\(\int \frac {(a+b x^2)^2 \sqrt {c+d x^2}}{x^{12}} \, dx\) [843]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{231 c^3 x^7}+\frac {4 d \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}-\frac {8 d^2 \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (c+d x^2\right )^{3/2}}{3465 c^5 x^3} \] Output:

-1/11*a^2*(d*x^2+c)^(3/2)/c/x^11-2/99*a*(-4*a*d+11*b*c)*(d*x^2+c)^(3/2)/c^ 
2/x^9-1/231*(16*a^2*d^2-44*a*b*c*d+33*b^2*c^2)*(d*x^2+c)^(3/2)/c^3/x^7+4/1 
155*d*(33*b^2*c^2-4*a*d*(-4*a*d+11*b*c))*(d*x^2+c)^(3/2)/c^4/x^5-8/3465*d^ 
2*(33*b^2*c^2-4*a*d*(-4*a*d+11*b*c))*(d*x^2+c)^(3/2)/c^5/x^3
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {\left (c+d x^2\right )^{3/2} \left (33 b^2 c^2 x^4 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+22 a b c x^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )+a^2 \left (315 c^4-280 c^3 d x^2+240 c^2 d^2 x^4-192 c d^3 x^6+128 d^4 x^8\right )\right )}{3465 c^5 x^{11}} \] Input:

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

Output:

-1/3465*((c + d*x^2)^(3/2)*(33*b^2*c^2*x^4*(15*c^2 - 12*c*d*x^2 + 8*d^2*x^ 
4) + 22*a*b*c*x^2*(35*c^3 - 30*c^2*d*x^2 + 24*c*d^2*x^4 - 16*d^3*x^6) + a^ 
2*(315*c^4 - 280*c^3*d*x^2 + 240*c^2*d^2*x^4 - 192*c*d^3*x^6 + 128*d^4*x^8 
)))/(c^5*x^11)
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {365, 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.

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

\(\Big \downarrow \) 365

\(\displaystyle \frac {\int \frac {\left (11 b^2 c x^2+2 a (11 b c-4 a d)\right ) \sqrt {d x^2+c}}{x^{10}}dx}{11 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}\)

\(\Big \downarrow \) 359

\(\displaystyle \frac {\frac {\left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \int \frac {\sqrt {d x^2+c}}{x^8}dx}{3 c}-\frac {2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{9 c x^9}}{11 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\frac {\left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (-\frac {4 d \int \frac {\sqrt {d x^2+c}}{x^6}dx}{7 c}-\frac {\left (c+d x^2\right )^{3/2}}{7 c x^7}\right )}{3 c}-\frac {2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{9 c x^9}}{11 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\frac {\left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (-\frac {4 d \left (-\frac {2 d \int \frac {\sqrt {d x^2+c}}{x^4}dx}{5 c}-\frac {\left (c+d x^2\right )^{3/2}}{5 c x^5}\right )}{7 c}-\frac {\left (c+d x^2\right )^{3/2}}{7 c x^7}\right )}{3 c}-\frac {2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{9 c x^9}}{11 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\frac {\left (-\frac {4 d \left (\frac {2 d \left (c+d x^2\right )^{3/2}}{15 c^2 x^3}-\frac {\left (c+d x^2\right )^{3/2}}{5 c x^5}\right )}{7 c}-\frac {\left (c+d x^2\right )^{3/2}}{7 c x^7}\right ) \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{3 c}-\frac {2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{9 c x^9}}{11 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}\)

Input:

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

Output:

-1/11*(a^2*(c + d*x^2)^(3/2))/(c*x^11) + ((-2*a*(11*b*c - 4*a*d)*(c + d*x^ 
2)^(3/2))/(9*c*x^9) + ((33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(-1/7*(c + d* 
x^2)^(3/2)/(c*x^7) - (4*d*(-1/5*(c + d*x^2)^(3/2)/(c*x^5) + (2*d*(c + d*x^ 
2)^(3/2))/(15*c^2*x^3)))/(7*c)))/(3*c))/(11*c)
 

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]
 

rule 365

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(-\frac {\left (\left (\frac {11}{7} b^{2} x^{4}+\frac {22}{9} a b \,x^{2}+a^{2}\right ) c^{4}-\frac {8 \left (\frac {99}{70} b^{2} x^{4}+\frac {33}{14} a b \,x^{2}+a^{2}\right ) d \,x^{2} c^{3}}{9}+\frac {16 d^{2} \left (\frac {11}{10} b^{2} x^{4}+\frac {11}{5} a b \,x^{2}+a^{2}\right ) x^{4} c^{2}}{21}-\frac {64 a \,d^{3} \left (\frac {11 b \,x^{2}}{6}+a \right ) x^{6} c}{105}+\frac {128 a^{2} d^{4} x^{8}}{315}\right ) \left (x^{2} d +c \right )^{\frac {3}{2}}}{11 x^{11} c^{5}}\)	\(129\)
gosper	\(-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}} \left (128 a^{2} d^{4} x^{8}-352 a b c \,d^{3} x^{8}+264 b^{2} c^{2} d^{2} x^{8}-192 a^{2} c \,d^{3} x^{6}+528 a b \,c^{2} d^{2} x^{6}-396 b^{2} c^{3} d \,x^{6}+240 a^{2} c^{2} d^{2} x^{4}-660 a b \,c^{3} d \,x^{4}+495 b^{2} c^{4} x^{4}-280 a^{2} c^{3} d \,x^{2}+770 a b \,c^{4} x^{2}+315 a^{2} c^{4}\right )}{3465 x^{11} c^{5}}\)	\(158\)
orering	\(-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}} \left (128 a^{2} d^{4} x^{8}-352 a b c \,d^{3} x^{8}+264 b^{2} c^{2} d^{2} x^{8}-192 a^{2} c \,d^{3} x^{6}+528 a b \,c^{2} d^{2} x^{6}-396 b^{2} c^{3} d \,x^{6}+240 a^{2} c^{2} d^{2} x^{4}-660 a b \,c^{3} d \,x^{4}+495 b^{2} c^{4} x^{4}-280 a^{2} c^{3} d \,x^{2}+770 a b \,c^{4} x^{2}+315 a^{2} c^{4}\right )}{3465 x^{11} c^{5}}\)	\(158\)
trager	\(-\frac {\left (128 a^{2} d^{5} x^{10}-352 a b c \,d^{4} x^{10}+264 b^{2} c^{2} d^{3} x^{10}-64 a^{2} c \,d^{4} x^{8}+176 a b \,c^{2} d^{3} x^{8}-132 b^{2} c^{3} d^{2} x^{8}+48 a^{2} c^{2} d^{3} x^{6}-132 a b \,c^{3} d^{2} x^{6}+99 b^{2} c^{4} d \,x^{6}-40 a^{2} c^{3} d^{2} x^{4}+110 a b \,c^{4} d \,x^{4}+495 b^{2} c^{5} x^{4}+35 a^{2} c^{4} d \,x^{2}+770 a b \,c^{5} x^{2}+315 a^{2} c^{5}\right ) \sqrt {x^{2} d +c}}{3465 x^{11} c^{5}}\)	\(199\)
risch	\(-\frac {\left (128 a^{2} d^{5} x^{10}-352 a b c \,d^{4} x^{10}+264 b^{2} c^{2} d^{3} x^{10}-64 a^{2} c \,d^{4} x^{8}+176 a b \,c^{2} d^{3} x^{8}-132 b^{2} c^{3} d^{2} x^{8}+48 a^{2} c^{2} d^{3} x^{6}-132 a b \,c^{3} d^{2} x^{6}+99 b^{2} c^{4} d \,x^{6}-40 a^{2} c^{3} d^{2} x^{4}+110 a b \,c^{4} d \,x^{4}+495 b^{2} c^{5} x^{4}+35 a^{2} c^{4} d \,x^{2}+770 a b \,c^{5} x^{2}+315 a^{2} c^{5}\right ) \sqrt {x^{2} d +c}}{3465 x^{11} c^{5}}\)	\(199\)
default	\(a^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{11 c \,x^{11}}-\frac {8 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{9 c \,x^{9}}-\frac {2 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )}{3 c}\right )}{11 c}\right )+b^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )+2 a b \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{9 c \,x^{9}}-\frac {2 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )}{3 c}\right )\)	\(266\)

Input:

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x,method=_RETURNVERBOSE)
 

Output:

-1/11*((11/7*b^2*x^4+22/9*a*b*x^2+a^2)*c^4-8/9*(99/70*b^2*x^4+33/14*a*b*x^ 
2+a^2)*d*x^2*c^3+16/21*d^2*(11/10*b^2*x^4+11/5*a*b*x^2+a^2)*x^4*c^2-64/105 
*a*d^3*(11/6*b*x^2+a)*x^6*c+128/315*a^2*d^4*x^8)*(d*x^2+c)^(3/2)/x^11/c^5
 

Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {{\left (8 \, {\left (33 \, b^{2} c^{2} d^{3} - 44 \, a b c d^{4} + 16 \, a^{2} d^{5}\right )} x^{10} - 4 \, {\left (33 \, b^{2} c^{3} d^{2} - 44 \, a b c^{2} d^{3} + 16 \, a^{2} c d^{4}\right )} x^{8} + 315 \, a^{2} c^{5} + 3 \, {\left (33 \, b^{2} c^{4} d - 44 \, a b c^{3} d^{2} + 16 \, a^{2} c^{2} d^{3}\right )} x^{6} + 5 \, {\left (99 \, b^{2} c^{5} + 22 \, a b c^{4} d - 8 \, a^{2} c^{3} d^{2}\right )} x^{4} + 35 \, {\left (22 \, a b c^{5} + a^{2} c^{4} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3465 \, c^{5} x^{11}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x, algorithm="fricas")
 

Output:

-1/3465*(8*(33*b^2*c^2*d^3 - 44*a*b*c*d^4 + 16*a^2*d^5)*x^10 - 4*(33*b^2*c 
^3*d^2 - 44*a*b*c^2*d^3 + 16*a^2*c*d^4)*x^8 + 315*a^2*c^5 + 3*(33*b^2*c^4* 
d - 44*a*b*c^3*d^2 + 16*a^2*c^2*d^3)*x^6 + 5*(99*b^2*c^5 + 22*a*b*c^4*d - 
8*a^2*c^3*d^2)*x^4 + 35*(22*a*b*c^5 + a^2*c^4*d)*x^2)*sqrt(d*x^2 + c)/(c^5 
*x^11)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1856 vs. \(2 (190) = 380\).

Time = 2.76 (sec) , antiderivative size = 1856, normalized size of antiderivative = 9.77 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=\text {Too large to display} \] Input:

integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**12,x)
 

Output:

-315*a**2*c**9*d**(33/2)*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 138 
60*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 34 
65*c**5*d**20*x**18) - 1295*a**2*c**8*d**(35/2)*x**2*sqrt(c/(d*x**2) + 1)/ 
(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 
 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 1990*a**2*c**7*d**(37/2 
)*x**4*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**1 
2 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**1 
8) - 1358*a**2*c**6*d**(39/2)*x**6*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x 
**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19* 
x**16 + 3465*c**5*d**20*x**18) - 343*a**2*c**5*d**(41/2)*x**8*sqrt(c/(d*x* 
*2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**1 
8*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 35*a**2*c**4*d 
**(43/2)*x**10*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d* 
*17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d* 
*20*x**18) - 280*a**2*c**3*d**(45/2)*x**12*sqrt(c/(d*x**2) + 1)/(3465*c**9 
*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c** 
6*d**19*x**16 + 3465*c**5*d**20*x**18) - 560*a**2*c**2*d**(47/2)*x**14*sqr 
t(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790* 
c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 448*a 
**2*c*d**(49/2)*x**16*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 138...
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{2}}{105 \, c^{3} x^{3}} + \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{3}}{315 \, c^{4} x^{3}} - \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{4}}{3465 \, c^{5} x^{3}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{35 \, c^{2} x^{5}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{105 \, c^{3} x^{5}} + \frac {64 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{1155 \, c^{4} x^{5}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{7 \, c x^{7}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{21 \, c^{2} x^{7}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{231 \, c^{3} x^{7}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{9 \, c x^{9}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{99 \, c^{2} x^{9}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{11 \, c x^{11}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x, algorithm="maxima")
 

Output:

-8/105*(d*x^2 + c)^(3/2)*b^2*d^2/(c^3*x^3) + 32/315*(d*x^2 + c)^(3/2)*a*b* 
d^3/(c^4*x^3) - 128/3465*(d*x^2 + c)^(3/2)*a^2*d^4/(c^5*x^3) + 4/35*(d*x^2 
 + c)^(3/2)*b^2*d/(c^2*x^5) - 16/105*(d*x^2 + c)^(3/2)*a*b*d^2/(c^3*x^5) + 
 64/1155*(d*x^2 + c)^(3/2)*a^2*d^3/(c^4*x^5) - 1/7*(d*x^2 + c)^(3/2)*b^2/( 
c*x^7) + 4/21*(d*x^2 + c)^(3/2)*a*b*d/(c^2*x^7) - 16/231*(d*x^2 + c)^(3/2) 
*a^2*d^2/(c^3*x^7) - 2/9*(d*x^2 + c)^(3/2)*a*b/(c*x^9) + 8/99*(d*x^2 + c)^ 
(3/2)*a^2*d/(c^2*x^9) - 1/11*(d*x^2 + c)^(3/2)*a^2/(c*x^11)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (170) = 340\).

Time = 0.14 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.52 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x, algorithm="giac")
 

Output:

16/3465*(2310*(sqrt(d)*x - sqrt(d*x^2 + c))^16*b^2*d^(7/2) - 8085*(sqrt(d) 
*x - sqrt(d*x^2 + c))^14*b^2*c*d^(7/2) + 13860*(sqrt(d)*x - sqrt(d*x^2 + c 
))^14*a*b*d^(9/2) + 9933*(sqrt(d)*x - sqrt(d*x^2 + c))^12*b^2*c^2*d^(7/2) 
- 19404*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a*b*c*d^(9/2) + 22176*(sqrt(d)*x 
- sqrt(d*x^2 + c))^12*a^2*d^(11/2) - 5313*(sqrt(d)*x - sqrt(d*x^2 + c))^10 
*b^2*c^3*d^(7/2) + 924*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b*c^2*d^(9/2) + 
14784*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a^2*c*d^(11/2) + 2805*(sqrt(d)*x - 
sqrt(d*x^2 + c))^8*b^2*c^4*d^(7/2) - 660*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a 
*b*c^3*d^(9/2) + 5280*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*c^2*d^(11/2) - 3 
135*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^5*d^(7/2) + 7260*(sqrt(d)*x - sq 
rt(d*x^2 + c))^6*a*b*c^4*d^(9/2) - 2640*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^ 
2*c^3*d^(11/2) + 1815*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^6*d^(7/2) - 24 
20*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^5*d^(9/2) + 880*(sqrt(d)*x - sqrt 
(d*x^2 + c))^4*a^2*c^4*d^(11/2) - 363*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2* 
c^7*d^(7/2) + 484*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^6*d^(9/2) - 176*(s 
qrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^5*d^(11/2) + 33*b^2*c^8*d^(7/2) - 44*a 
*b*c^7*d^(9/2) + 16*a^2*c^6*d^(11/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c 
)^11
 

Mupad [B] (verification not implemented)

Time = 2.84 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=\frac {8\,a^2\,d^2\,\sqrt {d\,x^2+c}}{693\,c^2\,x^7}-\frac {b^2\,\sqrt {d\,x^2+c}}{7\,x^7}-\frac {2\,a\,b\,\sqrt {d\,x^2+c}}{9\,x^9}-\frac {a^2\,\sqrt {d\,x^2+c}}{11\,x^{11}}-\frac {16\,a^2\,d^3\,\sqrt {d\,x^2+c}}{1155\,c^3\,x^5}+\frac {64\,a^2\,d^4\,\sqrt {d\,x^2+c}}{3465\,c^4\,x^3}-\frac {128\,a^2\,d^5\,\sqrt {d\,x^2+c}}{3465\,c^5\,x}+\frac {4\,b^2\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^3}-\frac {8\,b^2\,d^3\,\sqrt {d\,x^2+c}}{105\,c^3\,x}-\frac {a^2\,d\,\sqrt {d\,x^2+c}}{99\,c\,x^9}-\frac {b^2\,d\,\sqrt {d\,x^2+c}}{35\,c\,x^5}+\frac {4\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^5}-\frac {16\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{315\,c^3\,x^3}+\frac {32\,a\,b\,d^4\,\sqrt {d\,x^2+c}}{315\,c^4\,x}-\frac {2\,a\,b\,d\,\sqrt {d\,x^2+c}}{63\,c\,x^7} \] Input:

int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/x^12,x)
 

Output:

(8*a^2*d^2*(c + d*x^2)^(1/2))/(693*c^2*x^7) - (b^2*(c + d*x^2)^(1/2))/(7*x 
^7) - (2*a*b*(c + d*x^2)^(1/2))/(9*x^9) - (a^2*(c + d*x^2)^(1/2))/(11*x^11 
) - (16*a^2*d^3*(c + d*x^2)^(1/2))/(1155*c^3*x^5) + (64*a^2*d^4*(c + d*x^2 
)^(1/2))/(3465*c^4*x^3) - (128*a^2*d^5*(c + d*x^2)^(1/2))/(3465*c^5*x) + ( 
4*b^2*d^2*(c + d*x^2)^(1/2))/(105*c^2*x^3) - (8*b^2*d^3*(c + d*x^2)^(1/2)) 
/(105*c^3*x) - (a^2*d*(c + d*x^2)^(1/2))/(99*c*x^9) - (b^2*d*(c + d*x^2)^( 
1/2))/(35*c*x^5) + (4*a*b*d^2*(c + d*x^2)^(1/2))/(105*c^2*x^5) - (16*a*b*d 
^3*(c + d*x^2)^(1/2))/(315*c^3*x^3) + (32*a*b*d^4*(c + d*x^2)^(1/2))/(315* 
c^4*x) - (2*a*b*d*(c + d*x^2)^(1/2))/(63*c*x^7)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=\frac {-315 \sqrt {d \,x^{2}+c}\, a^{2} c^{5}-35 \sqrt {d \,x^{2}+c}\, a^{2} c^{4} d \,x^{2}+40 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} d^{2} x^{4}-48 \sqrt {d \,x^{2}+c}\, a^{2} c^{2} d^{3} x^{6}+64 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{4} x^{8}-128 \sqrt {d \,x^{2}+c}\, a^{2} d^{5} x^{10}-770 \sqrt {d \,x^{2}+c}\, a b \,c^{5} x^{2}-110 \sqrt {d \,x^{2}+c}\, a b \,c^{4} d \,x^{4}+132 \sqrt {d \,x^{2}+c}\, a b \,c^{3} d^{2} x^{6}-176 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d^{3} x^{8}+352 \sqrt {d \,x^{2}+c}\, a b c \,d^{4} x^{10}-495 \sqrt {d \,x^{2}+c}\, b^{2} c^{5} x^{4}-99 \sqrt {d \,x^{2}+c}\, b^{2} c^{4} d \,x^{6}+132 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} d^{2} x^{8}-264 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} d^{3} x^{10}+128 \sqrt {d}\, a^{2} d^{5} x^{11}-352 \sqrt {d}\, a b c \,d^{4} x^{11}+264 \sqrt {d}\, b^{2} c^{2} d^{3} x^{11}}{3465 c^{5} x^{11}} \] Input:

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x)
 

Output:

( - 315*sqrt(c + d*x**2)*a**2*c**5 - 35*sqrt(c + d*x**2)*a**2*c**4*d*x**2 
+ 40*sqrt(c + d*x**2)*a**2*c**3*d**2*x**4 - 48*sqrt(c + d*x**2)*a**2*c**2* 
d**3*x**6 + 64*sqrt(c + d*x**2)*a**2*c*d**4*x**8 - 128*sqrt(c + d*x**2)*a* 
*2*d**5*x**10 - 770*sqrt(c + d*x**2)*a*b*c**5*x**2 - 110*sqrt(c + d*x**2)* 
a*b*c**4*d*x**4 + 132*sqrt(c + d*x**2)*a*b*c**3*d**2*x**6 - 176*sqrt(c + d 
*x**2)*a*b*c**2*d**3*x**8 + 352*sqrt(c + d*x**2)*a*b*c*d**4*x**10 - 495*sq 
rt(c + d*x**2)*b**2*c**5*x**4 - 99*sqrt(c + d*x**2)*b**2*c**4*d*x**6 + 132 
*sqrt(c + d*x**2)*b**2*c**3*d**2*x**8 - 264*sqrt(c + d*x**2)*b**2*c**2*d** 
3*x**10 + 128*sqrt(d)*a**2*d**5*x**11 - 352*sqrt(d)*a*b*c*d**4*x**11 + 264 
*sqrt(d)*b**2*c**2*d**3*x**11)/(3465*c**5*x**11)