) + (2*b*(a + b*x^2)^( 
5/2))/(35*a^2*x^5)))/(9*a)))/(11*a)
 

Defintions of rubi rules used

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 245

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + 
 b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) 
   Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si 
mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63

method	result	size
pseudoelliptic	\(-\frac {\left (\left (\frac {11 x^{2} B}{9}+A \right ) a^{3}-\frac {2 b \left (\frac {22 x^{2} B}{21}+A \right ) x^{2} a^{2}}{3}+\frac {8 b^{2} \left (\frac {11 x^{2} B}{15}+A \right ) x^{4} a}{21}-\frac {16 A \,b^{3} x^{6}}{105}\right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{11 x^{11} a^{4}}\)	\(74\)
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (-48 A \,b^{3} x^{6}+88 B a \,b^{2} x^{6}+120 a A \,b^{2} x^{4}-220 B \,a^{2} b \,x^{4}-210 a^{2} A b \,x^{2}+385 B \,a^{3} x^{2}+315 a^{3} A \right )}{3465 x^{11} a^{4}}\)	\(83\)
orering	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (-48 A \,b^{3} x^{6}+88 B a \,b^{2} x^{6}+120 a A \,b^{2} x^{4}-220 B \,a^{2} b \,x^{4}-210 a^{2} A b \,x^{2}+385 B \,a^{3} x^{2}+315 a^{3} A \right )}{3465 x^{11} a^{4}}\)	\(83\)
trager	\(-\frac {\left (-48 A \,b^{5} x^{10}+88 B a \,b^{4} x^{10}+24 a A \,b^{4} x^{8}-44 B \,a^{2} b^{3} x^{8}-18 a^{2} A \,b^{3} x^{6}+33 B \,a^{3} b^{2} x^{6}+15 a^{3} A \,b^{2} x^{4}+550 B \,a^{4} b \,x^{4}+420 a^{4} A b \,x^{2}+385 B \,a^{5} x^{2}+315 a^{5} A \right ) \sqrt {b \,x^{2}+a}}{3465 x^{11} a^{4}}\)	\(131\)
risch	\(-\frac {\left (-48 A \,b^{5} x^{10}+88 B a \,b^{4} x^{10}+24 a A \,b^{4} x^{8}-44 B \,a^{2} b^{3} x^{8}-18 a^{2} A \,b^{3} x^{6}+33 B \,a^{3} b^{2} x^{6}+15 a^{3} A \,b^{2} x^{4}+550 B \,a^{4} b \,x^{4}+420 a^{4} A b \,x^{2}+385 B \,a^{5} x^{2}+315 a^{5} A \right ) \sqrt {b \,x^{2}+a}}{3465 x^{11} a^{4}}\)	\(131\)
default	\(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{11 a \,x^{11}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )}{11 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )\)	\(150\)

Input:

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

Output:

-1/11*((11/9*x^2*B+A)*a^3-2/3*b*(22/21*x^2*B+A)*x^2*a^2+8/21*b^2*(11/15*x^ 
2*B+A)*x^4*a-16/105*A*b^3*x^6)*(b*x^2+a)^(5/2)/x^11/a^4
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{12}} \, dx=-\frac {{\left (8 \, {\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} x^{10} - 4 \, {\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{8} + 3 \, {\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{6} + 315 \, A a^{5} + 5 \, {\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{4} + 35 \, {\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3465 \, a^{4} x^{11}} \] Input:

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

Output:

-1/3465*(8*(11*B*a*b^4 - 6*A*b^5)*x^10 - 4*(11*B*a^2*b^3 - 6*A*a*b^4)*x^8 
+ 3*(11*B*a^3*b^2 - 6*A*a^2*b^3)*x^6 + 315*A*a^5 + 5*(110*B*a^4*b + 3*A*a^ 
3*b^2)*x^4 + 35*(11*B*a^5 + 12*A*a^4*b)*x^2)*sqrt(b*x^2 + a)/(a^4*x^11)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2412 vs. \(2 (112) = 224\).

Time = 3.75 (sec) , antiderivative size = 2412, normalized size of antiderivative = 20.62 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{12}} \, dx=\text {Too large to display} \] Input:

integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**12,x)
 

Output:

-315*A*a**10*b**(33/2)*sqrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860 
*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465 
*a**5*b**20*x**18) - 1295*A*a**9*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(3465 
*a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 1386 
0*a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 1990*A*a**8*b**(37/2)*x**4*s 
qrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 2079 
0*a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 135 
8*A*a**7*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 1386 
0*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 346 
5*a**5*b**20*x**18) - 35*A*a**7*b**(21/2)*sqrt(a/(b*x**2) + 1)/(315*a**7*b 
**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x* 
*14) - 343*A*a**6*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x** 
10 + 13860*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860*a**6*b**19*x* 
*16 + 3465*a**5*b**20*x**18) - 110*A*a**6*b**(23/2)*x**2*sqrt(a/(b*x**2) + 
 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 31 
5*a**4*b**12*x**14) - 35*A*a**5*b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(3465 
*a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 1386 
0*a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 114*A*a**5*b**(25/2)*x**4*sq 
rt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b 
**11*x**12 + 315*a**4*b**12*x**14) - 280*A*a**4*b**(45/2)*x**12*sqrt(a/...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{12}} \, dx=-\frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{315 \, a^{3} x^{5}} + \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{1155 \, a^{4} x^{5}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{63 \, a^{2} x^{7}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{231 \, a^{3} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{9 \, a x^{9}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{33 \, a^{2} x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{11 \, a x^{11}} \] Input:

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

Output:

-8/315*(b*x^2 + a)^(5/2)*B*b^2/(a^3*x^5) + 16/1155*(b*x^2 + a)^(5/2)*A*b^3 
/(a^4*x^5) + 4/63*(b*x^2 + a)^(5/2)*B*b/(a^2*x^7) - 8/231*(b*x^2 + a)^(5/2 
)*A*b^2/(a^3*x^7) - 1/9*(b*x^2 + a)^(5/2)*B/(a*x^9) + 2/33*(b*x^2 + a)^(5/ 
2)*A*b/(a^2*x^9) - 1/11*(b*x^2 + a)^(5/2)*A/(a*x^11)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (101) = 202\).

Time = 0.14 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.90 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{12}} \, dx=\frac {16 \, {\left (2310 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} B b^{\frac {9}{2}} - 1155 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} B a b^{\frac {9}{2}} + 6930 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} A b^{\frac {11}{2}} + 231 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a^{2} b^{\frac {9}{2}} + 12474 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A a b^{\frac {11}{2}} - 4851 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{3} b^{\frac {9}{2}} + 15246 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a^{2} b^{\frac {11}{2}} + 2475 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{4} b^{\frac {9}{2}} + 4950 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{3} b^{\frac {11}{2}} + 495 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{5} b^{\frac {9}{2}} + 990 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{4} b^{\frac {11}{2}} + 605 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{6} b^{\frac {9}{2}} - 330 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{5} b^{\frac {11}{2}} - 121 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{7} b^{\frac {9}{2}} + 66 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{6} b^{\frac {11}{2}} + 11 \, B a^{8} b^{\frac {9}{2}} - 6 \, A a^{7} b^{\frac {11}{2}}\right )}}{3465 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{11}} \] Input:

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

Output:

16/3465*(2310*(sqrt(b)*x - sqrt(b*x^2 + a))^16*B*b^(9/2) - 1155*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^14*B*a*b^(9/2) + 6930*(sqrt(b)*x - sqrt(b*x^2 + a))^14 
*A*b^(11/2) + 231*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a^2*b^(9/2) + 12474*( 
sqrt(b)*x - sqrt(b*x^2 + a))^12*A*a*b^(11/2) - 4851*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^10*B*a^3*b^(9/2) + 15246*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a^2*b^ 
(11/2) + 2475*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^4*b^(9/2) + 4950*(sqrt(b 
)*x - sqrt(b*x^2 + a))^8*A*a^3*b^(11/2) + 495*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^6*B*a^5*b^(9/2) + 990*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^4*b^(11/2) + 6 
05*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^6*b^(9/2) - 330*(sqrt(b)*x - sqrt(b 
*x^2 + a))^4*A*a^5*b^(11/2) - 121*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^7*b^ 
(9/2) + 66*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^6*b^(11/2) + 11*B*a^8*b^(9/ 
2) - 6*A*a^7*b^(11/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^11
 

Mupad [B] (verification not implemented)

Time = 2.22 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{12}} \, dx=\frac {2\,A\,b^3\,\sqrt {b\,x^2+a}}{385\,a^2\,x^5}-\frac {4\,A\,b\,\sqrt {b\,x^2+a}}{33\,x^9}-\frac {B\,a\,\sqrt {b\,x^2+a}}{9\,x^9}-\frac {10\,B\,b\,\sqrt {b\,x^2+a}}{63\,x^7}-\frac {A\,b^2\,\sqrt {b\,x^2+a}}{231\,a\,x^7}-\frac {A\,a\,\sqrt {b\,x^2+a}}{11\,x^{11}}-\frac {8\,A\,b^4\,\sqrt {b\,x^2+a}}{1155\,a^3\,x^3}+\frac {16\,A\,b^5\,\sqrt {b\,x^2+a}}{1155\,a^4\,x}-\frac {B\,b^2\,\sqrt {b\,x^2+a}}{105\,a\,x^5}+\frac {4\,B\,b^3\,\sqrt {b\,x^2+a}}{315\,a^2\,x^3}-\frac {8\,B\,b^4\,\sqrt {b\,x^2+a}}{315\,a^3\,x} \] Input:

int(((A + B*x^2)*(a + b*x^2)^(3/2))/x^12,x)
 

Output:

(2*A*b^3*(a + b*x^2)^(1/2))/(385*a^2*x^5) - (4*A*b*(a + b*x^2)^(1/2))/(33* 
x^9) - (B*a*(a + b*x^2)^(1/2))/(9*x^9) - (10*B*b*(a + b*x^2)^(1/2))/(63*x^ 
7) - (A*b^2*(a + b*x^2)^(1/2))/(231*a*x^7) - (A*a*(a + b*x^2)^(1/2))/(11*x 
^11) - (8*A*b^4*(a + b*x^2)^(1/2))/(1155*a^3*x^3) + (16*A*b^5*(a + b*x^2)^ 
(1/2))/(1155*a^4*x) - (B*b^2*(a + b*x^2)^(1/2))/(105*a*x^5) + (4*B*b^3*(a 
+ b*x^2)^(1/2))/(315*a^2*x^3) - (8*B*b^4*(a + b*x^2)^(1/2))/(315*a^3*x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{12}} \, dx=\frac {-63 \sqrt {b \,x^{2}+a}\, a^{5}-161 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-113 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}-3 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}+4 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}-8 \sqrt {b \,x^{2}+a}\, b^{5} x^{10}+8 \sqrt {b}\, b^{5} x^{11}}{693 a^{3} x^{11}} \] Input:

int((b*x^2+a)^(3/2)*(B*x^2+A)/x^12,x)
 

Output:

( - 63*sqrt(a + b*x**2)*a**5 - 161*sqrt(a + b*x**2)*a**4*b*x**2 - 113*sqrt 
(a + b*x**2)*a**3*b**2*x**4 - 3*sqrt(a + b*x**2)*a**2*b**3*x**6 + 4*sqrt(a 
 + b*x**2)*a*b**4*x**8 - 8*sqrt(a + b*x**2)*b**5*x**10 + 8*sqrt(b)*b**5*x* 
*11)/(693*a**3*x**11)