Integrand size = 24, antiderivative size = 144 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^7}-\frac {\left (21 b^2 c^2-24 a b c d+8 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{105 c^3 x^5}+\frac {2 d \left (21 b^2 c^2-8 a d (3 b c-a d)\right ) \left (c+d x^2\right )^{3/2}}{315 c^4 x^3} \] Output:
-1/9*a^2*(d*x^2+c)^(3/2)/c/x^9-2/21*a*(-a*d+3*b*c)*(d*x^2+c)^(3/2)/c^2/x^7 -1/105*(8*a^2*d^2-24*a*b*c*d+21*b^2*c^2)*(d*x^2+c)^(3/2)/c^3/x^5+2/315*d*( 21*b^2*c^2-8*a*d*(-a*d+3*b*c))*(d*x^2+c)^(3/2)/c^4/x^3
Time = 0.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx=-\frac {\left (c+d x^2\right )^{3/2} \left (21 b^2 c^2 x^4 \left (3 c-2 d x^2\right )+6 a b c x^2 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+a^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )\right )}{315 c^4 x^9} \] Input:
Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]
Output:
-1/315*((c + d*x^2)^(3/2)*(21*b^2*c^2*x^4*(3*c - 2*d*x^2) + 6*a*b*c*x^2*(1 5*c^2 - 12*c*d*x^2 + 8*d^2*x^4) + a^2*(35*c^3 - 30*c^2*d*x^2 + 24*c*d^2*x^ 4 - 16*d^3*x^6)))/(c^4*x^9)
Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {365, 27, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {\int \frac {3 \left (3 b^2 c x^2+2 a (3 b c-a d)\right ) \sqrt {d x^2+c}}{x^8}dx}{9 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (3 b^2 c x^2+2 a (3 b c-a d)\right ) \sqrt {d x^2+c}}{x^8}dx}{3 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {\frac {\left (21 b^2 c^2-8 a d (3 b c-a d)\right ) \int \frac {\sqrt {d x^2+c}}{x^6}dx}{7 c}-\frac {2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{7 c x^7}}{3 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {\left (21 b^2 c^2-8 a d (3 b c-a d)\right ) \left (-\frac {2 d \int \frac {\sqrt {d x^2+c}}{x^4}dx}{5 c}-\frac {\left (c+d x^2\right )^{3/2}}{5 c x^5}\right )}{7 c}-\frac {2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{7 c x^7}}{3 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {\frac {\left (\frac {2 d \left (c+d x^2\right )^{3/2}}{15 c^2 x^3}-\frac {\left (c+d x^2\right )^{3/2}}{5 c x^5}\right ) \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{7 c}-\frac {2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{7 c x^7}}{3 c}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}\) |
Input:
Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]
Output:
-1/9*(a^2*(c + d*x^2)^(3/2))/(c*x^9) + ((-2*a*(3*b*c - a*d)*(c + d*x^2)^(3 /2))/(7*c*x^7) + ((21*b^2*c^2 - 8*a*d*(3*b*c - a*d))*(-1/5*(c + d*x^2)^(3/ 2)/(c*x^5) + (2*d*(c + d*x^2)^(3/2))/(15*c^2*x^3)))/(7*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 = 0.53 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (\frac {9}{5} b^{2} x^{4}+\frac {18}{7} a b \,x^{2}+a^{2}\right ) c^{3}-\frac {6 \left (b \,x^{2}+a \right ) \left (\frac {7 b \,x^{2}}{5}+a \right ) d \,x^{2} c^{2}}{7}+\frac {24 a \,d^{2} x^{4} \left (2 b \,x^{2}+a \right ) c}{35}-\frac {16 a^{2} d^{3} x^{6}}{35}\right ) \left (x^{2} d +c \right )^{\frac {3}{2}}}{9 x^{9} c^{4}}\) | \(95\) |
| gosper | \(-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}} \left (-16 a^{2} d^{3} x^{6}+48 a b c \,d^{2} x^{6}-42 b^{2} c^{2} d \,x^{6}+24 a^{2} c \,d^{2} x^{4}-72 a b \,c^{2} d \,x^{4}+63 b^{2} c^{3} x^{4}-30 a^{2} c^{2} d \,x^{2}+90 a b \,c^{3} x^{2}+35 a^{2} c^{3}\right )}{315 x^{9} c^{4}}\) | \(117\) |
| orering | \(-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}} \left (-16 a^{2} d^{3} x^{6}+48 a b c \,d^{2} x^{6}-42 b^{2} c^{2} d \,x^{6}+24 a^{2} c \,d^{2} x^{4}-72 a b \,c^{2} d \,x^{4}+63 b^{2} c^{3} x^{4}-30 a^{2} c^{2} d \,x^{2}+90 a b \,c^{3} x^{2}+35 a^{2} c^{3}\right )}{315 x^{9} c^{4}}\) | \(117\) |
| trager | \(-\frac {\left (-16 a^{2} d^{4} x^{8}+48 a b c \,d^{3} x^{8}-42 b^{2} c^{2} d^{2} x^{8}+8 a^{2} c \,d^{3} x^{6}-24 a b \,c^{2} d^{2} x^{6}+21 b^{2} c^{3} d \,x^{6}-6 a^{2} c^{2} d^{2} x^{4}+18 a b \,c^{3} d \,x^{4}+63 b^{2} c^{4} x^{4}+5 a^{2} c^{3} d \,x^{2}+90 a b \,c^{4} x^{2}+35 a^{2} c^{4}\right ) \sqrt {x^{2} d +c}}{315 x^{9} c^{4}}\) | \(158\) |
| risch | \(-\frac {\left (-16 a^{2} d^{4} x^{8}+48 a b c \,d^{3} x^{8}-42 b^{2} c^{2} d^{2} x^{8}+8 a^{2} c \,d^{3} x^{6}-24 a b \,c^{2} d^{2} x^{6}+21 b^{2} c^{3} d \,x^{6}-6 a^{2} c^{2} d^{2} x^{4}+18 a b \,c^{3} d \,x^{4}+63 b^{2} c^{4} x^{4}+5 a^{2} c^{3} d \,x^{2}+90 a b \,c^{4} x^{2}+35 a^{2} c^{4}\right ) \sqrt {x^{2} d +c}}{315 x^{9} c^{4}}\) | \(158\) |
| default | \(a^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{9 c \,x^{9}}-\frac {2 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )}{3 c}\right )+b^{2} \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )+2 a b \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )\) | \(194\) |
Input:
int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x,method=_RETURNVERBOSE)
Output:
-1/9*((9/5*b^2*x^4+18/7*a*b*x^2+a^2)*c^3-6/7*(b*x^2+a)*(7/5*b*x^2+a)*d*x^2 *c^2+24/35*a*d^2*x^4*(2*b*x^2+a)*c-16/35*a^2*d^3*x^6)*(d*x^2+c)^(3/2)/x^9/ c^4
Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx=\frac {{\left (2 \, {\left (21 \, b^{2} c^{2} d^{2} - 24 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{8} - {\left (21 \, b^{2} c^{3} d - 24 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{6} - 35 \, a^{2} c^{4} - 3 \, {\left (21 \, b^{2} c^{4} + 6 \, a b c^{3} d - 2 \, a^{2} c^{2} d^{2}\right )} x^{4} - 5 \, {\left (18 \, a b c^{4} + a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{315 \, c^{4} x^{9}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x, algorithm="fricas")
Output:
1/315*(2*(21*b^2*c^2*d^2 - 24*a*b*c*d^3 + 8*a^2*d^4)*x^8 - (21*b^2*c^3*d - 24*a*b*c^2*d^2 + 8*a^2*c*d^3)*x^6 - 35*a^2*c^4 - 3*(21*b^2*c^4 + 6*a*b*c^ 3*d - 2*a^2*c^2*d^2)*x^4 - 5*(18*a*b*c^4 + a^2*c^3*d)*x^2)*sqrt(d*x^2 + c) /(c^4*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 1061 vs. \(2 (139) = 278\).
Time = 2.09 (sec) , antiderivative size = 1061, normalized size of antiderivative = 7.37 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx =\text {Too large to display} \] Input:
integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**10,x)
Output:
-35*a**2*c**7*d**(19/2)*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c** 6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) - 110*a**2*c* *6*d**(21/2)*x**2*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**1 0*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) - 114*a**2*c**5*d** (23/2)*x**4*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**1 0 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) - 40*a**2*c**4*d**(25/2)* x**6*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945 *c**5*d**11*x**12 + 315*c**4*d**12*x**14) + 5*a**2*c**3*d**(27/2)*x**8*sqr t(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d* *11*x**12 + 315*c**4*d**12*x**14) + 30*a**2*c**2*d**(29/2)*x**10*sqrt(c/(d *x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x* *12 + 315*c**4*d**12*x**14) + 40*a**2*c*d**(31/2)*x**12*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315 *c**4*d**12*x**14) + 16*a**2*d**(33/2)*x**14*sqrt(c/(d*x**2) + 1)/(315*c** 7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12 *x**14) - 30*a*b*c**5*d**(9/2)*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) - 66*a*b*c**4*d**(11/2)*x**2*sqr t(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6 *x**10) - 34*a*b*c**3*d**(13/2)*x**4*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x **6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) - 6*a*b*c**2*d**(15/2)*...
Time = 0.04 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx=\frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{15 \, c^{2} x^{3}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{105 \, c^{3} x^{3}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{315 \, c^{4} x^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{5 \, c x^{5}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{35 \, c^{2} x^{5}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{105 \, c^{3} x^{5}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{7 \, c x^{7}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{21 \, c^{2} x^{7}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{9 \, c x^{9}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x, algorithm="maxima")
Output:
2/15*(d*x^2 + c)^(3/2)*b^2*d/(c^2*x^3) - 16/105*(d*x^2 + c)^(3/2)*a*b*d^2/ (c^3*x^3) + 16/315*(d*x^2 + c)^(3/2)*a^2*d^3/(c^4*x^3) - 1/5*(d*x^2 + c)^( 3/2)*b^2/(c*x^5) + 8/35*(d*x^2 + c)^(3/2)*a*b*d/(c^2*x^5) - 8/105*(d*x^2 + c)^(3/2)*a^2*d^2/(c^3*x^5) - 2/7*(d*x^2 + c)^(3/2)*a*b/(c*x^7) + 2/21*(d* x^2 + c)^(3/2)*a^2*d/(c^2*x^7) - 1/9*(d*x^2 + c)^(3/2)*a^2/(c*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (128) = 256\).
Time = 0.14 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.02 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx=\frac {4 \, {\left (315 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} b^{2} d^{\frac {5}{2}} - 1155 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} b^{2} c d^{\frac {5}{2}} + 1680 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a b d^{\frac {7}{2}} + 1575 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{2} c^{2} d^{\frac {5}{2}} - 2520 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b c d^{\frac {7}{2}} + 2520 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a^{2} d^{\frac {9}{2}} - 1071 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{3} d^{\frac {5}{2}} + 504 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c^{2} d^{\frac {7}{2}} + 1512 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} c d^{\frac {9}{2}} + 609 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{4} d^{\frac {5}{2}} - 336 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{3} d^{\frac {7}{2}} + 672 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c^{2} d^{\frac {9}{2}} - 441 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{5} d^{\frac {5}{2}} + 864 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{4} d^{\frac {7}{2}} - 288 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{3} d^{\frac {9}{2}} + 189 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{6} d^{\frac {5}{2}} - 216 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{5} d^{\frac {7}{2}} + 72 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{4} d^{\frac {9}{2}} - 21 \, b^{2} c^{7} d^{\frac {5}{2}} + 24 \, a b c^{6} d^{\frac {7}{2}} - 8 \, a^{2} c^{5} d^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{9}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x, algorithm="giac")
Output:
4/315*(315*(sqrt(d)*x - sqrt(d*x^2 + c))^14*b^2*d^(5/2) - 1155*(sqrt(d)*x - sqrt(d*x^2 + c))^12*b^2*c*d^(5/2) + 1680*(sqrt(d)*x - sqrt(d*x^2 + c))^1 2*a*b*d^(7/2) + 1575*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^2*c^2*d^(5/2) - 25 20*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b*c*d^(7/2) + 2520*(sqrt(d)*x - sqrt (d*x^2 + c))^10*a^2*d^(9/2) - 1071*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^3 *d^(5/2) + 504*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c^2*d^(7/2) + 1512*(sqr t(d)*x - sqrt(d*x^2 + c))^8*a^2*c*d^(9/2) + 609*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^4*d^(5/2) - 336*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b*c^3*d^(7/2 ) + 672*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*c^2*d^(9/2) - 441*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^5*d^(5/2) + 864*(sqrt(d)*x - sqrt(d*x^2 + c))^4* a*b*c^4*d^(7/2) - 288*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*c^3*d^(9/2) + 18 9*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^6*d^(5/2) - 216*(sqrt(d)*x - sqrt( d*x^2 + c))^2*a*b*c^5*d^(7/2) + 72*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^4 *d^(9/2) - 21*b^2*c^7*d^(5/2) + 24*a*b*c^6*d^(7/2) - 8*a^2*c^5*d^(9/2))/(( sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^9
Time = 2.20 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx=\frac {2\,a^2\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^5}-\frac {b^2\,\sqrt {d\,x^2+c}}{5\,x^5}-\frac {2\,a\,b\,\sqrt {d\,x^2+c}}{7\,x^7}-\frac {a^2\,\sqrt {d\,x^2+c}}{9\,x^9}-\frac {8\,a^2\,d^3\,\sqrt {d\,x^2+c}}{315\,c^3\,x^3}+\frac {16\,a^2\,d^4\,\sqrt {d\,x^2+c}}{315\,c^4\,x}+\frac {2\,b^2\,d^2\,\sqrt {d\,x^2+c}}{15\,c^2\,x}-\frac {a^2\,d\,\sqrt {d\,x^2+c}}{63\,c\,x^7}-\frac {b^2\,d\,\sqrt {d\,x^2+c}}{15\,c\,x^3}+\frac {8\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^3}-\frac {16\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{105\,c^3\,x}-\frac {2\,a\,b\,d\,\sqrt {d\,x^2+c}}{35\,c\,x^5} \] Input:
int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/x^10,x)
Output:
(2*a^2*d^2*(c + d*x^2)^(1/2))/(105*c^2*x^5) - (b^2*(c + d*x^2)^(1/2))/(5*x ^5) - (2*a*b*(c + d*x^2)^(1/2))/(7*x^7) - (a^2*(c + d*x^2)^(1/2))/(9*x^9) - (8*a^2*d^3*(c + d*x^2)^(1/2))/(315*c^3*x^3) + (16*a^2*d^4*(c + d*x^2)^(1 /2))/(315*c^4*x) + (2*b^2*d^2*(c + d*x^2)^(1/2))/(15*c^2*x) - (a^2*d*(c + d*x^2)^(1/2))/(63*c*x^7) - (b^2*d*(c + d*x^2)^(1/2))/(15*c*x^3) + (8*a*b*d ^2*(c + d*x^2)^(1/2))/(105*c^2*x^3) - (16*a*b*d^3*(c + d*x^2)^(1/2))/(105* c^3*x) - (2*a*b*d*(c + d*x^2)^(1/2))/(35*c*x^5)
Time = 0.22 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{10}} \, dx=\frac {-35 \sqrt {d \,x^{2}+c}\, a^{2} c^{4}-5 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} d \,x^{2}+6 \sqrt {d \,x^{2}+c}\, a^{2} c^{2} d^{2} x^{4}-8 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{3} x^{6}+16 \sqrt {d \,x^{2}+c}\, a^{2} d^{4} x^{8}-90 \sqrt {d \,x^{2}+c}\, a b \,c^{4} x^{2}-18 \sqrt {d \,x^{2}+c}\, a b \,c^{3} d \,x^{4}+24 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d^{2} x^{6}-48 \sqrt {d \,x^{2}+c}\, a b c \,d^{3} x^{8}-63 \sqrt {d \,x^{2}+c}\, b^{2} c^{4} x^{4}-21 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} d \,x^{6}+42 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} d^{2} x^{8}-16 \sqrt {d}\, a^{2} d^{4} x^{9}+48 \sqrt {d}\, a b c \,d^{3} x^{9}-42 \sqrt {d}\, b^{2} c^{2} d^{2} x^{9}}{315 c^{4} x^{9}} \] Input:
int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x)
Output:
( - 35*sqrt(c + d*x**2)*a**2*c**4 - 5*sqrt(c + d*x**2)*a**2*c**3*d*x**2 + 6*sqrt(c + d*x**2)*a**2*c**2*d**2*x**4 - 8*sqrt(c + d*x**2)*a**2*c*d**3*x* *6 + 16*sqrt(c + d*x**2)*a**2*d**4*x**8 - 90*sqrt(c + d*x**2)*a*b*c**4*x** 2 - 18*sqrt(c + d*x**2)*a*b*c**3*d*x**4 + 24*sqrt(c + d*x**2)*a*b*c**2*d** 2*x**6 - 48*sqrt(c + d*x**2)*a*b*c*d**3*x**8 - 63*sqrt(c + d*x**2)*b**2*c* *4*x**4 - 21*sqrt(c + d*x**2)*b**2*c**3*d*x**6 + 42*sqrt(c + d*x**2)*b**2* c**2*d**2*x**8 - 16*sqrt(d)*a**2*d**4*x**9 + 48*sqrt(d)*a*b*c*d**3*x**9 - 42*sqrt(d)*b**2*c**2*d**2*x**9)/(315*c**4*x**9)