Integrand size = 24, antiderivative size = 236 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx=-\frac {a^2 \left (c+d x^2\right )^{5/2}}{15 c x^{15}}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{5/2}}{39 c^2 x^{13}}-\frac {\left (39 b^2 c^2-48 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{5/2}}{429 c^3 x^{11}}+\frac {2 d \left (39 b^2 c^2-16 a d (3 b c-a d)\right ) \left (c+d x^2\right )^{5/2}}{1287 c^4 x^9}-\frac {8 d^2 \left (39 b^2 c^2-16 a d (3 b c-a d)\right ) \left (c+d x^2\right )^{5/2}}{9009 c^5 x^7}+\frac {16 d^3 \left (39 b^2 c^2-16 a d (3 b c-a d)\right ) \left (c+d x^2\right )^{5/2}}{45045 c^6 x^5} \] Output:
-1/15*a^2*(d*x^2+c)^(5/2)/c/x^15-2/39*a*(-a*d+3*b*c)*(d*x^2+c)^(5/2)/c^2/x ^13-1/429*(16*a^2*d^2-48*a*b*c*d+39*b^2*c^2)*(d*x^2+c)^(5/2)/c^3/x^11+2/12 87*d*(39*b^2*c^2-16*a*d*(-a*d+3*b*c))*(d*x^2+c)^(5/2)/c^4/x^9-8/9009*d^2*( 39*b^2*c^2-16*a*d*(-a*d+3*b*c))*(d*x^2+c)^(5/2)/c^5/x^7+16/45045*d^3*(39*b ^2*c^2-16*a*d*(-a*d+3*b*c))*(d*x^2+c)^(5/2)/c^6/x^5
Time = 0.45 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx=-\frac {\left (c+d x^2\right )^{5/2} \left (39 b^2 c^2 x^4 \left (105 c^3-70 c^2 d x^2+40 c d^2 x^4-16 d^3 x^6\right )+6 a b c x^2 \left (1155 c^4-840 c^3 d x^2+560 c^2 d^2 x^4-320 c d^3 x^6+128 d^4 x^8\right )+a^2 \left (3003 c^5-2310 c^4 d x^2+1680 c^3 d^2 x^4-1120 c^2 d^3 x^6+640 c d^4 x^8-256 d^5 x^{10}\right )\right )}{45045 c^6 x^{15}} \] Input:
Integrate[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^16,x]
Output:
-1/45045*((c + d*x^2)^(5/2)*(39*b^2*c^2*x^4*(105*c^3 - 70*c^2*d*x^2 + 40*c *d^2*x^4 - 16*d^3*x^6) + 6*a*b*c*x^2*(1155*c^4 - 840*c^3*d*x^2 + 560*c^2*d ^2*x^4 - 320*c*d^3*x^6 + 128*d^4*x^8) + a^2*(3003*c^5 - 2310*c^4*d*x^2 + 1 680*c^3*d^2*x^4 - 1120*c^2*d^3*x^6 + 640*c*d^4*x^8 - 256*d^5*x^10)))/(c^6* x^15)
Time = 0.30 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.83, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {365, 27, 359, 245, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {\int \frac {5 \left (3 b^2 c x^2+2 a (3 b c-a d)\right ) \left (d x^2+c\right )^{3/2}}{x^{14}}dx}{15 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{15 c x^{15}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (3 b^2 c x^2+2 a (3 b c-a d)\right ) \left (d x^2+c\right )^{3/2}}{x^{14}}dx}{3 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{15 c x^{15}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {\frac {\left (39 b^2 c^2-16 a d (3 b c-a d)\right ) \int \frac {\left (d x^2+c\right )^{3/2}}{x^{12}}dx}{13 c}-\frac {2 a \left (c+d x^2\right )^{5/2} (3 b c-a d)}{13 c x^{13}}}{3 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{15 c x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {\left (39 b^2 c^2-16 a d (3 b c-a d)\right ) \left (-\frac {6 d \int \frac {\left (d x^2+c\right )^{3/2}}{x^{10}}dx}{11 c}-\frac {\left (c+d x^2\right )^{5/2}}{11 c x^{11}}\right )}{13 c}-\frac {2 a \left (c+d x^2\right )^{5/2} (3 b c-a d)}{13 c x^{13}}}{3 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{15 c x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {\left (39 b^2 c^2-16 a d (3 b c-a d)\right ) \left (-\frac {6 d \left (-\frac {4 d \int \frac {\left (d x^2+c\right )^{3/2}}{x^8}dx}{9 c}-\frac {\left (c+d x^2\right )^{5/2}}{9 c x^9}\right )}{11 c}-\frac {\left (c+d x^2\right )^{5/2}}{11 c x^{11}}\right )}{13 c}-\frac {2 a \left (c+d x^2\right )^{5/2} (3 b c-a d)}{13 c x^{13}}}{3 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{15 c x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {\left (39 b^2 c^2-16 a d (3 b c-a d)\right ) \left (-\frac {6 d \left (-\frac {4 d \left (-\frac {2 d \int \frac {\left (d x^2+c\right )^{3/2}}{x^6}dx}{7 c}-\frac {\left (c+d x^2\right )^{5/2}}{7 c x^7}\right )}{9 c}-\frac {\left (c+d x^2\right )^{5/2}}{9 c x^9}\right )}{11 c}-\frac {\left (c+d x^2\right )^{5/2}}{11 c x^{11}}\right )}{13 c}-\frac {2 a \left (c+d x^2\right )^{5/2} (3 b c-a d)}{13 c x^{13}}}{3 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{15 c x^{15}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {\frac {\left (-\frac {6 d \left (-\frac {4 d \left (\frac {2 d \left (c+d x^2\right )^{5/2}}{35 c^2 x^5}-\frac {\left (c+d x^2\right )^{5/2}}{7 c x^7}\right )}{9 c}-\frac {\left (c+d x^2\right )^{5/2}}{9 c x^9}\right )}{11 c}-\frac {\left (c+d x^2\right )^{5/2}}{11 c x^{11}}\right ) \left (39 b^2 c^2-16 a d (3 b c-a d)\right )}{13 c}-\frac {2 a \left (c+d x^2\right )^{5/2} (3 b c-a d)}{13 c x^{13}}}{3 c}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{15 c x^{15}}\) |
Input:
Int[((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^16,x]
Output:
-1/15*(a^2*(c + d*x^2)^(5/2))/(c*x^15) + ((-2*a*(3*b*c - a*d)*(c + d*x^2)^ (5/2))/(13*c*x^13) + ((39*b^2*c^2 - 16*a*d*(3*b*c - a*d))*(-1/11*(c + d*x^ 2)^(5/2)/(c*x^11) - (6*d*(-1/9*(c + d*x^2)^(5/2)/(c*x^9) - (4*d*(-1/7*(c + d*x^2)^(5/2)/(c*x^7) + (2*d*(c + d*x^2)^(5/2))/(35*c^2*x^5)))/(9*c)))/(11 *c)))/(13*c))/(3*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Time = 0.65 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (\frac {15}{11} b^{2} x^{4}+\frac {30}{13} a b \,x^{2}+a^{2}\right ) c^{5}-\frac {10 \left (b \,x^{2}+a \right ) d \left (\frac {13 b \,x^{2}}{11}+a \right ) x^{2} c^{4}}{13}+\frac {80 d^{2} \left (\frac {13}{14} b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) x^{4} c^{3}}{143}-\frac {160 d^{3} \left (\frac {39}{70} b^{2} x^{4}+\frac {12}{7} a b \,x^{2}+a^{2}\right ) x^{6} c^{2}}{429}+\frac {640 a \,d^{4} x^{8} \left (\frac {6 b \,x^{2}}{5}+a \right ) c}{3003}-\frac {256 a^{2} d^{5} x^{10}}{3003}\right ) \left (x^{2} d +c \right )^{\frac {5}{2}}}{15 x^{15} c^{6}}\) | \(155\) |
| gosper | \(-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}} \left (-256 a^{2} d^{5} x^{10}+768 a b c \,d^{4} x^{10}-624 b^{2} c^{2} d^{3} x^{10}+640 a^{2} c \,d^{4} x^{8}-1920 a b \,c^{2} d^{3} x^{8}+1560 b^{2} c^{3} d^{2} x^{8}-1120 a^{2} c^{2} d^{3} x^{6}+3360 a b \,c^{3} d^{2} x^{6}-2730 b^{2} c^{4} d \,x^{6}+1680 a^{2} c^{3} d^{2} x^{4}-5040 a b \,c^{4} d \,x^{4}+4095 b^{2} c^{5} x^{4}-2310 a^{2} c^{4} d \,x^{2}+6930 a b \,c^{5} x^{2}+3003 a^{2} c^{5}\right )}{45045 x^{15} c^{6}}\) | \(199\) |
| orering | \(-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}} \left (-256 a^{2} d^{5} x^{10}+768 a b c \,d^{4} x^{10}-624 b^{2} c^{2} d^{3} x^{10}+640 a^{2} c \,d^{4} x^{8}-1920 a b \,c^{2} d^{3} x^{8}+1560 b^{2} c^{3} d^{2} x^{8}-1120 a^{2} c^{2} d^{3} x^{6}+3360 a b \,c^{3} d^{2} x^{6}-2730 b^{2} c^{4} d \,x^{6}+1680 a^{2} c^{3} d^{2} x^{4}-5040 a b \,c^{4} d \,x^{4}+4095 b^{2} c^{5} x^{4}-2310 a^{2} c^{4} d \,x^{2}+6930 a b \,c^{5} x^{2}+3003 a^{2} c^{5}\right )}{45045 x^{15} c^{6}}\) | \(199\) |
| trager | \(-\frac {\left (-256 a^{2} d^{7} x^{14}+768 a b c \,d^{6} x^{14}-624 b^{2} c^{2} d^{5} x^{14}+128 a^{2} c \,d^{6} x^{12}-384 a b \,c^{2} d^{5} x^{12}+312 b^{2} c^{3} d^{4} x^{12}-96 a^{2} c^{2} d^{5} x^{10}+288 a b \,c^{3} d^{4} x^{10}-234 b^{2} c^{4} d^{3} x^{10}+80 a^{2} c^{3} d^{4} x^{8}-240 a b \,c^{4} d^{3} x^{8}+195 b^{2} c^{5} d^{2} x^{8}-70 a^{2} c^{4} d^{3} x^{6}+210 a b \,c^{5} d^{2} x^{6}+5460 b^{2} c^{6} d \,x^{6}+63 a^{2} c^{5} d^{2} x^{4}+8820 a b \,c^{6} d \,x^{4}+4095 b^{2} c^{7} x^{4}+3696 a^{2} c^{6} d \,x^{2}+6930 a b \,c^{7} x^{2}+3003 a^{2} c^{7}\right ) \sqrt {x^{2} d +c}}{45045 x^{15} c^{6}}\) | \(281\) |
| risch | \(-\frac {\left (-256 a^{2} d^{7} x^{14}+768 a b c \,d^{6} x^{14}-624 b^{2} c^{2} d^{5} x^{14}+128 a^{2} c \,d^{6} x^{12}-384 a b \,c^{2} d^{5} x^{12}+312 b^{2} c^{3} d^{4} x^{12}-96 a^{2} c^{2} d^{5} x^{10}+288 a b \,c^{3} d^{4} x^{10}-234 b^{2} c^{4} d^{3} x^{10}+80 a^{2} c^{3} d^{4} x^{8}-240 a b \,c^{4} d^{3} x^{8}+195 b^{2} c^{5} d^{2} x^{8}-70 a^{2} c^{4} d^{3} x^{6}+210 a b \,c^{5} d^{2} x^{6}+5460 b^{2} c^{6} d \,x^{6}+63 a^{2} c^{5} d^{2} x^{4}+8820 a b \,c^{6} d \,x^{4}+4095 b^{2} c^{7} x^{4}+3696 a^{2} c^{6} d \,x^{2}+6930 a b \,c^{7} x^{2}+3003 a^{2} c^{7}\right ) \sqrt {x^{2} d +c}}{45045 x^{15} c^{6}}\) | \(281\) |
| default | \(a^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{15 c \,x^{15}}-\frac {2 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{13 c \,x^{13}}-\frac {8 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{11 c \,x^{11}}-\frac {6 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{9 c \,x^{9}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{7 c \,x^{7}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {5}{2}}}{35 c^{2} x^{5}}\right )}{9 c}\right )}{11 c}\right )}{13 c}\right )}{3 c}\right )+b^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{11 c \,x^{11}}-\frac {6 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{9 c \,x^{9}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{7 c \,x^{7}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {5}{2}}}{35 c^{2} x^{5}}\right )}{9 c}\right )}{11 c}\right )+2 a b \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{13 c \,x^{13}}-\frac {8 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{11 c \,x^{11}}-\frac {6 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{9 c \,x^{9}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {5}{2}}}{7 c \,x^{7}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {5}{2}}}{35 c^{2} x^{5}}\right )}{9 c}\right )}{11 c}\right )}{13 c}\right )\) | \(338\) |
Input:
int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^16,x,method=_RETURNVERBOSE)
Output:
-1/15*((15/11*b^2*x^4+30/13*a*b*x^2+a^2)*c^5-10/13*(b*x^2+a)*d*(13/11*b*x^ 2+a)*x^2*c^4+80/143*d^2*(13/14*b^2*x^4+2*a*b*x^2+a^2)*x^4*c^3-160/429*d^3* (39/70*b^2*x^4+12/7*a*b*x^2+a^2)*x^6*c^2+640/3003*a*d^4*x^8*(6/5*b*x^2+a)* c-256/3003*a^2*d^5*x^10)*(d*x^2+c)^(5/2)/x^15/c^6
Time = 0.47 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx=\frac {{\left (16 \, {\left (39 \, b^{2} c^{2} d^{5} - 48 \, a b c d^{6} + 16 \, a^{2} d^{7}\right )} x^{14} - 8 \, {\left (39 \, b^{2} c^{3} d^{4} - 48 \, a b c^{2} d^{5} + 16 \, a^{2} c d^{6}\right )} x^{12} + 6 \, {\left (39 \, b^{2} c^{4} d^{3} - 48 \, a b c^{3} d^{4} + 16 \, a^{2} c^{2} d^{5}\right )} x^{10} - 3003 \, a^{2} c^{7} - 5 \, {\left (39 \, b^{2} c^{5} d^{2} - 48 \, a b c^{4} d^{3} + 16 \, a^{2} c^{3} d^{4}\right )} x^{8} - 70 \, {\left (78 \, b^{2} c^{6} d + 3 \, a b c^{5} d^{2} - a^{2} c^{4} d^{3}\right )} x^{6} - 63 \, {\left (65 \, b^{2} c^{7} + 140 \, a b c^{6} d + a^{2} c^{5} d^{2}\right )} x^{4} - 462 \, {\left (15 \, a b c^{7} + 8 \, a^{2} c^{6} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{45045 \, c^{6} x^{15}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^16,x, algorithm="fricas")
Output:
1/45045*(16*(39*b^2*c^2*d^5 - 48*a*b*c*d^6 + 16*a^2*d^7)*x^14 - 8*(39*b^2* c^3*d^4 - 48*a*b*c^2*d^5 + 16*a^2*c*d^6)*x^12 + 6*(39*b^2*c^4*d^3 - 48*a*b *c^3*d^4 + 16*a^2*c^2*d^5)*x^10 - 3003*a^2*c^7 - 5*(39*b^2*c^5*d^2 - 48*a* b*c^4*d^3 + 16*a^2*c^3*d^4)*x^8 - 70*(78*b^2*c^6*d + 3*a*b*c^5*d^2 - a^2*c ^4*d^3)*x^6 - 63*(65*b^2*c^7 + 140*a*b*c^6*d + a^2*c^5*d^2)*x^4 - 462*(15* a*b*c^7 + 8*a^2*c^6*d)*x^2)*sqrt(d*x^2 + c)/(c^6*x^15)
Leaf count of result is larger than twice the leaf count of optimal. 6316 vs. \(2 (231) = 462\).
Time = 7.97 (sec) , antiderivative size = 6316, normalized size of antiderivative = 26.76 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/x**16,x)
Output:
-3003*a**2*c**14*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**13*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**12*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**11*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) - 693*a**2*c* *11*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**10*d**(81/2)*x**8*sqr t(c/(d*x**2) + 1)/(45045*c**13*d**36*x**14 + 270270*c**12*d**37*x**16 + 67 5675*c**11*d**38*x**18 + 900900*c**10*d**39*x**20 + 675675*c**9*d**40*x**2 2 + 270270*c**8*d**41*x**24 + 45045*c**7*d**42*x**26) - 3528*a**2*c**10*d* *(55/2)*x**2*sqrt(c/(d*x**2) + 1)/(9009*c**11*d**25*x**12 + 45045*c**10...
Time = 0.05 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx=\frac {16 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} d^{3}}{1155 \, c^{4} x^{5}} - \frac {256 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d^{4}}{15015 \, c^{5} x^{5}} + \frac {256 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{5}}{45045 \, c^{6} x^{5}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} d^{2}}{231 \, c^{3} x^{7}} + \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d^{3}}{3003 \, c^{4} x^{7}} - \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{4}}{9009 \, c^{5} x^{7}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} d}{33 \, c^{2} x^{9}} - \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d^{2}}{429 \, c^{3} x^{9}} + \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{3}}{1287 \, c^{4} x^{9}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2}}{11 \, c x^{11}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d}{143 \, c^{2} x^{11}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{2}}{429 \, c^{3} x^{11}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{13 \, c x^{13}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d}{39 \, c^{2} x^{13}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{15 \, c x^{15}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^16,x, algorithm="maxima")
Output:
16/1155*(d*x^2 + c)^(5/2)*b^2*d^3/(c^4*x^5) - 256/15015*(d*x^2 + c)^(5/2)* a*b*d^4/(c^5*x^5) + 256/45045*(d*x^2 + c)^(5/2)*a^2*d^5/(c^6*x^5) - 8/231* (d*x^2 + c)^(5/2)*b^2*d^2/(c^3*x^7) + 128/3003*(d*x^2 + c)^(5/2)*a*b*d^3/( c^4*x^7) - 128/9009*(d*x^2 + c)^(5/2)*a^2*d^4/(c^5*x^7) + 2/33*(d*x^2 + c) ^(5/2)*b^2*d/(c^2*x^9) - 32/429*(d*x^2 + c)^(5/2)*a*b*d^2/(c^3*x^9) + 32/1 287*(d*x^2 + c)^(5/2)*a^2*d^3/(c^4*x^9) - 1/11*(d*x^2 + c)^(5/2)*b^2/(c*x^ 11) + 16/143*(d*x^2 + c)^(5/2)*a*b*d/(c^2*x^11) - 16/429*(d*x^2 + c)^(5/2) *a^2*d^2/(c^3*x^11) - 2/13*(d*x^2 + c)^(5/2)*a*b/(c*x^13) + 2/39*(d*x^2 + c)^(5/2)*a^2*d/(c^2*x^13) - 1/15*(d*x^2 + c)^(5/2)*a^2/(c*x^15)
Leaf count of result is larger than twice the leaf count of optimal. 908 vs. \(2 (212) = 424\).
Time = 0.15 (sec) , antiderivative size = 908, normalized size of antiderivative = 3.85 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^16,x, algorithm="giac")
Output:
32/45045*(45045*(sqrt(d)*x - sqrt(d*x^2 + c))^22*b^2*d^(11/2) - 99099*(sqr t(d)*x - sqrt(d*x^2 + c))^20*b^2*c*d^(11/2) + 288288*(sqrt(d)*x - sqrt(d*x ^2 + c))^20*a*b*d^(13/2) + 45045*(sqrt(d)*x - sqrt(d*x^2 + c))^18*b^2*c^2* d^(11/2) + 480480*(sqrt(d)*x - sqrt(d*x^2 + c))^18*a^2*d^(15/2) - 57915*(s qrt(d)*x - sqrt(d*x^2 + c))^16*b^2*c^3*d^(11/2) - 205920*(sqrt(d)*x - sqrt (d*x^2 + c))^16*a*b*c^2*d^(13/2) + 1029600*(sqrt(d)*x - sqrt(d*x^2 + c))^1 6*a^2*c*d^(15/2) + 193050*(sqrt(d)*x - sqrt(d*x^2 + c))^14*b^2*c^4*d^(11/2 ) - 514800*(sqrt(d)*x - sqrt(d*x^2 + c))^14*a*b*c^3*d^(13/2) + 1132560*(sq rt(d)*x - sqrt(d*x^2 + c))^14*a^2*c^2*d^(15/2) - 150150*(sqrt(d)*x - sqrt( d*x^2 + c))^12*b^2*c^5*d^(11/2) + 240240*(sqrt(d)*x - sqrt(d*x^2 + c))^12* a*b*c^4*d^(13/2) + 400400*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a^2*c^3*d^(15/2 ) + 18018*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^2*c^6*d^(11/2) + 144144*(sqrt (d)*x - sqrt(d*x^2 + c))^10*a*b*c^5*d^(13/2) + 48048*(sqrt(d)*x - sqrt(d*x ^2 + c))^10*a^2*c^4*d^(15/2) - 8190*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^ 7*d^(11/2) + 65520*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c^6*d^(13/2) - 2184 0*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*c^5*d^(15/2) + 17745*(sqrt(d)*x - sq rt(d*x^2 + c))^6*b^2*c^8*d^(11/2) - 21840*(sqrt(d)*x - sqrt(d*x^2 + c))^6* a*b*c^7*d^(13/2) + 7280*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*c^6*d^(15/2) - 4095*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^9*d^(11/2) + 5040*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^8*d^(13/2) - 1680*(sqrt(d)*x - sqrt(d*x^2 + c...
Time = 5.98 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx=\frac {2\,a^2\,d^3\,\sqrt {d\,x^2+c}}{1287\,c^2\,x^9}-\frac {b^2\,c\,\sqrt {d\,x^2+c}}{11\,x^{11}}-\frac {16\,a^2\,d\,\sqrt {d\,x^2+c}}{195\,x^{13}}-\frac {4\,b^2\,d\,\sqrt {d\,x^2+c}}{33\,x^9}-\frac {2\,a\,b\,c\,\sqrt {d\,x^2+c}}{13\,x^{13}}-\frac {28\,a\,b\,d\,\sqrt {d\,x^2+c}}{143\,x^{11}}-\frac {a^2\,d^2\,\sqrt {d\,x^2+c}}{715\,c\,x^{11}}-\frac {a^2\,c\,\sqrt {d\,x^2+c}}{15\,x^{15}}-\frac {16\,a^2\,d^4\,\sqrt {d\,x^2+c}}{9009\,c^3\,x^7}+\frac {32\,a^2\,d^5\,\sqrt {d\,x^2+c}}{15015\,c^4\,x^5}-\frac {128\,a^2\,d^6\,\sqrt {d\,x^2+c}}{45045\,c^5\,x^3}+\frac {256\,a^2\,d^7\,\sqrt {d\,x^2+c}}{45045\,c^6\,x}-\frac {b^2\,d^2\,\sqrt {d\,x^2+c}}{231\,c\,x^7}+\frac {2\,b^2\,d^3\,\sqrt {d\,x^2+c}}{385\,c^2\,x^5}-\frac {8\,b^2\,d^4\,\sqrt {d\,x^2+c}}{1155\,c^3\,x^3}+\frac {16\,b^2\,d^5\,\sqrt {d\,x^2+c}}{1155\,c^4\,x}-\frac {2\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{429\,c\,x^9}+\frac {16\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{3003\,c^2\,x^7}-\frac {32\,a\,b\,d^4\,\sqrt {d\,x^2+c}}{5005\,c^3\,x^5}+\frac {128\,a\,b\,d^5\,\sqrt {d\,x^2+c}}{15015\,c^4\,x^3}-\frac {256\,a\,b\,d^6\,\sqrt {d\,x^2+c}}{15015\,c^5\,x} \] Input:
int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/x^16,x)
Output:
(2*a^2*d^3*(c + d*x^2)^(1/2))/(1287*c^2*x^9) - (b^2*c*(c + d*x^2)^(1/2))/( 11*x^11) - (16*a^2*d*(c + d*x^2)^(1/2))/(195*x^13) - (4*b^2*d*(c + d*x^2)^ (1/2))/(33*x^9) - (2*a*b*c*(c + d*x^2)^(1/2))/(13*x^13) - (28*a*b*d*(c + d *x^2)^(1/2))/(143*x^11) - (a^2*d^2*(c + d*x^2)^(1/2))/(715*c*x^11) - (a^2* c*(c + d*x^2)^(1/2))/(15*x^15) - (16*a^2*d^4*(c + d*x^2)^(1/2))/(9009*c^3* x^7) + (32*a^2*d^5*(c + d*x^2)^(1/2))/(15015*c^4*x^5) - (128*a^2*d^6*(c + d*x^2)^(1/2))/(45045*c^5*x^3) + (256*a^2*d^7*(c + d*x^2)^(1/2))/(45045*c^6 *x) - (b^2*d^2*(c + d*x^2)^(1/2))/(231*c*x^7) + (2*b^2*d^3*(c + d*x^2)^(1/ 2))/(385*c^2*x^5) - (8*b^2*d^4*(c + d*x^2)^(1/2))/(1155*c^3*x^3) + (16*b^2 *d^5*(c + d*x^2)^(1/2))/(1155*c^4*x) - (2*a*b*d^2*(c + d*x^2)^(1/2))/(429* c*x^9) + (16*a*b*d^3*(c + d*x^2)^(1/2))/(3003*c^2*x^7) - (32*a*b*d^4*(c + d*x^2)^(1/2))/(5005*c^3*x^5) + (128*a*b*d^5*(c + d*x^2)^(1/2))/(15015*c^4* x^3) - (256*a*b*d^6*(c + d*x^2)^(1/2))/(15015*c^5*x)
Time = 0.28 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^{16}} \, dx=\frac {-3003 \sqrt {d \,x^{2}+c}\, a^{2} c^{7}-3696 \sqrt {d \,x^{2}+c}\, a^{2} c^{6} d \,x^{2}-63 \sqrt {d \,x^{2}+c}\, a^{2} c^{5} d^{2} x^{4}+70 \sqrt {d \,x^{2}+c}\, a^{2} c^{4} d^{3} x^{6}-80 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} d^{4} x^{8}+96 \sqrt {d \,x^{2}+c}\, a^{2} c^{2} d^{5} x^{10}-128 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{6} x^{12}+256 \sqrt {d \,x^{2}+c}\, a^{2} d^{7} x^{14}-6930 \sqrt {d \,x^{2}+c}\, a b \,c^{7} x^{2}-8820 \sqrt {d \,x^{2}+c}\, a b \,c^{6} d \,x^{4}-210 \sqrt {d \,x^{2}+c}\, a b \,c^{5} d^{2} x^{6}+240 \sqrt {d \,x^{2}+c}\, a b \,c^{4} d^{3} x^{8}-288 \sqrt {d \,x^{2}+c}\, a b \,c^{3} d^{4} x^{10}+384 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d^{5} x^{12}-768 \sqrt {d \,x^{2}+c}\, a b c \,d^{6} x^{14}-4095 \sqrt {d \,x^{2}+c}\, b^{2} c^{7} x^{4}-5460 \sqrt {d \,x^{2}+c}\, b^{2} c^{6} d \,x^{6}-195 \sqrt {d \,x^{2}+c}\, b^{2} c^{5} d^{2} x^{8}+234 \sqrt {d \,x^{2}+c}\, b^{2} c^{4} d^{3} x^{10}-312 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} d^{4} x^{12}+624 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} d^{5} x^{14}-256 \sqrt {d}\, a^{2} d^{7} x^{15}+768 \sqrt {d}\, a b c \,d^{6} x^{15}-624 \sqrt {d}\, b^{2} c^{2} d^{5} x^{15}}{45045 c^{6} x^{15}} \] Input:
int((b*x^2+a)^2*(d*x^2+c)^(3/2)/x^16,x)
Output:
( - 3003*sqrt(c + d*x**2)*a**2*c**7 - 3696*sqrt(c + d*x**2)*a**2*c**6*d*x* *2 - 63*sqrt(c + d*x**2)*a**2*c**5*d**2*x**4 + 70*sqrt(c + d*x**2)*a**2*c* *4*d**3*x**6 - 80*sqrt(c + d*x**2)*a**2*c**3*d**4*x**8 + 96*sqrt(c + d*x** 2)*a**2*c**2*d**5*x**10 - 128*sqrt(c + d*x**2)*a**2*c*d**6*x**12 + 256*sqr t(c + d*x**2)*a**2*d**7*x**14 - 6930*sqrt(c + d*x**2)*a*b*c**7*x**2 - 8820 *sqrt(c + d*x**2)*a*b*c**6*d*x**4 - 210*sqrt(c + d*x**2)*a*b*c**5*d**2*x** 6 + 240*sqrt(c + d*x**2)*a*b*c**4*d**3*x**8 - 288*sqrt(c + d*x**2)*a*b*c** 3*d**4*x**10 + 384*sqrt(c + d*x**2)*a*b*c**2*d**5*x**12 - 768*sqrt(c + d*x **2)*a*b*c*d**6*x**14 - 4095*sqrt(c + d*x**2)*b**2*c**7*x**4 - 5460*sqrt(c + d*x**2)*b**2*c**6*d*x**6 - 195*sqrt(c + d*x**2)*b**2*c**5*d**2*x**8 + 2 34*sqrt(c + d*x**2)*b**2*c**4*d**3*x**10 - 312*sqrt(c + d*x**2)*b**2*c**3* d**4*x**12 + 624*sqrt(c + d*x**2)*b**2*c**2*d**5*x**14 - 256*sqrt(d)*a**2* d**7*x**15 + 768*sqrt(d)*a*b*c*d**6*x**15 - 624*sqrt(d)*b**2*c**2*d**5*x** 15)/(45045*c**6*x**15)