Integrand size = 22, antiderivative size = 150 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx=-\frac {A \left (a+b x^2\right )^{5/2}}{13 a x^{13}}+\frac {(8 A b-13 a B) \left (a+b x^2\right )^{5/2}}{143 a^2 x^{11}}-\frac {2 b (8 A b-13 a B) \left (a+b x^2\right )^{5/2}}{429 a^3 x^9}+\frac {8 b^2 (8 A b-13 a B) \left (a+b x^2\right )^{5/2}}{3003 a^4 x^7}-\frac {16 b^3 (8 A b-13 a B) \left (a+b x^2\right )^{5/2}}{15015 a^5 x^5} \] Output:
-1/13*A*(b*x^2+a)^(5/2)/a/x^13+1/143*(8*A*b-13*B*a)*(b*x^2+a)^(5/2)/a^2/x^ 11-2/429*b*(8*A*b-13*B*a)*(b*x^2+a)^(5/2)/a^3/x^9+8/3003*b^2*(8*A*b-13*B*a )*(b*x^2+a)^(5/2)/a^4/x^7-16/15015*b^3*(8*A*b-13*B*a)*(b*x^2+a)^(5/2)/a^5/ x^5
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx=\frac {\left (a+b x^2\right )^{5/2} \left (-128 A b^4 x^8-105 a^4 \left (11 A+13 B x^2\right )+70 a^3 b x^2 \left (12 A+13 B x^2\right )-40 a^2 b^2 x^4 \left (14 A+13 B x^2\right )+16 a b^3 x^6 \left (20 A+13 B x^2\right )\right )}{15015 a^5 x^{13}} \] Input:
Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^14,x]
Output:
((a + b*x^2)^(5/2)*(-128*A*b^4*x^8 - 105*a^4*(11*A + 13*B*x^2) + 70*a^3*b* x^2*(12*A + 13*B*x^2) - 40*a^2*b^2*x^4*(14*A + 13*B*x^2) + 16*a*b^3*x^6*(2 0*A + 13*B*x^2)))/(15015*a^5*x^13)
Time = 0.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {(8 A b-13 a B) \int \frac {\left (b x^2+a\right )^{3/2}}{x^{12}}dx}{13 a}-\frac {A \left (a+b x^2\right )^{5/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-13 a B) \left (-\frac {6 b \int \frac {\left (b x^2+a\right )^{3/2}}{x^{10}}dx}{11 a}-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}}\right )}{13 a}-\frac {A \left (a+b x^2\right )^{5/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-13 a B) \left (-\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^{3/2}}{x^8}dx}{9 a}-\frac {\left (a+b x^2\right )^{5/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}}\right )}{13 a}-\frac {A \left (a+b x^2\right )^{5/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-13 a B) \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx}{7 a}-\frac {\left (a+b x^2\right )^{5/2}}{7 a x^7}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}}\right )}{13 a}-\frac {A \left (a+b x^2\right )^{5/2}}{13 a x^{13}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {\left (-\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{5/2}}{35 a^2 x^5}-\frac {\left (a+b x^2\right )^{5/2}}{7 a x^7}\right )}{9 a}-\frac {\left (a+b x^2\right )^{5/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}}\right ) (8 A b-13 a B)}{13 a}-\frac {A \left (a+b x^2\right )^{5/2}}{13 a x^{13}}\) |
Input:
Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^14,x]
Output:
-1/13*(A*(a + b*x^2)^(5/2))/(a*x^13) - ((8*A*b - 13*a*B)*(-1/11*(a + b*x^2 )^(5/2)/(a*x^11) - (6*b*(-1/9*(a + b*x^2)^(5/2)/(a*x^9) - (4*b*(-1/7*(a + b*x^2)^(5/2)/(a*x^7) + (2*b*(a + b*x^2)^(5/2))/(35*a^2*x^5)))/(9*a)))/(11* a)))/(13*a)
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]
Time = 0.50 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (\frac {13 x^{2} B}{11}+A \right ) a^{4}-\frac {8 b \,x^{2} \left (\frac {13 x^{2} B}{12}+A \right ) a^{3}}{11}+\frac {16 b^{2} \left (\frac {13 x^{2} B}{14}+A \right ) x^{4} a^{2}}{33}-\frac {64 b^{3} \left (\frac {13 x^{2} B}{20}+A \right ) x^{6} a}{231}+\frac {128 A \,x^{8} b^{4}}{1155}\right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{13 x^{13} a^{5}}\) | \(93\) |
| gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (128 A \,x^{8} b^{4}-208 B \,x^{8} a \,b^{3}-320 A \,x^{6} a \,b^{3}+520 B \,x^{6} a^{2} b^{2}+560 A \,x^{4} a^{2} b^{2}-910 B \,x^{4} a^{3} b -840 A \,x^{2} a^{3} b +1365 B \,x^{2} a^{4}+1155 A \,a^{4}\right )}{15015 x^{13} a^{5}}\) | \(107\) |
| orering | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (128 A \,x^{8} b^{4}-208 B \,x^{8} a \,b^{3}-320 A \,x^{6} a \,b^{3}+520 B \,x^{6} a^{2} b^{2}+560 A \,x^{4} a^{2} b^{2}-910 B \,x^{4} a^{3} b -840 A \,x^{2} a^{3} b +1365 B \,x^{2} a^{4}+1155 A \,a^{4}\right )}{15015 x^{13} a^{5}}\) | \(107\) |
| trager | \(-\frac {\left (128 A \,b^{6} x^{12}-208 B a \,b^{5} x^{12}-64 A a \,b^{5} x^{10}+104 B \,a^{2} b^{4} x^{10}+48 A \,a^{2} b^{4} x^{8}-78 B \,a^{3} b^{3} x^{8}-40 A \,a^{3} b^{3} x^{6}+65 B \,a^{4} b^{2} x^{6}+35 A \,a^{4} b^{2} x^{4}+1820 B \,a^{5} b \,x^{4}+1470 A \,a^{5} b \,x^{2}+1365 B \,a^{6} x^{2}+1155 a^{6} A \right ) \sqrt {b \,x^{2}+a}}{15015 x^{13} a^{5}}\) | \(155\) |
| risch | \(-\frac {\left (128 A \,b^{6} x^{12}-208 B a \,b^{5} x^{12}-64 A a \,b^{5} x^{10}+104 B \,a^{2} b^{4} x^{10}+48 A \,a^{2} b^{4} x^{8}-78 B \,a^{3} b^{3} x^{8}-40 A \,a^{3} b^{3} x^{6}+65 B \,a^{4} b^{2} x^{6}+35 A \,a^{4} b^{2} x^{4}+1820 B \,a^{5} b \,x^{4}+1470 A \,a^{5} b \,x^{2}+1365 B \,a^{6} x^{2}+1155 a^{6} A \right ) \sqrt {b \,x^{2}+a}}{15015 x^{13} a^{5}}\) | \(155\) |
| default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{13 a \,x^{13}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{11 a \,x^{11}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{11 a \,x^{11}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )}{11 a}\right )\) | \(198\) |
Input:
int((b*x^2+a)^(3/2)*(B*x^2+A)/x^14,x,method=_RETURNVERBOSE)
Output:
-1/13*((13/11*x^2*B+A)*a^4-8/11*b*x^2*(13/12*x^2*B+A)*a^3+16/33*b^2*(13/14 *x^2*B+A)*x^4*a^2-64/231*b^3*(13/20*x^2*B+A)*x^6*a+128/1155*A*x^8*b^4)*(b* x^2+a)^(5/2)/x^13/a^5
Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx=\frac {{\left (16 \, {\left (13 \, B a b^{5} - 8 \, A b^{6}\right )} x^{12} - 8 \, {\left (13 \, B a^{2} b^{4} - 8 \, A a b^{5}\right )} x^{10} + 6 \, {\left (13 \, B a^{3} b^{3} - 8 \, A a^{2} b^{4}\right )} x^{8} - 1155 \, A a^{6} - 5 \, {\left (13 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3}\right )} x^{6} - 35 \, {\left (52 \, B a^{5} b + A a^{4} b^{2}\right )} x^{4} - 105 \, {\left (13 \, B a^{6} + 14 \, A a^{5} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15015 \, a^{5} x^{13}} \] Input:
integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^14,x, algorithm="fricas")
Output:
1/15015*(16*(13*B*a*b^5 - 8*A*b^6)*x^12 - 8*(13*B*a^2*b^4 - 8*A*a*b^5)*x^1 0 + 6*(13*B*a^3*b^3 - 8*A*a^2*b^4)*x^8 - 1155*A*a^6 - 5*(13*B*a^4*b^2 - 8* A*a^3*b^3)*x^6 - 35*(52*B*a^5*b + A*a^4*b^2)*x^4 - 105*(13*B*a^6 + 14*A*a^ 5*b)*x^2)*sqrt(b*x^2 + a)/(a^5*x^13)
Leaf count of result is larger than twice the leaf count of optimal. 3487 vs. \(2 (146) = 292\).
Time = 4.80 (sec) , antiderivative size = 3487, normalized size of antiderivative = 23.25 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**14,x)
Output:
-693*A*a**12*b**(51/2)*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x**12 + 4504 5*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**28*x**18 + 45 045*a**7*b**29*x**20 + 9009*a**6*b**30*x**22) - 3528*A*a**11*b**(53/2)*x** 2*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x**12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**28*x**18 + 45045*a**7*b**29*x**20 + 9009*a**6*b**30*x**22) - 7175*A*a**10*b**(55/2)*x**4*sqrt(a/(b*x**2) + 1 )/(9009*a**11*b**25*x**12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x** 16 + 90090*a**8*b**28*x**18 + 45045*a**7*b**29*x**20 + 9009*a**6*b**30*x** 22) - 7290*A*a**9*b**(57/2)*x**6*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x* *12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**28* x**18 + 45045*a**7*b**29*x**20 + 9009*a**6*b**30*x**22) - 315*A*a**9*b**(3 5/2)*sqrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860*a**8*b**17*x**12 + 20790*a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465*a**5*b**20*x**18) - 3699*A*a**8*b**(59/2)*x**8*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x**12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**28*x** 18 + 45045*a**7*b**29*x**20 + 9009*a**6*b**30*x**22) - 1295*A*a**8*b**(37/ 2)*x**2*sqrt(a/(b*x**2) + 1)/(3465*a**9*b**16*x**10 + 13860*a**8*b**17*x** 12 + 20790*a**7*b**18*x**14 + 13860*a**6*b**19*x**16 + 3465*a**5*b**20*x** 18) - 756*A*a**7*b**(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x* *12 + 45045*a**10*b**26*x**14 + 90090*a**9*b**27*x**16 + 90090*a**8*b**...
Time = 0.04 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx=\frac {16 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{3}}{1155 \, a^{4} x^{5}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4}}{15015 \, a^{5} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{231 \, a^{3} x^{7}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{3003 \, a^{4} x^{7}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{33 \, a^{2} x^{9}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{429 \, a^{3} x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{11 \, a x^{11}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{143 \, a^{2} x^{11}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{13 \, a x^{13}} \] Input:
integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^14,x, algorithm="maxima")
Output:
16/1155*(b*x^2 + a)^(5/2)*B*b^3/(a^4*x^5) - 128/15015*(b*x^2 + a)^(5/2)*A* b^4/(a^5*x^5) - 8/231*(b*x^2 + a)^(5/2)*B*b^2/(a^3*x^7) + 64/3003*(b*x^2 + a)^(5/2)*A*b^3/(a^4*x^7) + 2/33*(b*x^2 + a)^(5/2)*B*b/(a^2*x^9) - 16/429* (b*x^2 + a)^(5/2)*A*b^2/(a^3*x^9) - 1/11*(b*x^2 + a)^(5/2)*B/(a*x^11) + 8/ 143*(b*x^2 + a)^(5/2)*A*b/(a^2*x^11) - 1/13*(b*x^2 + a)^(5/2)*A/(a*x^13)
Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (130) = 260\).
Time = 0.14 (sec) , antiderivative size = 512, normalized size of antiderivative = 3.41 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx=\frac {32 \, {\left (15015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} B b^{\frac {11}{2}} - 3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} B a b^{\frac {11}{2}} + 48048 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} A b^{\frac {13}{2}} - 6006 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} B a^{2} b^{\frac {11}{2}} + 96096 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} A a b^{\frac {13}{2}} - 28314 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a^{3} b^{\frac {11}{2}} + 109824 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A a^{2} b^{\frac {13}{2}} + 13728 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{4} b^{\frac {11}{2}} + 37752 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a^{3} b^{\frac {13}{2}} + 5720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{5} b^{\frac {11}{2}} + 5720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{4} b^{\frac {13}{2}} + 3718 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{6} b^{\frac {11}{2}} - 2288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{5} b^{\frac {13}{2}} - 1014 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{7} b^{\frac {11}{2}} + 624 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{6} b^{\frac {13}{2}} + 169 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{8} b^{\frac {11}{2}} - 104 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{7} b^{\frac {13}{2}} - 13 \, B a^{9} b^{\frac {11}{2}} + 8 \, A a^{8} b^{\frac {13}{2}}\right )}}{15015 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{13}} \] Input:
integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^14,x, algorithm="giac")
Output:
32/15015*(15015*(sqrt(b)*x - sqrt(b*x^2 + a))^18*B*b^(11/2) - 3003*(sqrt(b )*x - sqrt(b*x^2 + a))^16*B*a*b^(11/2) + 48048*(sqrt(b)*x - sqrt(b*x^2 + a ))^16*A*b^(13/2) - 6006*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^2*b^(11/2) + 96096*(sqrt(b)*x - sqrt(b*x^2 + a))^14*A*a*b^(13/2) - 28314*(sqrt(b)*x - s qrt(b*x^2 + a))^12*B*a^3*b^(11/2) + 109824*(sqrt(b)*x - sqrt(b*x^2 + a))^1 2*A*a^2*b^(13/2) + 13728*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^4*b^(11/2) + 37752*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a^3*b^(13/2) + 5720*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^5*b^(11/2) + 5720*(sqrt(b)*x - sqrt(b*x^2 + a))^8* A*a^4*b^(13/2) + 3718*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^6*b^(11/2) - 228 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^5*b^(13/2) - 1014*(sqrt(b)*x - sqrt( b*x^2 + a))^4*B*a^7*b^(11/2) + 624*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^6*b ^(13/2) + 169*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^8*b^(11/2) - 104*(sqrt(b )*x - sqrt(b*x^2 + a))^2*A*a^7*b^(13/2) - 13*B*a^9*b^(11/2) + 8*A*a^8*b^(1 3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^13
Time = 2.91 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx=\frac {8\,A\,b^3\,\sqrt {b\,x^2+a}}{3003\,a^2\,x^7}-\frac {14\,A\,b\,\sqrt {b\,x^2+a}}{143\,x^{11}}-\frac {B\,a\,\sqrt {b\,x^2+a}}{11\,x^{11}}-\frac {4\,B\,b\,\sqrt {b\,x^2+a}}{33\,x^9}-\frac {A\,b^2\,\sqrt {b\,x^2+a}}{429\,a\,x^9}-\frac {A\,a\,\sqrt {b\,x^2+a}}{13\,x^{13}}-\frac {16\,A\,b^4\,\sqrt {b\,x^2+a}}{5005\,a^3\,x^5}+\frac {64\,A\,b^5\,\sqrt {b\,x^2+a}}{15015\,a^4\,x^3}-\frac {128\,A\,b^6\,\sqrt {b\,x^2+a}}{15015\,a^5\,x}-\frac {B\,b^2\,\sqrt {b\,x^2+a}}{231\,a\,x^7}+\frac {2\,B\,b^3\,\sqrt {b\,x^2+a}}{385\,a^2\,x^5}-\frac {8\,B\,b^4\,\sqrt {b\,x^2+a}}{1155\,a^3\,x^3}+\frac {16\,B\,b^5\,\sqrt {b\,x^2+a}}{1155\,a^4\,x} \] Input:
int(((A + B*x^2)*(a + b*x^2)^(3/2))/x^14,x)
Output:
(8*A*b^3*(a + b*x^2)^(1/2))/(3003*a^2*x^7) - (14*A*b*(a + b*x^2)^(1/2))/(1 43*x^11) - (B*a*(a + b*x^2)^(1/2))/(11*x^11) - (4*B*b*(a + b*x^2)^(1/2))/( 33*x^9) - (A*b^2*(a + b*x^2)^(1/2))/(429*a*x^9) - (A*a*(a + b*x^2)^(1/2))/ (13*x^13) - (16*A*b^4*(a + b*x^2)^(1/2))/(5005*a^3*x^5) + (64*A*b^5*(a + b *x^2)^(1/2))/(15015*a^4*x^3) - (128*A*b^6*(a + b*x^2)^(1/2))/(15015*a^5*x) - (B*b^2*(a + b*x^2)^(1/2))/(231*a*x^7) + (2*B*b^3*(a + b*x^2)^(1/2))/(38 5*a^2*x^5) - (8*B*b^4*(a + b*x^2)^(1/2))/(1155*a^3*x^3) + (16*B*b^5*(a + b *x^2)^(1/2))/(1155*a^4*x)
Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{14}} \, dx=\frac {-231 \sqrt {b \,x^{2}+a}\, a^{6}-567 \sqrt {b \,x^{2}+a}\, a^{5} b \,x^{2}-371 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} x^{4}-5 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} x^{6}+6 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{8}-8 \sqrt {b \,x^{2}+a}\, a \,b^{5} x^{10}+16 \sqrt {b \,x^{2}+a}\, b^{6} x^{12}-16 \sqrt {b}\, b^{6} x^{13}}{3003 a^{4} x^{13}} \] Input:
int((b*x^2+a)^(3/2)*(B*x^2+A)/x^14,x)
Output:
( - 231*sqrt(a + b*x**2)*a**6 - 567*sqrt(a + b*x**2)*a**5*b*x**2 - 371*sqr t(a + b*x**2)*a**4*b**2*x**4 - 5*sqrt(a + b*x**2)*a**3*b**3*x**6 + 6*sqrt( a + b*x**2)*a**2*b**4*x**8 - 8*sqrt(a + b*x**2)*a*b**5*x**10 + 16*sqrt(a + b*x**2)*b**6*x**12 - 16*sqrt(b)*b**6*x**13)/(3003*a**4*x**13)