\(\int \frac {(a+b x^2)^2 (c+d x^2)^{5/2}}{x^{14}} \, dx\) [877]

Optimal result

Integrand size = 24, antiderivative size = 143 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{14}} \, dx=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{13 c x^{13}}-\frac {2 a (13 b c-3 a d) \left (c+d x^2\right )^{7/2}}{143 c^2 x^{11}}-\frac {\left (143 b^2-\frac {8 a d (13 b c-3 a d)}{c^2}\right ) \left (c+d x^2\right )^{7/2}}{1287 c x^9}+\frac {2 d \left (143 b^2 c^2-8 a d (13 b c-3 a d)\right ) \left (c+d x^2\right )^{7/2}}{9009 c^4 x^7} \] Output:

-1/13*a^2*(d*x^2+c)^(7/2)/c/x^13-2/143*a*(-3*a*d+13*b*c)*(d*x^2+c)^(7/2)/c 
^2/x^11-1/1287*(143*b^2-8*a*d*(-3*a*d+13*b*c)/c^2)*(d*x^2+c)^(7/2)/c/x^9+2 
/9009*d*(143*b^2*c^2-8*a*d*(-3*a*d+13*b*c))*(d*x^2+c)^(7/2)/c^4/x^7
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{14}} \, dx=-\frac {\left (c+d x^2\right )^{7/2} \left (143 b^2 c^2 x^4 \left (7 c-2 d x^2\right )+26 a b c x^2 \left (63 c^2-28 c d x^2+8 d^2 x^4\right )+3 a^2 \left (231 c^3-126 c^2 d x^2+56 c d^2 x^4-16 d^3 x^6\right )\right )}{9009 c^4 x^{13}} \] Input:

Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^14,x]
 

Output:

-1/9009*((c + d*x^2)^(7/2)*(143*b^2*c^2*x^4*(7*c - 2*d*x^2) + 26*a*b*c*x^2 
*(63*c^2 - 28*c*d*x^2 + 8*d^2*x^4) + 3*a^2*(231*c^3 - 126*c^2*d*x^2 + 56*c 
*d^2*x^4 - 16*d^3*x^6)))/(c^4*x^13)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {365, 359, 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.

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

\(\Big \downarrow \) 365

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

\(\Big \downarrow \) 359

\(\displaystyle \frac {\frac {\left (143 b^2 c^2-8 a d (13 b c-3 a d)\right ) \int \frac {\left (d x^2+c\right )^{5/2}}{x^{10}}dx}{11 c}-\frac {2 a \left (c+d x^2\right )^{7/2} (13 b c-3 a d)}{11 c x^{11}}}{13 c}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{13 c x^{13}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\frac {\left (143 b^2 c^2-8 a d (13 b c-3 a d)\right ) \left (-\frac {2 d \int \frac {\left (d x^2+c\right )^{5/2}}{x^8}dx}{9 c}-\frac {\left (c+d x^2\right )^{7/2}}{9 c x^9}\right )}{11 c}-\frac {2 a \left (c+d x^2\right )^{7/2} (13 b c-3 a d)}{11 c x^{11}}}{13 c}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{13 c x^{13}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\frac {\left (\frac {2 d \left (c+d x^2\right )^{7/2}}{63 c^2 x^7}-\frac {\left (c+d x^2\right )^{7/2}}{9 c x^9}\right ) \left (143 b^2 c^2-8 a d (13 b c-3 a d)\right )}{11 c}-\frac {2 a \left (c+d x^2\right )^{7/2} (13 b c-3 a d)}{11 c x^{11}}}{13 c}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{13 c x^{13}}\)

Input:

Int[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^14,x]
 

Output:

-1/13*(a^2*(c + d*x^2)^(7/2))/(c*x^13) + ((-2*a*(13*b*c - 3*a*d)*(c + d*x^ 
2)^(7/2))/(11*c*x^11) + ((143*b^2*c^2 - 8*a*d*(13*b*c - 3*a*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)
 

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 = 0.74 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(-\frac {\left (\left (\frac {13}{9} b^{2} x^{4}+\frac {26}{11} a b \,x^{2}+a^{2}\right ) c^{3}-\frac {6 d \left (\frac {143}{189} b^{2} x^{4}+\frac {52}{27} a b \,x^{2}+a^{2}\right ) x^{2} c^{2}}{11}+\frac {8 a \,d^{2} x^{4} \left (\frac {26 b \,x^{2}}{21}+a \right ) c}{33}-\frac {16 a^{2} d^{3} x^{6}}{231}\right ) \left (x^{2} d +c \right )^{\frac {7}{2}}}{13 x^{13} c^{4}}\)	\(99\)
gosper	\(-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}} \left (-48 a^{2} d^{3} x^{6}+208 a b c \,d^{2} x^{6}-286 b^{2} c^{2} d \,x^{6}+168 a^{2} c \,d^{2} x^{4}-728 a b \,c^{2} d \,x^{4}+1001 b^{2} c^{3} x^{4}-378 a^{2} c^{2} d \,x^{2}+1638 a b \,c^{3} x^{2}+693 a^{2} c^{3}\right )}{9009 x^{13} c^{4}}\)	\(117\)
orering	\(-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}} \left (-48 a^{2} d^{3} x^{6}+208 a b c \,d^{2} x^{6}-286 b^{2} c^{2} d \,x^{6}+168 a^{2} c \,d^{2} x^{4}-728 a b \,c^{2} d \,x^{4}+1001 b^{2} c^{3} x^{4}-378 a^{2} c^{2} d \,x^{2}+1638 a b \,c^{3} x^{2}+693 a^{2} c^{3}\right )}{9009 x^{13} c^{4}}\)	\(117\)
default	\(a^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{13 c \,x^{13}}-\frac {6 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{11 c \,x^{11}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{9 c \,x^{9}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 c^{2} x^{7}}\right )}{11 c}\right )}{13 c}\right )+b^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{9 c \,x^{9}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 c^{2} x^{7}}\right )+2 a b \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{11 c \,x^{11}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}}}{9 c \,x^{9}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 c^{2} x^{7}}\right )}{11 c}\right )\)	\(194\)
trager	\(-\frac {\left (-48 a^{2} d^{6} x^{12}+208 a b c \,d^{5} x^{12}-286 b^{2} c^{2} d^{4} x^{12}+24 a^{2} c \,d^{5} x^{10}-104 a b \,c^{2} d^{4} x^{10}+143 b^{2} c^{3} d^{3} x^{10}-18 a^{2} c^{2} d^{4} x^{8}+78 a b \,c^{3} d^{3} x^{8}+2145 b^{2} c^{4} d^{2} x^{8}+15 a^{2} c^{3} d^{3} x^{6}+2938 a b \,c^{4} d^{2} x^{6}+2717 b^{2} c^{5} d \,x^{6}+1113 a^{2} c^{4} d^{2} x^{4}+4186 a b \,c^{5} d \,x^{4}+1001 b^{2} c^{6} x^{4}+1701 a^{2} c^{5} d \,x^{2}+1638 a b \,c^{6} x^{2}+693 a^{2} c^{6}\right ) \sqrt {x^{2} d +c}}{9009 x^{13} c^{4}}\)	\(240\)
risch	\(-\frac {\left (-48 a^{2} d^{6} x^{12}+208 a b c \,d^{5} x^{12}-286 b^{2} c^{2} d^{4} x^{12}+24 a^{2} c \,d^{5} x^{10}-104 a b \,c^{2} d^{4} x^{10}+143 b^{2} c^{3} d^{3} x^{10}-18 a^{2} c^{2} d^{4} x^{8}+78 a b \,c^{3} d^{3} x^{8}+2145 b^{2} c^{4} d^{2} x^{8}+15 a^{2} c^{3} d^{3} x^{6}+2938 a b \,c^{4} d^{2} x^{6}+2717 b^{2} c^{5} d \,x^{6}+1113 a^{2} c^{4} d^{2} x^{4}+4186 a b \,c^{5} d \,x^{4}+1001 b^{2} c^{6} x^{4}+1701 a^{2} c^{5} d \,x^{2}+1638 a b \,c^{6} x^{2}+693 a^{2} c^{6}\right ) \sqrt {x^{2} d +c}}{9009 x^{13} c^{4}}\)	\(240\)

Input:

int((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^14,x,method=_RETURNVERBOSE)
 

Output:

-1/13*((13/9*b^2*x^4+26/11*a*b*x^2+a^2)*c^3-6/11*d*(143/189*b^2*x^4+52/27* 
a*b*x^2+a^2)*x^2*c^2+8/33*a*d^2*x^4*(26/21*b*x^2+a)*c-16/231*a^2*d^3*x^6)* 
(d*x^2+c)^(7/2)/x^13/c^4
 

Fricas [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{14}} \, dx=\frac {{\left (2 \, {\left (143 \, b^{2} c^{2} d^{4} - 104 \, a b c d^{5} + 24 \, a^{2} d^{6}\right )} x^{12} - {\left (143 \, b^{2} c^{3} d^{3} - 104 \, a b c^{2} d^{4} + 24 \, a^{2} c d^{5}\right )} x^{10} - 3 \, {\left (715 \, b^{2} c^{4} d^{2} + 26 \, a b c^{3} d^{3} - 6 \, a^{2} c^{2} d^{4}\right )} x^{8} - 693 \, a^{2} c^{6} - {\left (2717 \, b^{2} c^{5} d + 2938 \, a b c^{4} d^{2} + 15 \, a^{2} c^{3} d^{3}\right )} x^{6} - 7 \, {\left (143 \, b^{2} c^{6} + 598 \, a b c^{5} d + 159 \, a^{2} c^{4} d^{2}\right )} x^{4} - 63 \, {\left (26 \, a b c^{6} + 27 \, a^{2} c^{5} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{9009 \, c^{4} x^{13}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^14,x, algorithm="fricas")
 

Output:

1/9009*(2*(143*b^2*c^2*d^4 - 104*a*b*c*d^5 + 24*a^2*d^6)*x^12 - (143*b^2*c 
^3*d^3 - 104*a*b*c^2*d^4 + 24*a^2*c*d^5)*x^10 - 3*(715*b^2*c^4*d^2 + 26*a* 
b*c^3*d^3 - 6*a^2*c^2*d^4)*x^8 - 693*a^2*c^6 - (2717*b^2*c^5*d + 2938*a*b* 
c^4*d^2 + 15*a^2*c^3*d^3)*x^6 - 7*(143*b^2*c^6 + 598*a*b*c^5*d + 159*a^2*c 
^4*d^2)*x^4 - 63*(26*a*b*c^6 + 27*a^2*c^5*d)*x^2)*sqrt(d*x^2 + c)/(c^4*x^1 
3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5618 vs. \(2 (138) = 276\).

Time = 7.66 (sec) , antiderivative size = 5618, normalized size of antiderivative = 39.29 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{14}} \, dx=\text {Too large to display} \] Input:

integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x**14,x)
 

Output:

-693*a**2*c**13*d**(51/2)*sqrt(c/(d*x**2) + 1)/(9009*c**11*d**25*x**12 + 4 
5045*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) - 3528*a**2*c**12*d**(53/ 
2)*x**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) - 7175*a**2*c**11*d**(55/2)*x**4*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) - 7290*a**2*c**10*d**(57/2)*x**6*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 + 9009 
0*c**8*d**28*x**18 + 45045*c**7*d**29*x**20 + 9009*c**6*d**30*x**22) - 630 
*a**2*c**10*d**(35/2)*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) - 3699*a**2*c**9*d**(59/2)*x**8*sqrt(c/(d*x**2) + 1)/(90 
09*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) - 
 2590*a**2*c**9*d**(37/2)*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**1 
6 + 3465*c**5*d**20*x**18) - 756*a**2*c**8*d**(61/2)*x**10*sqrt(c/(d*x**2) 
 + 1)/(9009*c**11*d**25*x**12 + 45045*c**10*d**26*x**14 + 90090*c**9*d*...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{14}} \, dx=\frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} d}{63 \, c^{2} x^{7}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d^{2}}{693 \, c^{3} x^{7}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{3}}{3003 \, c^{4} x^{7}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2}}{9 \, c x^{9}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d}{99 \, c^{2} x^{9}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2}}{429 \, c^{3} x^{9}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b}{11 \, c x^{11}} + \frac {6 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d}{143 \, c^{2} x^{11}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{13 \, c x^{13}} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^14,x, algorithm="maxima")
 

Output:

2/63*(d*x^2 + c)^(7/2)*b^2*d/(c^2*x^7) - 16/693*(d*x^2 + c)^(7/2)*a*b*d^2/ 
(c^3*x^7) + 16/3003*(d*x^2 + c)^(7/2)*a^2*d^3/(c^4*x^7) - 1/9*(d*x^2 + c)^ 
(7/2)*b^2/(c*x^9) + 8/99*(d*x^2 + c)^(7/2)*a*b*d/(c^2*x^9) - 8/429*(d*x^2 
+ c)^(7/2)*a^2*d^2/(c^3*x^9) - 2/11*(d*x^2 + c)^(7/2)*a*b/(c*x^11) + 6/143 
*(d*x^2 + c)^(7/2)*a^2*d/(c^2*x^11) - 1/13*(d*x^2 + c)^(7/2)*a^2/(c*x^13)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 935 vs. \(2 (127) = 254\).

Time = 0.16 (sec) , antiderivative size = 935, normalized size of antiderivative = 6.54 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{14}} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^14,x, algorithm="giac")
 

Output:

4/9009*(9009*(sqrt(d)*x - sqrt(d*x^2 + c))^22*b^2*d^(9/2) - 21021*(sqrt(d) 
*x - sqrt(d*x^2 + c))^20*b^2*c*d^(9/2) + 48048*(sqrt(d)*x - sqrt(d*x^2 + c 
))^20*a*b*d^(11/2) + 39039*(sqrt(d)*x - sqrt(d*x^2 + c))^18*b^2*c^2*d^(9/2 
) + 24024*(sqrt(d)*x - sqrt(d*x^2 + c))^18*a*b*c*d^(11/2) + 72072*(sqrt(d) 
*x - sqrt(d*x^2 + c))^18*a^2*d^(13/2) - 99099*(sqrt(d)*x - sqrt(d*x^2 + c) 
)^16*b^2*c^3*d^(9/2) + 72072*(sqrt(d)*x - sqrt(d*x^2 + c))^16*a*b*c^2*d^(1 
1/2) + 216216*(sqrt(d)*x - sqrt(d*x^2 + c))^16*a^2*c*d^(13/2) + 138138*(sq 
rt(d)*x - sqrt(d*x^2 + c))^14*b^2*c^4*d^(9/2) - 192192*(sqrt(d)*x - sqrt(d 
*x^2 + c))^14*a*b*c^3*d^(11/2) + 432432*(sqrt(d)*x - sqrt(d*x^2 + c))^14*a 
^2*c^2*d^(13/2) - 107250*(sqrt(d)*x - sqrt(d*x^2 + c))^12*b^2*c^5*d^(9/2) 
- 13728*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a*b*c^4*d^(11/2) + 391248*(sqrt(d 
)*x - sqrt(d*x^2 + c))^12*a^2*c^3*d^(13/2) + 84942*(sqrt(d)*x - sqrt(d*x^2 
 + c))^10*b^2*c^6*d^(9/2) - 61776*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b*c^5 
*d^(11/2) + 247104*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a^2*c^4*d^(13/2) - 632 
06*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^7*d^(9/2) + 98384*(sqrt(d)*x - sq 
rt(d*x^2 + c))^8*a*b*c^6*d^(11/2) + 54912*(sqrt(d)*x - sqrt(d*x^2 + c))^8* 
a^2*c^5*d^(13/2) + 19877*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^8*d^(9/2) + 
 18304*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b*c^7*d^(11/2) + 6864*(sqrt(d)*x 
- sqrt(d*x^2 + c))^6*a^2*c^6*d^(13/2) - 2145*(sqrt(d)*x - sqrt(d*x^2 + c)) 
^4*b^2*c^9*d^(9/2) + 8112*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^8*d^(11...
 

Mupad [B] (verification not implemented)

Time = 6.31 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.65 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{14}} \, dx=\frac {2\,a^2\,d^4\,\sqrt {d\,x^2+c}}{1001\,c^2\,x^5}-\frac {53\,a^2\,d^2\,\sqrt {d\,x^2+c}}{429\,x^9}-\frac {b^2\,c^2\,\sqrt {d\,x^2+c}}{9\,x^9}-\frac {5\,b^2\,d^2\,\sqrt {d\,x^2+c}}{21\,x^5}-\frac {5\,a^2\,d^3\,\sqrt {d\,x^2+c}}{3003\,c\,x^7}-\frac {a^2\,c^2\,\sqrt {d\,x^2+c}}{13\,x^{13}}-\frac {8\,a^2\,d^5\,\sqrt {d\,x^2+c}}{3003\,c^3\,x^3}+\frac {16\,a^2\,d^6\,\sqrt {d\,x^2+c}}{3003\,c^4\,x}-\frac {b^2\,d^3\,\sqrt {d\,x^2+c}}{63\,c\,x^3}+\frac {2\,b^2\,d^4\,\sqrt {d\,x^2+c}}{63\,c^2\,x}-\frac {2\,a\,b\,c^2\,\sqrt {d\,x^2+c}}{11\,x^{11}}-\frac {226\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{693\,x^7}-\frac {27\,a^2\,c\,d\,\sqrt {d\,x^2+c}}{143\,x^{11}}-\frac {19\,b^2\,c\,d\,\sqrt {d\,x^2+c}}{63\,x^7}-\frac {2\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{231\,c\,x^5}+\frac {8\,a\,b\,d^4\,\sqrt {d\,x^2+c}}{693\,c^2\,x^3}-\frac {16\,a\,b\,d^5\,\sqrt {d\,x^2+c}}{693\,c^3\,x}-\frac {46\,a\,b\,c\,d\,\sqrt {d\,x^2+c}}{99\,x^9} \] Input:

int(((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^14,x)
 

Output:

(2*a^2*d^4*(c + d*x^2)^(1/2))/(1001*c^2*x^5) - (53*a^2*d^2*(c + d*x^2)^(1/ 
2))/(429*x^9) - (b^2*c^2*(c + d*x^2)^(1/2))/(9*x^9) - (5*b^2*d^2*(c + d*x^ 
2)^(1/2))/(21*x^5) - (5*a^2*d^3*(c + d*x^2)^(1/2))/(3003*c*x^7) - (a^2*c^2 
*(c + d*x^2)^(1/2))/(13*x^13) - (8*a^2*d^5*(c + d*x^2)^(1/2))/(3003*c^3*x^ 
3) + (16*a^2*d^6*(c + d*x^2)^(1/2))/(3003*c^4*x) - (b^2*d^3*(c + d*x^2)^(1 
/2))/(63*c*x^3) + (2*b^2*d^4*(c + d*x^2)^(1/2))/(63*c^2*x) - (2*a*b*c^2*(c 
 + d*x^2)^(1/2))/(11*x^11) - (226*a*b*d^2*(c + d*x^2)^(1/2))/(693*x^7) - ( 
27*a^2*c*d*(c + d*x^2)^(1/2))/(143*x^11) - (19*b^2*c*d*(c + d*x^2)^(1/2))/ 
(63*x^7) - (2*a*b*d^3*(c + d*x^2)^(1/2))/(231*c*x^5) + (8*a*b*d^4*(c + d*x 
^2)^(1/2))/(693*c^2*x^3) - (16*a*b*d^5*(c + d*x^2)^(1/2))/(693*c^3*x) - (4 
6*a*b*c*d*(c + d*x^2)^(1/2))/(99*x^9)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.91 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^{14}} \, dx=\frac {-693 \sqrt {d \,x^{2}+c}\, a^{2} c^{6}-1701 \sqrt {d \,x^{2}+c}\, a^{2} c^{5} d \,x^{2}-1113 \sqrt {d \,x^{2}+c}\, a^{2} c^{4} d^{2} x^{4}-15 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} d^{3} x^{6}+18 \sqrt {d \,x^{2}+c}\, a^{2} c^{2} d^{4} x^{8}-24 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{5} x^{10}+48 \sqrt {d \,x^{2}+c}\, a^{2} d^{6} x^{12}-1638 \sqrt {d \,x^{2}+c}\, a b \,c^{6} x^{2}-4186 \sqrt {d \,x^{2}+c}\, a b \,c^{5} d \,x^{4}-2938 \sqrt {d \,x^{2}+c}\, a b \,c^{4} d^{2} x^{6}-78 \sqrt {d \,x^{2}+c}\, a b \,c^{3} d^{3} x^{8}+104 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d^{4} x^{10}-208 \sqrt {d \,x^{2}+c}\, a b c \,d^{5} x^{12}-1001 \sqrt {d \,x^{2}+c}\, b^{2} c^{6} x^{4}-2717 \sqrt {d \,x^{2}+c}\, b^{2} c^{5} d \,x^{6}-2145 \sqrt {d \,x^{2}+c}\, b^{2} c^{4} d^{2} x^{8}-143 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} d^{3} x^{10}+286 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} d^{4} x^{12}-48 \sqrt {d}\, a^{2} d^{6} x^{13}+208 \sqrt {d}\, a b c \,d^{5} x^{13}-286 \sqrt {d}\, b^{2} c^{2} d^{4} x^{13}}{9009 c^{4} x^{13}} \] Input:

int((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^14,x)
 

Output:

( - 693*sqrt(c + d*x**2)*a**2*c**6 - 1701*sqrt(c + d*x**2)*a**2*c**5*d*x** 
2 - 1113*sqrt(c + d*x**2)*a**2*c**4*d**2*x**4 - 15*sqrt(c + d*x**2)*a**2*c 
**3*d**3*x**6 + 18*sqrt(c + d*x**2)*a**2*c**2*d**4*x**8 - 24*sqrt(c + d*x* 
*2)*a**2*c*d**5*x**10 + 48*sqrt(c + d*x**2)*a**2*d**6*x**12 - 1638*sqrt(c 
+ d*x**2)*a*b*c**6*x**2 - 4186*sqrt(c + d*x**2)*a*b*c**5*d*x**4 - 2938*sqr 
t(c + d*x**2)*a*b*c**4*d**2*x**6 - 78*sqrt(c + d*x**2)*a*b*c**3*d**3*x**8 
+ 104*sqrt(c + d*x**2)*a*b*c**2*d**4*x**10 - 208*sqrt(c + d*x**2)*a*b*c*d* 
*5*x**12 - 1001*sqrt(c + d*x**2)*b**2*c**6*x**4 - 2717*sqrt(c + d*x**2)*b* 
*2*c**5*d*x**6 - 2145*sqrt(c + d*x**2)*b**2*c**4*d**2*x**8 - 143*sqrt(c + 
d*x**2)*b**2*c**3*d**3*x**10 + 286*sqrt(c + d*x**2)*b**2*c**2*d**4*x**12 - 
 48*sqrt(d)*a**2*d**6*x**13 + 208*sqrt(d)*a*b*c*d**5*x**13 - 286*sqrt(d)*b 
**2*c**2*d**4*x**13)/(9009*c**4*x**13)