3.953 \(\int (a+\frac{b}{x^2}) (c+\frac{d}{x^2})^{3/2} x^{12} \, dx\)
Optimal. Leaf size=150 \[ \frac{8 d^2 x^7 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{3003 c^4}-\frac{16 d^3 x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{15015 c^5}+\frac{x^{11} \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{143 c^2}-\frac{2 d x^9 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{429 c^3}+\frac{a x^{13} \left (c+\frac{d}{x^2}\right )^{5/2}}{13 c} \]
[Out]
(-16*d^3*(13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^5)/(15015*c^5) + (8*d^2*(13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^7)/
(3003*c^4) - (2*d*(13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^9)/(429*c^3) + ((13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^11
)/(143*c^2) + (a*(c + d/x^2)^(5/2)*x^13)/(13*c)
________________________________________________________________________________________
Rubi [A] time = 0.0717679, antiderivative size = 150, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used =
{453, 271, 264} \[ \frac{8 d^2 x^7 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{3003 c^4}-\frac{16 d^3 x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{15015 c^5}+\frac{x^{11} \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{143 c^2}-\frac{2 d x^9 \left (c+\frac{d}{x^2}\right )^{5/2} (13 b c-8 a d)}{429 c^3}+\frac{a x^{13} \left (c+\frac{d}{x^2}\right )^{5/2}}{13 c} \]
Antiderivative was successfully verified.
[In]
Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x]
[Out]
(-16*d^3*(13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^5)/(15015*c^5) + (8*d^2*(13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^7)/
(3003*c^4) - (2*d*(13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^9)/(429*c^3) + ((13*b*c - 8*a*d)*(c + d/x^2)^(5/2)*x^11
)/(143*c^2) + (a*(c + d/x^2)^(5/2)*x^13)/(13*c)
Rule 453
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] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(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]
Rule 271
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(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]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Rule 264
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]
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2} x^{12} \, dx &=\frac{a \left (c+\frac{d}{x^2}\right )^{5/2} x^{13}}{13 c}+\frac{(13 b c-8 a d) \int \left (c+\frac{d}{x^2}\right )^{3/2} x^{10} \, dx}{13 c}\\ &=\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}-\frac{(6 d (13 b c-8 a d)) \int \left (c+\frac{d}{x^2}\right )^{3/2} x^8 \, dx}{143 c^2}\\ &=-\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}+\frac{\left (8 d^2 (13 b c-8 a d)\right ) \int \left (c+\frac{d}{x^2}\right )^{3/2} x^6 \, dx}{429 c^3}\\ &=\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}-\frac{\left (16 d^3 (13 b c-8 a d)\right ) \int \left (c+\frac{d}{x^2}\right )^{3/2} x^4 \, dx}{3003 c^4}\\ &=-\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}\\ \end{align*}
Mathematica [A] time = 0.0740671, size = 110, normalized size = 0.73 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (a \left (560 c^2 d^2 x^4-840 c^3 d x^6+1155 c^4 x^8-320 c d^3 x^2+128 d^4\right )+13 b c \left (-70 c^2 d x^4+105 c^3 x^6+40 c d^2 x^2-16 d^3\right )\right )}{15015 c^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^12,x]
[Out]
(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)
________________________________________________________________________________________
Maple [A] time = 0.009, size = 115, normalized size = 0.8 \begin{align*}{\frac{{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\,ac{d}^{3}{x}^{2}+520\,b{c}^{2}{d}^{2}{x}^{2}+128\,a{d}^{4}-208\,bc{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{15015\,{c}^{5}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x)
[Out]
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
________________________________________________________________________________________
Maxima [A] time = 0.959037, size = 213, normalized size = 1.42 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x, algorithm="maxima")
[Out]
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
________________________________________________________________________________________
Fricas [A] time = 1.34779, size = 352, normalized size = 2.35 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x, algorithm="fricas")
[Out]
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] time = 19.192, size = 3351, normalized size = 22.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**12,x)
[Out]
693*a*c**12*d**(51/2)*x**22*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) + 3528*a*c**11*d**(53/2)*x**20*sq
rt(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 + 9009*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 + 1386
0*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) + 1295*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)/(900
9*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**2
9*x**2 + 9009*c**6*d**30) + 1990*a*c**7*d**(39/2)*x**14*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) - 63*a*c**6*d**(63/2)*x**10*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) + 1358*a*c**6*d**(41/2)*x**12*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) - 630*a*c**5*
d**(65/2)*x**8*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 + 9
0090*c**8*d**28*x**4 + 45045*c**7*d**29*x**2 + 9009*c**6*d**30) + 343*a*c**5*d**(43/2)*x**10*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) - 1680*a*c**4*d**(67/2)*x**6*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) + 35*a*c**4*d**(45/2)*x**8*
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) - 2016*a*c**3*d**(69/2)*x**4*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) + 280*a*c*
*3*d**(47/2)*x**6*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 + 1
3860*c**6*d**19*x**2 + 3465*c**5*d**20) - 1152*a*c**2*d**(71/2)*x**2*sqrt(c*x**2/d + 1)/(9009*c**11*d**25*x**1
0 + 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) + 560*a*c**2*d**(49/2)*x**4*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) - 256*a*c*d**(73/2)*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) + 448*a*c*d**(51/2)*x**2*sqrt(c*x**2/d + 1)/(3465*c**9*d**16*x**8 + 13860*c**8*d**17*x**6 + 2
0790*c**7*d**18*x**4 + 13860*c**6*d**19*x**2 + 3465*c**5*d**20) + 128*a*d**(53/2)*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) + 315*
b*c**10*d**(33/2)*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) + 1295*b*c**9*d**(35/2)*x**16*sqrt(c*x**2/d + 1)/(3465*c**9*d**1
6*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) + 1990*b*c**
8*d**(37/2)*x**14*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 + 1
3860*c**6*d**19*x**2 + 3465*c**5*d**20) + 1358*b*c**7*d**(39/2)*x**12*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) + 35*b*c**7*d**(21
/2)*x**14*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12)
+ 343*b*c**6*d**(41/2)*x**10*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) + 110*b*c**6*d**(23/2)*x**12*sqrt(c*x**2/d + 1)/(315*c**7*
d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 35*b*c**5*d**(43/2)*x**8*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) + 114*b*c**5*d**(25/2)*x**10*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**
5*d**11*x**2 + 315*c**4*d**12) + 280*b*c**4*d**(45/2)*x**6*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) + 40*b*c**4*d**(27/2)*x**8*sq
rt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 560*b*c**
3*d**(47/2)*x**4*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 + 13
860*c**6*d**19*x**2 + 3465*c**5*d**20) - 5*b*c**3*d**(29/2)*x**6*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*
c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 448*b*c**2*d**(49/2)*x**2*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) - 30
*b*c**2*d**(31/2)*x**4*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 31
5*c**4*d**12) + 128*b*c*d**(51/2)*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) - 40*b*c*d**(33/2)*x**2*sqrt(c*x**2/d + 1)/(315*c**7*d
**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) - 16*b*d**(35/2)*sqrt(c*x**2/d + 1)/(31
5*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12)
________________________________________________________________________________________
Giac [B] time = 1.10463, size = 440, normalized size = 2.93 \begin{align*} \frac{\frac{143 \,{\left (35 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d + 189 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{3}\right )} b d \mathrm{sgn}\left (x\right )}{c^{3}} + \frac{13 \,{\left (315 \,{\left (c x^{2} + d\right )}^{\frac{11}{2}} - 1540 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{4}\right )} b \mathrm{sgn}\left (x\right )}{c^{3}} + \frac{13 \,{\left (315 \,{\left (c x^{2} + d\right )}^{\frac{11}{2}} - 1540 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{4}\right )} a d \mathrm{sgn}\left (x\right )}{c^{4}} + \frac{5 \,{\left (693 \,{\left (c x^{2} + d\right )}^{\frac{13}{2}} - 4095 \,{\left (c x^{2} + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{5}\right )} a \mathrm{sgn}\left (x\right )}{c^{4}}}{45045 \, c} + \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^12,x, algorithm="giac")
[Out]
1/45045*(143*(35*(c*x^2 + d)^(9/2) - 135*(c*x^2 + d)^(7/2)*d + 189*(c*x^2 + d)^(5/2)*d^2 - 105*(c*x^2 + d)^(3/
2)*d^3)*b*d*sgn(x)/c^3 + 13*(315*(c*x^2 + d)^(11/2) - 1540*(c*x^2 + d)^(9/2)*d + 2970*(c*x^2 + d)^(7/2)*d^2 -
2772*(c*x^2 + d)^(5/2)*d^3 + 1155*(c*x^2 + d)^(3/2)*d^4)*b*sgn(x)/c^3 + 13*(315*(c*x^2 + d)^(11/2) - 1540*(c*x
^2 + d)^(9/2)*d + 2970*(c*x^2 + d)^(7/2)*d^2 - 2772*(c*x^2 + d)^(5/2)*d^3 + 1155*(c*x^2 + d)^(3/2)*d^4)*a*d*sg
n(x)/c^4 + 5*(693*(c*x^2 + d)^(13/2) - 4095*(c*x^2 + d)^(11/2)*d + 10010*(c*x^2 + d)^(9/2)*d^2 - 12870*(c*x^2
+ d)^(7/2)*d^3 + 9009*(c*x^2 + d)^(5/2)*d^4 - 3003*(c*x^2 + d)^(3/2)*d^5)*a*sgn(x)/c^4)/c + 16/15015*(13*b*c*d
^(11/2) - 8*a*d^(13/2))*sgn(x)/c^5