Integrand size = 24, antiderivative size = 157 \[ \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {c^2 (b c-a d)^2 \left (c+d x^2\right )^{5/2}}{5 d^5}-\frac {2 c (b c-a d) (2 b c-a d) \left (c+d x^2\right )^{7/2}}{7 d^5}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) \left (c+d x^2\right )^{9/2}}{9 d^5}-\frac {2 b (2 b c-a d) \left (c+d x^2\right )^{11/2}}{11 d^5}+\frac {b^2 \left (c+d x^2\right )^{13/2}}{13 d^5} \] Output:
1/5*c^2*(-a*d+b*c)^2*(d*x^2+c)^(5/2)/d^5-2/7*c*(-a*d+b*c)*(-a*d+2*b*c)*(d* x^2+c)^(7/2)/d^5+1/9*(a^2*d^2-6*a*b*c*d+6*b^2*c^2)*(d*x^2+c)^(9/2)/d^5-2/1 1*b*(-a*d+2*b*c)*(d*x^2+c)^(11/2)/d^5+1/13*b^2*(d*x^2+c)^(13/2)/d^5
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85 \[ \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {\left (c+d x^2\right )^{5/2} \left (143 a^2 d^2 \left (8 c^2-20 c d x^2+35 d^2 x^4\right )+78 a b d \left (-16 c^3+40 c^2 d x^2-70 c d^2 x^4+105 d^3 x^6\right )+3 b^2 \left (128 c^4-320 c^3 d x^2+560 c^2 d^2 x^4-840 c d^3 x^6+1155 d^4 x^8\right )\right )}{45045 d^5} \] Input:
Integrate[x^5*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]
Output:
((c + d*x^2)^(5/2)*(143*a^2*d^2*(8*c^2 - 20*c*d*x^2 + 35*d^2*x^4) + 78*a*b *d*(-16*c^3 + 40*c^2*d*x^2 - 70*c*d^2*x^4 + 105*d^3*x^6) + 3*b^2*(128*c^4 - 320*c^3*d*x^2 + 560*c^2*d^2*x^4 - 840*c*d^3*x^6 + 1155*d^4*x^8)))/(45045 *d^5)
Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {354, 99, 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.
| \(\displaystyle \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int x^4 \left (b x^2+a\right )^2 \left (d x^2+c\right )^{3/2}dx^2\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {b^2 \left (d x^2+c\right )^{11/2}}{d^4}-\frac {2 b (2 b c-a d) \left (d x^2+c\right )^{9/2}}{d^4}+\frac {\left (6 b^2 c^2-6 a b d c+a^2 d^2\right ) \left (d x^2+c\right )^{7/2}}{d^4}+\frac {2 c (b c-a d) (a d-2 b c) \left (d x^2+c\right )^{5/2}}{d^4}+\frac {c^2 (b c-a d)^2 \left (d x^2+c\right )^{3/2}}{d^4}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (c+d x^2\right )^{9/2} \left (a^2 d^2-6 a b c d+6 b^2 c^2\right )}{9 d^5}+\frac {2 c^2 \left (c+d x^2\right )^{5/2} (b c-a d)^2}{5 d^5}-\frac {4 b \left (c+d x^2\right )^{11/2} (2 b c-a d)}{11 d^5}-\frac {4 c \left (c+d x^2\right )^{7/2} (b c-a d) (2 b c-a d)}{7 d^5}+\frac {2 b^2 \left (c+d x^2\right )^{13/2}}{13 d^5}\right )\) |
Input:
Int[x^5*(a + b*x^2)^2*(c + d*x^2)^(3/2),x]
Output:
((2*c^2*(b*c - a*d)^2*(c + d*x^2)^(5/2))/(5*d^5) - (4*c*(b*c - a*d)*(2*b*c - a*d)*(c + d*x^2)^(7/2))/(7*d^5) + (2*(6*b^2*c^2 - 6*a*b*c*d + a^2*d^2)* (c + d*x^2)^(9/2))/(9*d^5) - (4*b*(2*b*c - a*d)*(c + d*x^2)^(11/2))/(11*d^ 5) + (2*b^2*(c + d*x^2)^(13/2))/(13*d^5))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 = 0.59 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.76
| method | result | size |
| pseudoelliptic | \(\frac {8 \left (\frac {35 \left (\frac {9}{13} b^{2} x^{4}+\frac {18}{11} a b \,x^{2}+a^{2}\right ) x^{4} d^{4}}{8}-\frac {5 c \left (\frac {126}{143} b^{2} x^{4}+\frac {21}{11} a b \,x^{2}+a^{2}\right ) x^{2} d^{3}}{2}+c^{2} \left (\frac {210}{143} b^{2} x^{4}+\frac {30}{11} a b \,x^{2}+a^{2}\right ) d^{2}-\frac {12 \left (\frac {10 b \,x^{2}}{13}+a \right ) c^{3} b d}{11}+\frac {48 b^{2} c^{4}}{143}\right ) \left (x^{2} d +c \right )^{\frac {5}{2}}}{315 d^{5}}\) | \(120\) |
| gosper | \(\frac {\left (x^{2} d +c \right )^{\frac {5}{2}} \left (3465 b^{2} x^{8} d^{4}+8190 a b \,d^{4} x^{6}-2520 b^{2} c \,d^{3} x^{6}+5005 a^{2} d^{4} x^{4}-5460 a b c \,d^{3} x^{4}+1680 b^{2} c^{2} d^{2} x^{4}-2860 a^{2} c \,d^{3} x^{2}+3120 a b \,c^{2} d^{2} x^{2}-960 b^{2} c^{3} d \,x^{2}+1144 a^{2} c^{2} d^{2}-1248 a b \,c^{3} d +384 b^{2} c^{4}\right )}{45045 d^{5}}\) | \(149\) |
| orering | \(\frac {\left (x^{2} d +c \right )^{\frac {5}{2}} \left (3465 b^{2} x^{8} d^{4}+8190 a b \,d^{4} x^{6}-2520 b^{2} c \,d^{3} x^{6}+5005 a^{2} d^{4} x^{4}-5460 a b c \,d^{3} x^{4}+1680 b^{2} c^{2} d^{2} x^{4}-2860 a^{2} c \,d^{3} x^{2}+3120 a b \,c^{2} d^{2} x^{2}-960 b^{2} c^{3} d \,x^{2}+1144 a^{2} c^{2} d^{2}-1248 a b \,c^{3} d +384 b^{2} c^{4}\right )}{45045 d^{5}}\) | \(149\) |
| trager | \(\frac {\left (3465 d^{6} b^{2} x^{12}+8190 a b \,d^{6} x^{10}+4410 b^{2} c \,d^{5} x^{10}+5005 a^{2} d^{6} x^{8}+10920 a b c \,d^{5} x^{8}+105 b^{2} c^{2} d^{4} x^{8}+7150 a^{2} c \,d^{5} x^{6}+390 a b \,c^{2} d^{4} x^{6}-120 b^{2} c^{3} d^{3} x^{6}+429 a^{2} c^{2} d^{4} x^{4}-468 a b \,c^{3} d^{3} x^{4}+144 b^{2} c^{4} d^{2} x^{4}-572 a^{2} c^{3} d^{3} x^{2}+624 a b \,c^{4} d^{2} x^{2}-192 b^{2} c^{5} d \,x^{2}+1144 a^{2} c^{4} d^{2}-1248 a b \,c^{5} d +384 b^{2} c^{6}\right ) \sqrt {x^{2} d +c}}{45045 d^{5}}\) | \(231\) |
| risch | \(\frac {\left (3465 d^{6} b^{2} x^{12}+8190 a b \,d^{6} x^{10}+4410 b^{2} c \,d^{5} x^{10}+5005 a^{2} d^{6} x^{8}+10920 a b c \,d^{5} x^{8}+105 b^{2} c^{2} d^{4} x^{8}+7150 a^{2} c \,d^{5} x^{6}+390 a b \,c^{2} d^{4} x^{6}-120 b^{2} c^{3} d^{3} x^{6}+429 a^{2} c^{2} d^{4} x^{4}-468 a b \,c^{3} d^{3} x^{4}+144 b^{2} c^{4} d^{2} x^{4}-572 a^{2} c^{3} d^{3} x^{2}+624 a b \,c^{4} d^{2} x^{2}-192 b^{2} c^{5} d \,x^{2}+1144 a^{2} c^{4} d^{2}-1248 a b \,c^{5} d +384 b^{2} c^{6}\right ) \sqrt {x^{2} d +c}}{45045 d^{5}}\) | \(231\) |
| default | \(a^{2} \left (\frac {x^{4} \left (x^{2} d +c \right )^{\frac {5}{2}}}{9 d}-\frac {4 c \left (\frac {x^{2} \left (x^{2} d +c \right )^{\frac {5}{2}}}{7 d}-\frac {2 c \left (x^{2} d +c \right )^{\frac {5}{2}}}{35 d^{2}}\right )}{9 d}\right )+b^{2} \left (\frac {x^{8} \left (x^{2} d +c \right )^{\frac {5}{2}}}{13 d}-\frac {8 c \left (\frac {x^{6} \left (x^{2} d +c \right )^{\frac {5}{2}}}{11 d}-\frac {6 c \left (\frac {x^{4} \left (x^{2} d +c \right )^{\frac {5}{2}}}{9 d}-\frac {4 c \left (\frac {x^{2} \left (x^{2} d +c \right )^{\frac {5}{2}}}{7 d}-\frac {2 c \left (x^{2} d +c \right )^{\frac {5}{2}}}{35 d^{2}}\right )}{9 d}\right )}{11 d}\right )}{13 d}\right )+2 a b \left (\frac {x^{6} \left (x^{2} d +c \right )^{\frac {5}{2}}}{11 d}-\frac {6 c \left (\frac {x^{4} \left (x^{2} d +c \right )^{\frac {5}{2}}}{9 d}-\frac {4 c \left (\frac {x^{2} \left (x^{2} d +c \right )^{\frac {5}{2}}}{7 d}-\frac {2 c \left (x^{2} d +c \right )^{\frac {5}{2}}}{35 d^{2}}\right )}{9 d}\right )}{11 d}\right )\) | \(257\) |
Input:
int(x^5*(b*x^2+a)^2*(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
8/315*(35/8*(9/13*b^2*x^4+18/11*a*b*x^2+a^2)*x^4*d^4-5/2*c*(126/143*b^2*x^ 4+21/11*a*b*x^2+a^2)*x^2*d^3+c^2*(210/143*b^2*x^4+30/11*a*b*x^2+a^2)*d^2-1 2/11*(10/13*b*x^2+a)*c^3*b*d+48/143*b^2*c^4)*(d*x^2+c)^(5/2)/d^5
Time = 0.09 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.39 \[ \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {{\left (3465 \, b^{2} d^{6} x^{12} + 630 \, {\left (7 \, b^{2} c d^{5} + 13 \, a b d^{6}\right )} x^{10} + 35 \, {\left (3 \, b^{2} c^{2} d^{4} + 312 \, a b c d^{5} + 143 \, a^{2} d^{6}\right )} x^{8} + 384 \, b^{2} c^{6} - 1248 \, a b c^{5} d + 1144 \, a^{2} c^{4} d^{2} - 10 \, {\left (12 \, b^{2} c^{3} d^{3} - 39 \, a b c^{2} d^{4} - 715 \, a^{2} c d^{5}\right )} x^{6} + 3 \, {\left (48 \, b^{2} c^{4} d^{2} - 156 \, a b c^{3} d^{3} + 143 \, a^{2} c^{2} d^{4}\right )} x^{4} - 4 \, {\left (48 \, b^{2} c^{5} d - 156 \, a b c^{4} d^{2} + 143 \, a^{2} c^{3} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{45045 \, d^{5}} \] Input:
integrate(x^5*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
1/45045*(3465*b^2*d^6*x^12 + 630*(7*b^2*c*d^5 + 13*a*b*d^6)*x^10 + 35*(3*b ^2*c^2*d^4 + 312*a*b*c*d^5 + 143*a^2*d^6)*x^8 + 384*b^2*c^6 - 1248*a*b*c^5 *d + 1144*a^2*c^4*d^2 - 10*(12*b^2*c^3*d^3 - 39*a*b*c^2*d^4 - 715*a^2*c*d^ 5)*x^6 + 3*(48*b^2*c^4*d^2 - 156*a*b*c^3*d^3 + 143*a^2*c^2*d^4)*x^4 - 4*(4 8*b^2*c^5*d - 156*a*b*c^4*d^2 + 143*a^2*c^3*d^3)*x^2)*sqrt(d*x^2 + c)/d^5
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (148) = 296\).
Time = 0.50 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.97 \[ \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\begin {cases} \frac {8 a^{2} c^{4} \sqrt {c + d x^{2}}}{315 d^{3}} - \frac {4 a^{2} c^{3} x^{2} \sqrt {c + d x^{2}}}{315 d^{2}} + \frac {a^{2} c^{2} x^{4} \sqrt {c + d x^{2}}}{105 d} + \frac {10 a^{2} c x^{6} \sqrt {c + d x^{2}}}{63} + \frac {a^{2} d x^{8} \sqrt {c + d x^{2}}}{9} - \frac {32 a b c^{5} \sqrt {c + d x^{2}}}{1155 d^{4}} + \frac {16 a b c^{4} x^{2} \sqrt {c + d x^{2}}}{1155 d^{3}} - \frac {4 a b c^{3} x^{4} \sqrt {c + d x^{2}}}{385 d^{2}} + \frac {2 a b c^{2} x^{6} \sqrt {c + d x^{2}}}{231 d} + \frac {8 a b c x^{8} \sqrt {c + d x^{2}}}{33} + \frac {2 a b d x^{10} \sqrt {c + d x^{2}}}{11} + \frac {128 b^{2} c^{6} \sqrt {c + d x^{2}}}{15015 d^{5}} - \frac {64 b^{2} c^{5} x^{2} \sqrt {c + d x^{2}}}{15015 d^{4}} + \frac {16 b^{2} c^{4} x^{4} \sqrt {c + d x^{2}}}{5005 d^{3}} - \frac {8 b^{2} c^{3} x^{6} \sqrt {c + d x^{2}}}{3003 d^{2}} + \frac {b^{2} c^{2} x^{8} \sqrt {c + d x^{2}}}{429 d} + \frac {14 b^{2} c x^{10} \sqrt {c + d x^{2}}}{143} + \frac {b^{2} d x^{12} \sqrt {c + d x^{2}}}{13} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\frac {a^{2} x^{6}}{6} + \frac {a b x^{8}}{4} + \frac {b^{2} x^{10}}{10}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**5*(b*x**2+a)**2*(d*x**2+c)**(3/2),x)
Output:
Piecewise((8*a**2*c**4*sqrt(c + d*x**2)/(315*d**3) - 4*a**2*c**3*x**2*sqrt (c + d*x**2)/(315*d**2) + a**2*c**2*x**4*sqrt(c + d*x**2)/(105*d) + 10*a** 2*c*x**6*sqrt(c + d*x**2)/63 + a**2*d*x**8*sqrt(c + d*x**2)/9 - 32*a*b*c** 5*sqrt(c + d*x**2)/(1155*d**4) + 16*a*b*c**4*x**2*sqrt(c + d*x**2)/(1155*d **3) - 4*a*b*c**3*x**4*sqrt(c + d*x**2)/(385*d**2) + 2*a*b*c**2*x**6*sqrt( c + d*x**2)/(231*d) + 8*a*b*c*x**8*sqrt(c + d*x**2)/33 + 2*a*b*d*x**10*sqr t(c + d*x**2)/11 + 128*b**2*c**6*sqrt(c + d*x**2)/(15015*d**5) - 64*b**2*c **5*x**2*sqrt(c + d*x**2)/(15015*d**4) + 16*b**2*c**4*x**4*sqrt(c + d*x**2 )/(5005*d**3) - 8*b**2*c**3*x**6*sqrt(c + d*x**2)/(3003*d**2) + b**2*c**2* x**8*sqrt(c + d*x**2)/(429*d) + 14*b**2*c*x**10*sqrt(c + d*x**2)/143 + b** 2*d*x**12*sqrt(c + d*x**2)/13, Ne(d, 0)), (c**(3/2)*(a**2*x**6/6 + a*b*x** 8/4 + b**2*x**10/10), True))
Time = 0.04 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.59 \[ \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x^{8}}{13 \, d} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x^{6}}{143 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x^{6}}{11 \, d} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} x^{4}}{429 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c x^{4}}{33 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} x^{4}}{9 \, d} - \frac {64 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} x^{2}}{3003 \, d^{4}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} x^{2}}{231 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x^{2}}{63 \, d^{2}} + \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{4}}{15015 \, d^{5}} - \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{3}}{1155 \, d^{4}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c^{2}}{315 \, d^{3}} \] Input:
integrate(x^5*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
1/13*(d*x^2 + c)^(5/2)*b^2*x^8/d - 8/143*(d*x^2 + c)^(5/2)*b^2*c*x^6/d^2 + 2/11*(d*x^2 + c)^(5/2)*a*b*x^6/d + 16/429*(d*x^2 + c)^(5/2)*b^2*c^2*x^4/d ^3 - 4/33*(d*x^2 + c)^(5/2)*a*b*c*x^4/d^2 + 1/9*(d*x^2 + c)^(5/2)*a^2*x^4/ d - 64/3003*(d*x^2 + c)^(5/2)*b^2*c^3*x^2/d^4 + 16/231*(d*x^2 + c)^(5/2)*a *b*c^2*x^2/d^3 - 4/63*(d*x^2 + c)^(5/2)*a^2*c*x^2/d^2 + 128/15015*(d*x^2 + c)^(5/2)*b^2*c^4/d^5 - 32/1155*(d*x^2 + c)^(5/2)*a*b*c^3/d^4 + 8/315*(d*x ^2 + c)^(5/2)*a^2*c^2/d^3
Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.30 \[ \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {3465 \, {\left (d x^{2} + c\right )}^{\frac {13}{2}} b^{2} - 16380 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} b^{2} c + 30030 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} c^{2} - 25740 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{3} + 9009 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{4} + 8190 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} a b d - 30030 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a b c d + 38610 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c^{2} d - 18018 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{3} d + 5005 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a^{2} d^{2} - 12870 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} c d^{2} + 9009 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c^{2} d^{2}}{45045 \, d^{5}} \] Input:
integrate(x^5*(b*x^2+a)^2*(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
1/45045*(3465*(d*x^2 + c)^(13/2)*b^2 - 16380*(d*x^2 + c)^(11/2)*b^2*c + 30 030*(d*x^2 + c)^(9/2)*b^2*c^2 - 25740*(d*x^2 + c)^(7/2)*b^2*c^3 + 9009*(d* x^2 + c)^(5/2)*b^2*c^4 + 8190*(d*x^2 + c)^(11/2)*a*b*d - 30030*(d*x^2 + c) ^(9/2)*a*b*c*d + 38610*(d*x^2 + c)^(7/2)*a*b*c^2*d - 18018*(d*x^2 + c)^(5/ 2)*a*b*c^3*d + 5005*(d*x^2 + c)^(9/2)*a^2*d^2 - 12870*(d*x^2 + c)^(7/2)*a^ 2*c*d^2 + 9009*(d*x^2 + c)^(5/2)*a^2*c^2*d^2)/d^5
Time = 1.03 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.30 \[ \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\sqrt {d\,x^2+c}\,\left (\frac {1144\,a^2\,c^4\,d^2-1248\,a\,b\,c^5\,d+384\,b^2\,c^6}{45045\,d^5}+\frac {x^8\,\left (5005\,a^2\,d^6+10920\,a\,b\,c\,d^5+105\,b^2\,c^2\,d^4\right )}{45045\,d^5}+\frac {2\,b\,x^{10}\,\left (13\,a\,d+7\,b\,c\right )}{143}+\frac {b^2\,d\,x^{12}}{13}+\frac {2\,c\,x^6\,\left (715\,a^2\,d^2+39\,a\,b\,c\,d-12\,b^2\,c^2\right )}{9009\,d^2}+\frac {c^2\,x^4\,\left (143\,a^2\,d^2-156\,a\,b\,c\,d+48\,b^2\,c^2\right )}{15015\,d^3}-\frac {4\,c^3\,x^2\,\left (143\,a^2\,d^2-156\,a\,b\,c\,d+48\,b^2\,c^2\right )}{45045\,d^4}\right ) \] Input:
int(x^5*(a + b*x^2)^2*(c + d*x^2)^(3/2),x)
Output:
(c + d*x^2)^(1/2)*((384*b^2*c^6 + 1144*a^2*c^4*d^2 - 1248*a*b*c^5*d)/(4504 5*d^5) + (x^8*(5005*a^2*d^6 + 105*b^2*c^2*d^4 + 10920*a*b*c*d^5))/(45045*d ^5) + (2*b*x^10*(13*a*d + 7*b*c))/143 + (b^2*d*x^12)/13 + (2*c*x^6*(715*a^ 2*d^2 - 12*b^2*c^2 + 39*a*b*c*d))/(9009*d^2) + (c^2*x^4*(143*a^2*d^2 + 48* b^2*c^2 - 156*a*b*c*d))/(15015*d^3) - (4*c^3*x^2*(143*a^2*d^2 + 48*b^2*c^2 - 156*a*b*c*d))/(45045*d^4))
Time = 0.22 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.46 \[ \int x^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \left (3465 b^{2} d^{6} x^{12}+8190 a b \,d^{6} x^{10}+4410 b^{2} c \,d^{5} x^{10}+5005 a^{2} d^{6} x^{8}+10920 a b c \,d^{5} x^{8}+105 b^{2} c^{2} d^{4} x^{8}+7150 a^{2} c \,d^{5} x^{6}+390 a b \,c^{2} d^{4} x^{6}-120 b^{2} c^{3} d^{3} x^{6}+429 a^{2} c^{2} d^{4} x^{4}-468 a b \,c^{3} d^{3} x^{4}+144 b^{2} c^{4} d^{2} x^{4}-572 a^{2} c^{3} d^{3} x^{2}+624 a b \,c^{4} d^{2} x^{2}-192 b^{2} c^{5} d \,x^{2}+1144 a^{2} c^{4} d^{2}-1248 a b \,c^{5} d +384 b^{2} c^{6}\right )}{45045 d^{5}} \] Input:
int(x^5*(b*x^2+a)^2*(d*x^2+c)^(3/2),x)
Output:
(sqrt(c + d*x**2)*(1144*a**2*c**4*d**2 - 572*a**2*c**3*d**3*x**2 + 429*a** 2*c**2*d**4*x**4 + 7150*a**2*c*d**5*x**6 + 5005*a**2*d**6*x**8 - 1248*a*b* c**5*d + 624*a*b*c**4*d**2*x**2 - 468*a*b*c**3*d**3*x**4 + 390*a*b*c**2*d* *4*x**6 + 10920*a*b*c*d**5*x**8 + 8190*a*b*d**6*x**10 + 384*b**2*c**6 - 19 2*b**2*c**5*d*x**2 + 144*b**2*c**4*d**2*x**4 - 120*b**2*c**3*d**3*x**6 + 1 05*b**2*c**2*d**4*x**8 + 4410*b**2*c*d**5*x**10 + 3465*b**2*d**6*x**12))/( 45045*d**5)