Integrand size = 24, antiderivative size = 114 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=-\frac {c (b c-a d)^2 \left (c+d x^2\right )^{7/2}}{7 d^4}+\frac {(b c-a d) (3 b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^4}-\frac {b (3 b c-2 a d) \left (c+d x^2\right )^{11/2}}{11 d^4}+\frac {b^2 \left (c+d x^2\right )^{13/2}}{13 d^4} \] Output:
-1/7*c*(-a*d+b*c)^2*(d*x^2+c)^(7/2)/d^4+1/9*(-a*d+b*c)*(-a*d+3*b*c)*(d*x^2 +c)^(9/2)/d^4-1/11*b*(-2*a*d+3*b*c)*(d*x^2+c)^(11/2)/d^4+1/13*b^2*(d*x^2+c )^(13/2)/d^4
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.87 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {\left (c+d x^2\right )^{7/2} \left (143 a^2 d^2 \left (-2 c+7 d x^2\right )+26 a b d \left (8 c^2-28 c d x^2+63 d^2 x^4\right )+b^2 \left (-48 c^3+168 c^2 d x^2-378 c d^2 x^4+693 d^3 x^6\right )\right )}{9009 d^4} \] Input:
Integrate[x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]
Output:
((c + d*x^2)^(7/2)*(143*a^2*d^2*(-2*c + 7*d*x^2) + 26*a*b*d*(8*c^2 - 28*c* d*x^2 + 63*d^2*x^4) + b^2*(-48*c^3 + 168*c^2*d*x^2 - 378*c*d^2*x^4 + 693*d ^3*x^6)))/(9009*d^4)
Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {354, 86, 2009}
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.
| \(\displaystyle \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int x^2 \left (b x^2+a\right )^2 \left (d x^2+c\right )^{5/2}dx^2\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {b^2 \left (d x^2+c\right )^{11/2}}{d^3}-\frac {b (3 b c-2 a d) \left (d x^2+c\right )^{9/2}}{d^3}+\frac {(b c-a d) (3 b c-a d) \left (d x^2+c\right )^{7/2}}{d^3}-\frac {c (b c-a d)^2 \left (d x^2+c\right )^{5/2}}{d^3}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 b \left (c+d x^2\right )^{11/2} (3 b c-2 a d)}{11 d^4}+\frac {2 \left (c+d x^2\right )^{9/2} (b c-a d) (3 b c-a d)}{9 d^4}-\frac {2 c \left (c+d x^2\right )^{7/2} (b c-a d)^2}{7 d^4}+\frac {2 b^2 \left (c+d x^2\right )^{13/2}}{13 d^4}\right )\) |
Input:
Int[x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]
Output:
((-2*c*(b*c - a*d)^2*(c + d*x^2)^(7/2))/(7*d^4) + (2*(b*c - a*d)*(3*b*c - a*d)*(c + d*x^2)^(9/2))/(9*d^4) - (2*b*(3*b*c - 2*a*d)*(c + d*x^2)^(11/2)) /(11*d^4) + (2*b^2*(c + d*x^2)^(13/2))/(13*d^4))/2
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.50 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.80
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (x^{2} d +c \right )^{\frac {7}{2}} \left (\left (-\frac {63}{26} d^{3} x^{6}+\frac {189}{143} c \,d^{2} x^{4}-\frac {84}{143} c^{2} d \,x^{2}+\frac {24}{143} c^{3}\right ) b^{2}-\frac {8 a d \left (\frac {63}{8} d^{2} x^{4}-\frac {7}{2} c d \,x^{2}+c^{2}\right ) b}{11}+a^{2} d^{2} \left (-\frac {7 x^{2} d}{2}+c \right )\right )}{63 d^{4}}\) | \(91\) |
| gosper | \(-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}} \left (-693 b^{2} d^{3} x^{6}-1638 a b \,d^{3} x^{4}+378 b^{2} c \,d^{2} x^{4}-1001 a^{2} d^{3} x^{2}+728 a b c \,d^{2} x^{2}-168 b^{2} c^{2} d \,x^{2}+286 c \,a^{2} d^{2}-208 a b \,c^{2} d +48 b^{2} c^{3}\right )}{9009 d^{4}}\) | \(108\) |
| orering | \(-\frac {\left (x^{2} d +c \right )^{\frac {7}{2}} \left (-693 b^{2} d^{3} x^{6}-1638 a b \,d^{3} x^{4}+378 b^{2} c \,d^{2} x^{4}-1001 a^{2} d^{3} x^{2}+728 a b c \,d^{2} x^{2}-168 b^{2} c^{2} d \,x^{2}+286 c \,a^{2} d^{2}-208 a b \,c^{2} d +48 b^{2} c^{3}\right )}{9009 d^{4}}\) | \(108\) |
| default | \(a^{2} \left (\frac {x^{2} \left (x^{2} d +c \right )^{\frac {7}{2}}}{9 d}-\frac {2 c \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 d^{2}}\right )+b^{2} \left (\frac {x^{6} \left (x^{2} d +c \right )^{\frac {7}{2}}}{13 d}-\frac {6 c \left (\frac {x^{4} \left (x^{2} d +c \right )^{\frac {7}{2}}}{11 d}-\frac {4 c \left (\frac {x^{2} \left (x^{2} d +c \right )^{\frac {7}{2}}}{9 d}-\frac {2 c \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 d^{2}}\right )}{11 d}\right )}{13 d}\right )+2 a b \left (\frac {x^{4} \left (x^{2} d +c \right )^{\frac {7}{2}}}{11 d}-\frac {4 c \left (\frac {x^{2} \left (x^{2} d +c \right )^{\frac {7}{2}}}{9 d}-\frac {2 c \left (x^{2} d +c \right )^{\frac {7}{2}}}{63 d^{2}}\right )}{11 d}\right )\) | \(185\) |
| trager | \(-\frac {\left (-693 b^{2} d^{6} x^{12}-1638 a b \,d^{6} x^{10}-1701 b^{2} c \,d^{5} x^{10}-1001 a^{2} d^{6} x^{8}-4186 a b c \,d^{5} x^{8}-1113 b^{2} c^{2} d^{4} x^{8}-2717 a^{2} c \,d^{5} x^{6}-2938 a b \,c^{2} d^{4} x^{6}-15 b^{2} c^{3} d^{3} x^{6}-2145 a^{2} c^{2} d^{4} x^{4}-78 a b \,c^{3} d^{3} x^{4}+18 b^{2} c^{4} d^{2} x^{4}-143 a^{2} c^{3} d^{3} x^{2}+104 a b \,c^{4} d^{2} x^{2}-24 b^{2} c^{5} d \,x^{2}+286 a^{2} c^{4} d^{2}-208 a b \,c^{5} d +48 b^{2} c^{6}\right ) \sqrt {x^{2} d +c}}{9009 d^{4}}\) | \(231\) |
| risch | \(-\frac {\left (-693 b^{2} d^{6} x^{12}-1638 a b \,d^{6} x^{10}-1701 b^{2} c \,d^{5} x^{10}-1001 a^{2} d^{6} x^{8}-4186 a b c \,d^{5} x^{8}-1113 b^{2} c^{2} d^{4} x^{8}-2717 a^{2} c \,d^{5} x^{6}-2938 a b \,c^{2} d^{4} x^{6}-15 b^{2} c^{3} d^{3} x^{6}-2145 a^{2} c^{2} d^{4} x^{4}-78 a b \,c^{3} d^{3} x^{4}+18 b^{2} c^{4} d^{2} x^{4}-143 a^{2} c^{3} d^{3} x^{2}+104 a b \,c^{4} d^{2} x^{2}-24 b^{2} c^{5} d \,x^{2}+286 a^{2} c^{4} d^{2}-208 a b \,c^{5} d +48 b^{2} c^{6}\right ) \sqrt {x^{2} d +c}}{9009 d^{4}}\) | \(231\) |
Input:
int(x^3*(b*x^2+a)^2*(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/63*(d*x^2+c)^(7/2)*((-63/26*d^3*x^6+189/143*c*d^2*x^4-84/143*c^2*d*x^2+ 24/143*c^3)*b^2-8/11*a*d*(63/8*d^2*x^4-7/2*c*d*x^2+c^2)*b+a^2*d^2*(-7/2*x^ 2*d+c))/d^4
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (98) = 196\).
Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.89 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {{\left (693 \, b^{2} d^{6} x^{12} + 63 \, {\left (27 \, b^{2} c d^{5} + 26 \, a b d^{6}\right )} x^{10} + 7 \, {\left (159 \, b^{2} c^{2} d^{4} + 598 \, a b c d^{5} + 143 \, a^{2} d^{6}\right )} x^{8} - 48 \, b^{2} c^{6} + 208 \, a b c^{5} d - 286 \, a^{2} c^{4} d^{2} + {\left (15 \, b^{2} c^{3} d^{3} + 2938 \, a b c^{2} d^{4} + 2717 \, a^{2} c d^{5}\right )} x^{6} - 3 \, {\left (6 \, b^{2} c^{4} d^{2} - 26 \, a b c^{3} d^{3} - 715 \, a^{2} c^{2} d^{4}\right )} x^{4} + {\left (24 \, b^{2} c^{5} d - 104 \, a b c^{4} d^{2} + 143 \, a^{2} c^{3} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{9009 \, d^{4}} \] Input:
integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^(5/2),x, algorithm="fricas")
Output:
1/9009*(693*b^2*d^6*x^12 + 63*(27*b^2*c*d^5 + 26*a*b*d^6)*x^10 + 7*(159*b^ 2*c^2*d^4 + 598*a*b*c*d^5 + 143*a^2*d^6)*x^8 - 48*b^2*c^6 + 208*a*b*c^5*d - 286*a^2*c^4*d^2 + (15*b^2*c^3*d^3 + 2938*a*b*c^2*d^4 + 2717*a^2*c*d^5)*x ^6 - 3*(6*b^2*c^4*d^2 - 26*a*b*c^3*d^3 - 715*a^2*c^2*d^4)*x^4 + (24*b^2*c^ 5*d - 104*a*b*c^4*d^2 + 143*a^2*c^3*d^3)*x^2)*sqrt(d*x^2 + c)/d^4
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (102) = 204\).
Time = 0.62 (sec) , antiderivative size = 468, normalized size of antiderivative = 4.11 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\begin {cases} - \frac {2 a^{2} c^{4} \sqrt {c + d x^{2}}}{63 d^{2}} + \frac {a^{2} c^{3} x^{2} \sqrt {c + d x^{2}}}{63 d} + \frac {5 a^{2} c^{2} x^{4} \sqrt {c + d x^{2}}}{21} + \frac {19 a^{2} c d x^{6} \sqrt {c + d x^{2}}}{63} + \frac {a^{2} d^{2} x^{8} \sqrt {c + d x^{2}}}{9} + \frac {16 a b c^{5} \sqrt {c + d x^{2}}}{693 d^{3}} - \frac {8 a b c^{4} x^{2} \sqrt {c + d x^{2}}}{693 d^{2}} + \frac {2 a b c^{3} x^{4} \sqrt {c + d x^{2}}}{231 d} + \frac {226 a b c^{2} x^{6} \sqrt {c + d x^{2}}}{693} + \frac {46 a b c d x^{8} \sqrt {c + d x^{2}}}{99} + \frac {2 a b d^{2} x^{10} \sqrt {c + d x^{2}}}{11} - \frac {16 b^{2} c^{6} \sqrt {c + d x^{2}}}{3003 d^{4}} + \frac {8 b^{2} c^{5} x^{2} \sqrt {c + d x^{2}}}{3003 d^{3}} - \frac {2 b^{2} c^{4} x^{4} \sqrt {c + d x^{2}}}{1001 d^{2}} + \frac {5 b^{2} c^{3} x^{6} \sqrt {c + d x^{2}}}{3003 d} + \frac {53 b^{2} c^{2} x^{8} \sqrt {c + d x^{2}}}{429} + \frac {27 b^{2} c d x^{10} \sqrt {c + d x^{2}}}{143} + \frac {b^{2} d^{2} x^{12} \sqrt {c + d x^{2}}}{13} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\frac {a^{2} x^{4}}{4} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(b*x**2+a)**2*(d*x**2+c)**(5/2),x)
Output:
Piecewise((-2*a**2*c**4*sqrt(c + d*x**2)/(63*d**2) + a**2*c**3*x**2*sqrt(c + d*x**2)/(63*d) + 5*a**2*c**2*x**4*sqrt(c + d*x**2)/21 + 19*a**2*c*d*x** 6*sqrt(c + d*x**2)/63 + a**2*d**2*x**8*sqrt(c + d*x**2)/9 + 16*a*b*c**5*sq rt(c + d*x**2)/(693*d**3) - 8*a*b*c**4*x**2*sqrt(c + d*x**2)/(693*d**2) + 2*a*b*c**3*x**4*sqrt(c + d*x**2)/(231*d) + 226*a*b*c**2*x**6*sqrt(c + d*x* *2)/693 + 46*a*b*c*d*x**8*sqrt(c + d*x**2)/99 + 2*a*b*d**2*x**10*sqrt(c + d*x**2)/11 - 16*b**2*c**6*sqrt(c + d*x**2)/(3003*d**4) + 8*b**2*c**5*x**2* sqrt(c + d*x**2)/(3003*d**3) - 2*b**2*c**4*x**4*sqrt(c + d*x**2)/(1001*d** 2) + 5*b**2*c**3*x**6*sqrt(c + d*x**2)/(3003*d) + 53*b**2*c**2*x**8*sqrt(c + d*x**2)/429 + 27*b**2*c*d*x**10*sqrt(c + d*x**2)/143 + b**2*d**2*x**12* sqrt(c + d*x**2)/13, Ne(d, 0)), (c**(5/2)*(a**2*x**4/4 + a*b*x**6/3 + b**2 *x**8/8), True))
Time = 0.06 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.59 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} x^{6}}{13 \, d} - \frac {6 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c x^{4}}{143 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b x^{4}}{11 \, d} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} x^{2}}{429 \, d^{3}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c x^{2}}{99 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} x^{2}}{9 \, d} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{3}}{3003 \, d^{4}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c^{2}}{693 \, d^{3}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} c}{63 \, d^{2}} \] Input:
integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^(5/2),x, algorithm="maxima")
Output:
1/13*(d*x^2 + c)^(7/2)*b^2*x^6/d - 6/143*(d*x^2 + c)^(7/2)*b^2*c*x^4/d^2 + 2/11*(d*x^2 + c)^(7/2)*a*b*x^4/d + 8/429*(d*x^2 + c)^(7/2)*b^2*c^2*x^2/d^ 3 - 8/99*(d*x^2 + c)^(7/2)*a*b*c*x^2/d^2 + 1/9*(d*x^2 + c)^(7/2)*a^2*x^2/d - 16/3003*(d*x^2 + c)^(7/2)*b^2*c^3/d^4 + 16/693*(d*x^2 + c)^(7/2)*a*b*c^ 2/d^3 - 2/63*(d*x^2 + c)^(7/2)*a^2*c/d^2
Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.32 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {693 \, {\left (d x^{2} + c\right )}^{\frac {13}{2}} b^{2} - 2457 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} b^{2} c + 3003 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} c^{2} - 1287 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{3} + 1638 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} a b d - 4004 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a b c d + 2574 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c^{2} d + 1001 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a^{2} d^{2} - 1287 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} c d^{2}}{9009 \, d^{4}} \] Input:
integrate(x^3*(b*x^2+a)^2*(d*x^2+c)^(5/2),x, algorithm="giac")
Output:
1/9009*(693*(d*x^2 + c)^(13/2)*b^2 - 2457*(d*x^2 + c)^(11/2)*b^2*c + 3003* (d*x^2 + c)^(9/2)*b^2*c^2 - 1287*(d*x^2 + c)^(7/2)*b^2*c^3 + 1638*(d*x^2 + c)^(11/2)*a*b*d - 4004*(d*x^2 + c)^(9/2)*a*b*c*d + 2574*(d*x^2 + c)^(7/2) *a*b*c^2*d + 1001*(d*x^2 + c)^(9/2)*a^2*d^2 - 1287*(d*x^2 + c)^(7/2)*a^2*c *d^2)/d^4
Time = 1.05 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.82 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\sqrt {d\,x^2+c}\,\left (\frac {x^8\,\left (1001\,a^2\,d^6+4186\,a\,b\,c\,d^5+1113\,b^2\,c^2\,d^4\right )}{9009\,d^4}-\frac {286\,a^2\,c^4\,d^2-208\,a\,b\,c^5\,d+48\,b^2\,c^6}{9009\,d^4}+\frac {b^2\,d^2\,x^{12}}{13}+\frac {c\,x^6\,\left (2717\,a^2\,d^2+2938\,a\,b\,c\,d+15\,b^2\,c^2\right )}{9009\,d}+\frac {b\,d\,x^{10}\,\left (26\,a\,d+27\,b\,c\right )}{143}+\frac {c^3\,x^2\,\left (143\,a^2\,d^2-104\,a\,b\,c\,d+24\,b^2\,c^2\right )}{9009\,d^3}+\frac {c^2\,x^4\,\left (715\,a^2\,d^2+26\,a\,b\,c\,d-6\,b^2\,c^2\right )}{3003\,d^2}\right ) \] Input:
int(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2),x)
Output:
(c + d*x^2)^(1/2)*((x^8*(1001*a^2*d^6 + 1113*b^2*c^2*d^4 + 4186*a*b*c*d^5) )/(9009*d^4) - (48*b^2*c^6 + 286*a^2*c^4*d^2 - 208*a*b*c^5*d)/(9009*d^4) + (b^2*d^2*x^12)/13 + (c*x^6*(2717*a^2*d^2 + 15*b^2*c^2 + 2938*a*b*c*d))/(9 009*d) + (b*d*x^10*(26*a*d + 27*b*c))/143 + (c^3*x^2*(143*a^2*d^2 + 24*b^2 *c^2 - 104*a*b*c*d))/(9009*d^3) + (c^2*x^4*(715*a^2*d^2 - 6*b^2*c^2 + 26*a *b*c*d))/(3003*d^2))
Time = 0.24 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.01 \[ \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \left (693 b^{2} d^{6} x^{12}+1638 a b \,d^{6} x^{10}+1701 b^{2} c \,d^{5} x^{10}+1001 a^{2} d^{6} x^{8}+4186 a b c \,d^{5} x^{8}+1113 b^{2} c^{2} d^{4} x^{8}+2717 a^{2} c \,d^{5} x^{6}+2938 a b \,c^{2} d^{4} x^{6}+15 b^{2} c^{3} d^{3} x^{6}+2145 a^{2} c^{2} d^{4} x^{4}+78 a b \,c^{3} d^{3} x^{4}-18 b^{2} c^{4} d^{2} x^{4}+143 a^{2} c^{3} d^{3} x^{2}-104 a b \,c^{4} d^{2} x^{2}+24 b^{2} c^{5} d \,x^{2}-286 a^{2} c^{4} d^{2}+208 a b \,c^{5} d -48 b^{2} c^{6}\right )}{9009 d^{4}} \] Input:
int(x^3*(b*x^2+a)^2*(d*x^2+c)^(5/2),x)
Output:
(sqrt(c + d*x**2)*( - 286*a**2*c**4*d**2 + 143*a**2*c**3*d**3*x**2 + 2145* a**2*c**2*d**4*x**4 + 2717*a**2*c*d**5*x**6 + 1001*a**2*d**6*x**8 + 208*a* b*c**5*d - 104*a*b*c**4*d**2*x**2 + 78*a*b*c**3*d**3*x**4 + 2938*a*b*c**2* d**4*x**6 + 4186*a*b*c*d**5*x**8 + 1638*a*b*d**6*x**10 - 48*b**2*c**6 + 24 *b**2*c**5*d*x**2 - 18*b**2*c**4*d**2*x**4 + 15*b**2*c**3*d**3*x**6 + 1113 *b**2*c**2*d**4*x**8 + 1701*b**2*c*d**5*x**10 + 693*b**2*d**6*x**12))/(900 9*d**4)