\(\int \frac {(a+b x^2)^2 (c+d x^2)^{5/2}}{x^{16}} \, dx\) [878]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{15 c x^{15}}-\frac {2 a (15 b c-4 a d) \left (c+d x^2\right )^{7/2}}{195 c^2 x^{13}}-\frac {\left (65 b^2 c^2-60 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{7/2}}{715 c^3 x^{11}}+\frac {4 d \left (65 b^2 c^2-4 a d (15 b c-4 a d)\right ) \left (c+d x^2\right )^{7/2}}{6435 c^4 x^9}-\frac {8 d^2 \left (65 b^2 c^2-4 a d (15 b c-4 a d)\right ) \left (c+d x^2\right )^{7/2}}{45045 c^5 x^7} \] Output:

-1/15*a^2*(d*x^2+c)^(7/2)/c/x^15-2/195*a*(-4*a*d+15*b*c)*(d*x^2+c)^(7/2)/c 
^2/x^13-1/715*(16*a^2*d^2-60*a*b*c*d+65*b^2*c^2)*(d*x^2+c)^(7/2)/c^3/x^11+ 
4/6435*d*(65*b^2*c^2-4*a*d*(-4*a*d+15*b*c))*(d*x^2+c)^(7/2)/c^4/x^9-8/4504 
5*d^2*(65*b^2*c^2-4*a*d*(-4*a*d+15*b*c))*(d*x^2+c)^(7/2)/c^5/x^7
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx=-\frac {\left (c+d x^2\right )^{7/2} \left (65 b^2 c^2 x^4 \left (63 c^2-28 c d x^2+8 d^2 x^4\right )+30 a b c x^2 \left (231 c^3-126 c^2 d x^2+56 c d^2 x^4-16 d^3 x^6\right )+a^2 \left (3003 c^4-1848 c^3 d x^2+1008 c^2 d^2 x^4-448 c d^3 x^6+128 d^4 x^8\right )\right )}{45045 c^5 x^{15}} \] Input:

Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^16,x]
 

Output:

-1/45045*((c + d*x^2)^(7/2)*(65*b^2*c^2*x^4*(63*c^2 - 28*c*d*x^2 + 8*d^2*x 
^4) + 30*a*b*c*x^2*(231*c^3 - 126*c^2*d*x^2 + 56*c*d^2*x^4 - 16*d^3*x^6) + 
 a^2*(3003*c^4 - 1848*c^3*d*x^2 + 1008*c^2*d^2*x^4 - 448*c*d^3*x^6 + 128*d 
^4*x^8)))/(c^5*x^15)
 

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx\)

\(\Big \downarrow \) 365

\(\displaystyle \frac {\int \frac {\left (15 b^2 c x^2+2 a (15 b c-4 a d)\right ) \left (d x^2+c\right )^{5/2}}{x^{14}}dx}{15 c}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{15 c x^{15}}\)

\(\Big \downarrow \) 359

\(\displaystyle \frac {\frac {3 \left (65 b^2 c^2-4 a d (15 b c-4 a d)\right ) \int \frac {\left (d x^2+c\right )^{5/2}}{x^{12}}dx}{13 c}-\frac {2 a \left (c+d x^2\right )^{7/2} (15 b c-4 a d)}{13 c x^{13}}}{15 c}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{15 c x^{15}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\frac {3 \left (65 b^2 c^2-4 a d (15 b c-4 a d)\right ) \left (-\frac {4 d \int \frac {\left (d x^2+c\right )^{5/2}}{x^{10}}dx}{11 c}-\frac {\left (c+d x^2\right )^{7/2}}{11 c x^{11}}\right )}{13 c}-\frac {2 a \left (c+d x^2\right )^{7/2} (15 b c-4 a d)}{13 c x^{13}}}{15 c}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{15 c x^{15}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\frac {3 \left (65 b^2 c^2-4 a d (15 b c-4 a d)\right ) \left (-\frac {4 d \left (-\frac {2 d \int \frac {\left (d x^2+c\right )^{5/2}}{x^8}dx}{9 c}-\frac {\left (c+d x^2\right )^{7/2}}{9 c x^9}\right )}{11 c}-\frac {\left (c+d x^2\right )^{7/2}}{11 c x^{11}}\right )}{13 c}-\frac {2 a \left (c+d x^2\right )^{7/2} (15 b c-4 a d)}{13 c x^{13}}}{15 c}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{15 c x^{15}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\frac {3 \left (-\frac {4 d \left (\frac {2 d \left (c+d x^2\right )^{7/2}}{63 c^2 x^7}-\frac {\left (c+d x^2\right )^{7/2}}{9 c x^9}\right )}{11 c}-\frac {\left (c+d x^2\right )^{7/2}}{11 c x^{11}}\right ) \left (65 b^2 c^2-4 a d (15 b c-4 a d)\right )}{13 c}-\frac {2 a \left (c+d x^2\right )^{7/2} (15 b c-4 a d)}{13 c x^{13}}}{15 c}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{15 c x^{15}}\)

Input:

Int[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^16,x]
 

Output:

-1/15*(a^2*(c + d*x^2)^(7/2))/(c*x^15) + ((-2*a*(15*b*c - 4*a*d)*(c + d*x^ 
2)^(7/2))/(13*c*x^13) + (3*(65*b^2*c^2 - 4*a*d*(15*b*c - 4*a*d))*(-1/11*(c 
 + d*x^2)^(7/2)/(c*x^11) - (4*d*(-1/9*(c + d*x^2)^(7/2)/(c*x^9) + (2*d*(c 
+ d*x^2)^(7/2))/(63*c^2*x^7)))/(11*c)))/(13*c))/(15*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 = 0.85 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(-\frac {\left (\left (\frac {15}{11} b^{2} x^{4}+\frac {30}{13} a b \,x^{2}+a^{2}\right ) c^{4}-\frac {8 d \left (\frac {65}{66} b^{2} x^{4}+\frac {45}{22} a b \,x^{2}+a^{2}\right ) x^{2} c^{3}}{13}+\frac {48 \left (\frac {65}{126} b^{2} x^{4}+\frac {5}{3} a b \,x^{2}+a^{2}\right ) d^{2} x^{4} c^{2}}{143}-\frac {64 a \,d^{3} \left (\frac {15 b \,x^{2}}{14}+a \right ) x^{6} c}{429}+\frac {128 a^{2} d^{4} x^{8}}{3003}\right ) \left (x^{2} d +c \right )^{\frac {7}{2}}}{15 x^{15} c^{5}}\)	\(129\)
gosper	\(-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}} \left (128 a^{2} d^{4} x^{8}-480 a b c \,d^{3} x^{8}+520 b^{2} c^{2} d^{2} x^{8}-448 a^{2} c \,d^{3} x^{6}+1680 a b \,c^{2} d^{2} x^{6}-1820 b^{2} c^{3} d \,x^{6}+1008 a^{2} c^{2} d^{2} x^{4}-3780 a b \,c^{3} d \,x^{4}+4095 b^{2} c^{4} x^{4}-1848 a^{2} c^{3} d \,x^{2}+6930 a b \,c^{4} x^{2}+3003 a^{2} c^{4}\right )}{45045 x^{15} c^{5}}\)	\(158\)
orering	\(-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}} \left (128 a^{2} d^{4} x^{8}-480 a b c \,d^{3} x^{8}+520 b^{2} c^{2} d^{2} x^{8}-448 a^{2} c \,d^{3} x^{6}+1680 a b \,c^{2} d^{2} x^{6}-1820 b^{2} c^{3} d \,x^{6}+1008 a^{2} c^{2} d^{2} x^{4}-3780 a b \,c^{3} d \,x^{4}+4095 b^{2} c^{4} x^{4}-1848 a^{2} c^{3} d \,x^{2}+6930 a b \,c^{4} x^{2}+3003 a^{2} c^{4}\right )}{45045 x^{15} c^{5}}\)	\(158\)
default	\(a^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{15 c \,x^{15}}-\frac {8 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{13 c \,x^{13}}-\frac {6 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{11 c \,x^{11}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{9 c \,x^{9}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 c^{2} x^{7}}\right )}{11 c}\right )}{13 c}\right )}{15 c}\right )+b^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{11 c \,x^{11}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{9 c \,x^{9}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 c^{2} x^{7}}\right )}{11 c}\right )+2 a b \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{13 c \,x^{13}}-\frac {6 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{11 c \,x^{11}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{9 c \,x^{9}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 c^{2} x^{7}}\right )}{11 c}\right )}{13 c}\right )\)	\(266\)
trager	\(-\frac {\left (128 a^{2} d^{7} x^{14}-480 a b c \,d^{6} x^{14}+520 b^{2} c^{2} d^{5} x^{14}-64 a^{2} c \,d^{6} x^{12}+240 a b \,c^{2} d^{5} x^{12}-260 b^{2} c^{3} d^{4} x^{12}+48 a^{2} c^{2} d^{5} x^{10}-180 a b \,c^{3} d^{4} x^{10}+195 b^{2} c^{4} d^{3} x^{10}-40 a^{2} c^{3} d^{4} x^{8}+150 a b \,c^{4} d^{3} x^{8}+7345 b^{2} c^{5} d^{2} x^{8}+35 a^{2} c^{4} d^{3} x^{6}+11130 a b \,c^{5} d^{2} x^{6}+10465 b^{2} c^{6} d \,x^{6}+4473 a^{2} c^{5} d^{2} x^{4}+17010 a b \,c^{6} d \,x^{4}+4095 b^{2} c^{7} x^{4}+7161 a^{2} c^{6} d \,x^{2}+6930 a b \,c^{7} x^{2}+3003 a^{2} c^{7}\right ) \sqrt {x^{2} d +c}}{45045 x^{15} c^{5}}\)	\(281\)
risch	\(-\frac {\left (128 a^{2} d^{7} x^{14}-480 a b c \,d^{6} x^{14}+520 b^{2} c^{2} d^{5} x^{14}-64 a^{2} c \,d^{6} x^{12}+240 a b \,c^{2} d^{5} x^{12}-260 b^{2} c^{3} d^{4} x^{12}+48 a^{2} c^{2} d^{5} x^{10}-180 a b \,c^{3} d^{4} x^{10}+195 b^{2} c^{4} d^{3} x^{10}-40 a^{2} c^{3} d^{4} x^{8}+150 a b \,c^{4} d^{3} x^{8}+7345 b^{2} c^{5} d^{2} x^{8}+35 a^{2} c^{4} d^{3} x^{6}+11130 a b \,c^{5} d^{2} x^{6}+10465 b^{2} c^{6} d \,x^{6}+4473 a^{2} c^{5} d^{2} x^{4}+17010 a b \,c^{6} d \,x^{4}+4095 b^{2} c^{7} x^{4}+7161 a^{2} c^{6} d \,x^{2}+6930 a b \,c^{7} x^{2}+3003 a^{2} c^{7}\right ) \sqrt {x^{2} d +c}}{45045 x^{15} c^{5}}\)	\(281\)

Input:

int((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^16,x,method=_RETURNVERBOSE)
 

Output:

-1/15*((15/11*b^2*x^4+30/13*a*b*x^2+a^2)*c^4-8/13*d*(65/66*b^2*x^4+45/22*a 
*b*x^2+a^2)*x^2*c^3+48/143*(65/126*b^2*x^4+5/3*a*b*x^2+a^2)*d^2*x^4*c^2-64 
/429*a*d^3*(15/14*b*x^2+a)*x^6*c+128/3003*a^2*d^4*x^8)*(d*x^2+c)^(7/2)/x^1 
5/c^5
 

Fricas [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx=-\frac {{\left (8 \, {\left (65 \, b^{2} c^{2} d^{5} - 60 \, a b c d^{6} + 16 \, a^{2} d^{7}\right )} x^{14} - 4 \, {\left (65 \, b^{2} c^{3} d^{4} - 60 \, a b c^{2} d^{5} + 16 \, a^{2} c d^{6}\right )} x^{12} + 3 \, {\left (65 \, b^{2} c^{4} d^{3} - 60 \, a b c^{3} d^{4} + 16 \, a^{2} c^{2} d^{5}\right )} x^{10} + 3003 \, a^{2} c^{7} + 5 \, {\left (1469 \, b^{2} c^{5} d^{2} + 30 \, a b c^{4} d^{3} - 8 \, a^{2} c^{3} d^{4}\right )} x^{8} + 35 \, {\left (299 \, b^{2} c^{6} d + 318 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )} x^{6} + 63 \, {\left (65 \, b^{2} c^{7} + 270 \, a b c^{6} d + 71 \, a^{2} c^{5} d^{2}\right )} x^{4} + 231 \, {\left (30 \, a b c^{7} + 31 \, a^{2} c^{6} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{45045 \, c^{5} x^{15}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^16,x, algorithm="fricas")
 

Output:

-1/45045*(8*(65*b^2*c^2*d^5 - 60*a*b*c*d^6 + 16*a^2*d^7)*x^14 - 4*(65*b^2* 
c^3*d^4 - 60*a*b*c^2*d^5 + 16*a^2*c*d^6)*x^12 + 3*(65*b^2*c^4*d^3 - 60*a*b 
*c^3*d^4 + 16*a^2*c^2*d^5)*x^10 + 3003*a^2*c^7 + 5*(1469*b^2*c^5*d^2 + 30* 
a*b*c^4*d^3 - 8*a^2*c^3*d^4)*x^8 + 35*(299*b^2*c^6*d + 318*a*b*c^5*d^2 + a 
^2*c^4*d^3)*x^6 + 63*(65*b^2*c^7 + 270*a*b*c^6*d + 71*a^2*c^5*d^2)*x^4 + 2 
31*(30*a*b*c^7 + 31*a^2*c^6*d)*x^2)*sqrt(d*x^2 + c)/(c^5*x^15)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8189 vs. \(2 (190) = 380\).

Time = 10.20 (sec) , antiderivative size = 8189, normalized size of antiderivative = 43.10 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx=\text {Too large to display} \] Input:

integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x**16,x)
 

Output:

-3003*a**2*c**15*d**(73/2)*sqrt(c/(d*x**2) + 1)/(45045*c**13*d**36*x**14 + 
 270270*c**12*d**37*x**16 + 675675*c**11*d**38*x**18 + 900900*c**10*d**39* 
x**20 + 675675*c**9*d**40*x**22 + 270270*c**8*d**41*x**24 + 45045*c**7*d** 
42*x**26) - 18249*a**2*c**14*d**(75/2)*x**2*sqrt(c/(d*x**2) + 1)/(45045*c* 
*13*d**36*x**14 + 270270*c**12*d**37*x**16 + 675675*c**11*d**38*x**18 + 90 
0900*c**10*d**39*x**20 + 675675*c**9*d**40*x**22 + 270270*c**8*d**41*x**24 
 + 45045*c**7*d**42*x**26) - 46179*a**2*c**13*d**(77/2)*x**4*sqrt(c/(d*x** 
2) + 1)/(45045*c**13*d**36*x**14 + 270270*c**12*d**37*x**16 + 675675*c**11 
*d**38*x**18 + 900900*c**10*d**39*x**20 + 675675*c**9*d**40*x**22 + 270270 
*c**8*d**41*x**24 + 45045*c**7*d**42*x**26) - 62293*a**2*c**12*d**(79/2)*x 
**6*sqrt(c/(d*x**2) + 1)/(45045*c**13*d**36*x**14 + 270270*c**12*d**37*x** 
16 + 675675*c**11*d**38*x**18 + 900900*c**10*d**39*x**20 + 675675*c**9*d** 
40*x**22 + 270270*c**8*d**41*x**24 + 45045*c**7*d**42*x**26) - 1386*a**2*c 
**12*d**(53/2)*sqrt(c/(d*x**2) + 1)/(9009*c**11*d**25*x**12 + 45045*c**10* 
d**26*x**14 + 90090*c**9*d**27*x**16 + 90090*c**8*d**28*x**18 + 45045*c**7 
*d**29*x**20 + 9009*c**6*d**30*x**22) - 47245*a**2*c**11*d**(81/2)*x**8*sq 
rt(c/(d*x**2) + 1)/(45045*c**13*d**36*x**14 + 270270*c**12*d**37*x**16 + 6 
75675*c**11*d**38*x**18 + 900900*c**10*d**39*x**20 + 675675*c**9*d**40*x** 
22 + 270270*c**8*d**41*x**24 + 45045*c**7*d**42*x**26) - 7056*a**2*c**11*d 
**(55/2)*x**2*sqrt(c/(d*x**2) + 1)/(9009*c**11*d**25*x**12 + 45045*c**1...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx=-\frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} d^{2}}{693 \, c^{3} x^{7}} + \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d^{3}}{3003 \, c^{4} x^{7}} - \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{4}}{45045 \, c^{5} x^{7}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} d}{99 \, c^{2} x^{9}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d^{2}}{429 \, c^{3} x^{9}} + \frac {64 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{3}}{6435 \, c^{4} x^{9}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2}}{11 \, c x^{11}} + \frac {12 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d}{143 \, c^{2} x^{11}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2}}{715 \, c^{3} x^{11}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b}{13 \, c x^{13}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d}{195 \, c^{2} x^{13}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{15 \, c x^{15}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^16,x, algorithm="maxima")
 

Output:

-8/693*(d*x^2 + c)^(7/2)*b^2*d^2/(c^3*x^7) + 32/3003*(d*x^2 + c)^(7/2)*a*b 
*d^3/(c^4*x^7) - 128/45045*(d*x^2 + c)^(7/2)*a^2*d^4/(c^5*x^7) + 4/99*(d*x 
^2 + c)^(7/2)*b^2*d/(c^2*x^9) - 16/429*(d*x^2 + c)^(7/2)*a*b*d^2/(c^3*x^9) 
 + 64/6435*(d*x^2 + c)^(7/2)*a^2*d^3/(c^4*x^9) - 1/11*(d*x^2 + c)^(7/2)*b^ 
2/(c*x^11) + 12/143*(d*x^2 + c)^(7/2)*a*b*d/(c^2*x^11) - 16/715*(d*x^2 + c 
)^(7/2)*a^2*d^2/(c^3*x^11) - 2/13*(d*x^2 + c)^(7/2)*a*b/(c*x^13) + 8/195*( 
d*x^2 + c)^(7/2)*a^2*d/(c^2*x^13) - 1/15*(d*x^2 + c)^(7/2)*a^2/(c*x^15)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1024 vs. \(2 (170) = 340\).

Time = 0.16 (sec) , antiderivative size = 1024, normalized size of antiderivative = 5.39 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^16,x, algorithm="giac")
 

Output:

16/45045*(30030*(sqrt(d)*x - sqrt(d*x^2 + c))^24*b^2*d^(11/2) - 45045*(sqr 
t(d)*x - sqrt(d*x^2 + c))^22*b^2*c*d^(11/2) + 180180*(sqrt(d)*x - sqrt(d*x 
^2 + c))^22*a*b*d^(13/2) + 45045*(sqrt(d)*x - sqrt(d*x^2 + c))^20*b^2*c^2* 
d^(11/2) + 180180*(sqrt(d)*x - sqrt(d*x^2 + c))^20*a*b*c*d^(13/2) + 288288 
*(sqrt(d)*x - sqrt(d*x^2 + c))^20*a^2*d^(15/2) - 195195*(sqrt(d)*x - sqrt( 
d*x^2 + c))^18*b^2*c^3*d^(11/2) + 180180*(sqrt(d)*x - sqrt(d*x^2 + c))^18* 
a*b*c^2*d^(13/2) + 960960*(sqrt(d)*x - sqrt(d*x^2 + c))^18*a^2*c*d^(15/2) 
+ 276705*(sqrt(d)*x - sqrt(d*x^2 + c))^16*b^2*c^4*d^(11/2) - 643500*(sqrt( 
d)*x - sqrt(d*x^2 + c))^16*a*b*c^3*d^(13/2) + 1853280*(sqrt(d)*x - sqrt(d* 
x^2 + c))^16*a^2*c^2*d^(15/2) - 141570*(sqrt(d)*x - sqrt(d*x^2 + c))^14*b^ 
2*c^5*d^(11/2) - 257400*(sqrt(d)*x - sqrt(d*x^2 + c))^14*a*b*c^4*d^(13/2) 
+ 1750320*(sqrt(d)*x - sqrt(d*x^2 + c))^14*a^2*c^3*d^(15/2) + 130130*(sqrt 
(d)*x - sqrt(d*x^2 + c))^12*b^2*c^6*d^(11/2) - 120120*(sqrt(d)*x - sqrt(d* 
x^2 + c))^12*a*b*c^5*d^(13/2) + 1041040*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a 
^2*c^4*d^(15/2) - 150150*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^2*c^7*d^(11/2) 
 + 360360*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b*c^6*d^(13/2) + 240240*(sqrt 
(d)*x - sqrt(d*x^2 + c))^10*a^2*c^5*d^(15/2) + 43680*(sqrt(d)*x - sqrt(d*x 
^2 + c))^8*b^2*c^8*d^(11/2) + 98280*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c^ 
7*d^(13/2) + 21840*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*c^6*d^(15/2) + 455* 
(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^9*d^(11/2) + 27300*(sqrt(d)*x - s...
 

Mupad [B] (verification not implemented)

Time = 7.81 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx=\frac {8\,a^2\,d^4\,\sqrt {d\,x^2+c}}{9009\,c^2\,x^7}-\frac {71\,a^2\,d^2\,\sqrt {d\,x^2+c}}{715\,x^{11}}-\frac {b^2\,c^2\,\sqrt {d\,x^2+c}}{11\,x^{11}}-\frac {113\,b^2\,d^2\,\sqrt {d\,x^2+c}}{693\,x^7}-\frac {a^2\,d^3\,\sqrt {d\,x^2+c}}{1287\,c\,x^9}-\frac {a^2\,c^2\,\sqrt {d\,x^2+c}}{15\,x^{15}}-\frac {16\,a^2\,d^5\,\sqrt {d\,x^2+c}}{15015\,c^3\,x^5}+\frac {64\,a^2\,d^6\,\sqrt {d\,x^2+c}}{45045\,c^4\,x^3}-\frac {128\,a^2\,d^7\,\sqrt {d\,x^2+c}}{45045\,c^5\,x}-\frac {b^2\,d^3\,\sqrt {d\,x^2+c}}{231\,c\,x^5}+\frac {4\,b^2\,d^4\,\sqrt {d\,x^2+c}}{693\,c^2\,x^3}-\frac {8\,b^2\,d^5\,\sqrt {d\,x^2+c}}{693\,c^3\,x}-\frac {2\,a\,b\,c^2\,\sqrt {d\,x^2+c}}{13\,x^{13}}-\frac {106\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{429\,x^9}-\frac {31\,a^2\,c\,d\,\sqrt {d\,x^2+c}}{195\,x^{13}}-\frac {23\,b^2\,c\,d\,\sqrt {d\,x^2+c}}{99\,x^9}-\frac {10\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{3003\,c\,x^7}+\frac {4\,a\,b\,d^4\,\sqrt {d\,x^2+c}}{1001\,c^2\,x^5}-\frac {16\,a\,b\,d^5\,\sqrt {d\,x^2+c}}{3003\,c^3\,x^3}+\frac {32\,a\,b\,d^6\,\sqrt {d\,x^2+c}}{3003\,c^4\,x}-\frac {54\,a\,b\,c\,d\,\sqrt {d\,x^2+c}}{143\,x^{11}} \] Input:

int(((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^16,x)
 

Output:

(8*a^2*d^4*(c + d*x^2)^(1/2))/(9009*c^2*x^7) - (71*a^2*d^2*(c + d*x^2)^(1/ 
2))/(715*x^11) - (b^2*c^2*(c + d*x^2)^(1/2))/(11*x^11) - (113*b^2*d^2*(c + 
 d*x^2)^(1/2))/(693*x^7) - (a^2*d^3*(c + d*x^2)^(1/2))/(1287*c*x^9) - (a^2 
*c^2*(c + d*x^2)^(1/2))/(15*x^15) - (16*a^2*d^5*(c + d*x^2)^(1/2))/(15015* 
c^3*x^5) + (64*a^2*d^6*(c + d*x^2)^(1/2))/(45045*c^4*x^3) - (128*a^2*d^7*( 
c + d*x^2)^(1/2))/(45045*c^5*x) - (b^2*d^3*(c + d*x^2)^(1/2))/(231*c*x^5) 
+ (4*b^2*d^4*(c + d*x^2)^(1/2))/(693*c^2*x^3) - (8*b^2*d^5*(c + d*x^2)^(1/ 
2))/(693*c^3*x) - (2*a*b*c^2*(c + d*x^2)^(1/2))/(13*x^13) - (106*a*b*d^2*( 
c + d*x^2)^(1/2))/(429*x^9) - (31*a^2*c*d*(c + d*x^2)^(1/2))/(195*x^13) - 
(23*b^2*c*d*(c + d*x^2)^(1/2))/(99*x^9) - (10*a*b*d^3*(c + d*x^2)^(1/2))/( 
3003*c*x^7) + (4*a*b*d^4*(c + d*x^2)^(1/2))/(1001*c^2*x^5) - (16*a*b*d^5*( 
c + d*x^2)^(1/2))/(3003*c^3*x^3) + (32*a*b*d^6*(c + d*x^2)^(1/2))/(3003*c^ 
4*x) - (54*a*b*c*d*(c + d*x^2)^(1/2))/(143*x^11)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.53 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{16}} \, dx=\frac {-3003 \sqrt {d \,x^{2}+c}\, a^{2} c^{7}-7161 \sqrt {d \,x^{2}+c}\, a^{2} c^{6} d \,x^{2}-4473 \sqrt {d \,x^{2}+c}\, a^{2} c^{5} d^{2} x^{4}-35 \sqrt {d \,x^{2}+c}\, a^{2} c^{4} d^{3} x^{6}+40 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} d^{4} x^{8}-48 \sqrt {d \,x^{2}+c}\, a^{2} c^{2} d^{5} x^{10}+64 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{6} x^{12}-128 \sqrt {d \,x^{2}+c}\, a^{2} d^{7} x^{14}-6930 \sqrt {d \,x^{2}+c}\, a b \,c^{7} x^{2}-17010 \sqrt {d \,x^{2}+c}\, a b \,c^{6} d \,x^{4}-11130 \sqrt {d \,x^{2}+c}\, a b \,c^{5} d^{2} x^{6}-150 \sqrt {d \,x^{2}+c}\, a b \,c^{4} d^{3} x^{8}+180 \sqrt {d \,x^{2}+c}\, a b \,c^{3} d^{4} x^{10}-240 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d^{5} x^{12}+480 \sqrt {d \,x^{2}+c}\, a b c \,d^{6} x^{14}-4095 \sqrt {d \,x^{2}+c}\, b^{2} c^{7} x^{4}-10465 \sqrt {d \,x^{2}+c}\, b^{2} c^{6} d \,x^{6}-7345 \sqrt {d \,x^{2}+c}\, b^{2} c^{5} d^{2} x^{8}-195 \sqrt {d \,x^{2}+c}\, b^{2} c^{4} d^{3} x^{10}+260 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} d^{4} x^{12}-520 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} d^{5} x^{14}+128 \sqrt {d}\, a^{2} d^{7} x^{15}-480 \sqrt {d}\, a b c \,d^{6} x^{15}+520 \sqrt {d}\, b^{2} c^{2} d^{5} x^{15}}{45045 c^{5} x^{15}} \] Input:

int((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^16,x)
 

Output:

( - 3003*sqrt(c + d*x**2)*a**2*c**7 - 7161*sqrt(c + d*x**2)*a**2*c**6*d*x* 
*2 - 4473*sqrt(c + d*x**2)*a**2*c**5*d**2*x**4 - 35*sqrt(c + d*x**2)*a**2* 
c**4*d**3*x**6 + 40*sqrt(c + d*x**2)*a**2*c**3*d**4*x**8 - 48*sqrt(c + d*x 
**2)*a**2*c**2*d**5*x**10 + 64*sqrt(c + d*x**2)*a**2*c*d**6*x**12 - 128*sq 
rt(c + d*x**2)*a**2*d**7*x**14 - 6930*sqrt(c + d*x**2)*a*b*c**7*x**2 - 170 
10*sqrt(c + d*x**2)*a*b*c**6*d*x**4 - 11130*sqrt(c + d*x**2)*a*b*c**5*d**2 
*x**6 - 150*sqrt(c + d*x**2)*a*b*c**4*d**3*x**8 + 180*sqrt(c + d*x**2)*a*b 
*c**3*d**4*x**10 - 240*sqrt(c + d*x**2)*a*b*c**2*d**5*x**12 + 480*sqrt(c + 
 d*x**2)*a*b*c*d**6*x**14 - 4095*sqrt(c + d*x**2)*b**2*c**7*x**4 - 10465*s 
qrt(c + d*x**2)*b**2*c**6*d*x**6 - 7345*sqrt(c + d*x**2)*b**2*c**5*d**2*x* 
*8 - 195*sqrt(c + d*x**2)*b**2*c**4*d**3*x**10 + 260*sqrt(c + d*x**2)*b**2 
*c**3*d**4*x**12 - 520*sqrt(c + d*x**2)*b**2*c**2*d**5*x**14 + 128*sqrt(d) 
*a**2*d**7*x**15 - 480*sqrt(d)*a*b*c*d**6*x**15 + 520*sqrt(d)*b**2*c**2*d* 
*5*x**15)/(45045*c**5*x**15)