Integrand size = 22, antiderivative size = 150 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx=-\frac {A \left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac {(8 A b-15 a B) \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac {2 b (8 A b-15 a B) \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac {8 b^2 (8 A b-15 a B) \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}-\frac {16 b^3 (8 A b-15 a B) \left (a+b x^2\right )^{7/2}}{45045 a^5 x^7} \] Output:
-1/15*A*(b*x^2+a)^(7/2)/a/x^15+1/195*(8*A*b-15*B*a)*(b*x^2+a)^(7/2)/a^2/x^ 13-2/715*b*(8*A*b-15*B*a)*(b*x^2+a)^(7/2)/a^3/x^11+8/6435*b^2*(8*A*b-15*B* a)*(b*x^2+a)^(7/2)/a^4/x^9-16/45045*b^3*(8*A*b-15*B*a)*(b*x^2+a)^(7/2)/a^5 /x^7
Time = 0.38 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx=\frac {\left (a+b x^2\right )^{7/2} \left (-128 A b^4 x^8-168 a^2 b^2 x^4 \left (6 A+5 B x^2\right )-231 a^4 \left (13 A+15 B x^2\right )+16 a b^3 x^6 \left (28 A+15 B x^2\right )+42 a^3 b x^2 \left (44 A+45 B x^2\right )\right )}{45045 a^5 x^{15}} \] Input:
Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^16,x]
Output:
((a + b*x^2)^(7/2)*(-128*A*b^4*x^8 - 168*a^2*b^2*x^4*(6*A + 5*B*x^2) - 231 *a^4*(13*A + 15*B*x^2) + 16*a*b^3*x^6*(28*A + 15*B*x^2) + 42*a^3*b*x^2*(44 *A + 45*B*x^2)))/(45045*a^5*x^15)
Time = 0.24 (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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {(8 A b-15 a B) \int \frac {\left (b x^2+a\right )^{5/2}}{x^{14}}dx}{15 a}-\frac {A \left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-15 a B) \left (-\frac {6 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{12}}dx}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {A \left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-15 a B) \left (-\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{10}}dx}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {A \left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {(8 A b-15 a B) \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^8}dx}{9 a}-\frac {\left (a+b x^2\right )^{7/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right )}{15 a}-\frac {A \left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {\left (-\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac {\left (a+b x^2\right )^{7/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\right )}{13 a}-\frac {\left (a+b x^2\right )^{7/2}}{13 a x^{13}}\right ) (8 A b-15 a B)}{15 a}-\frac {A \left (a+b x^2\right )^{7/2}}{15 a x^{15}}\) |
Input:
Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^16,x]
Output:
-1/15*(A*(a + b*x^2)^(7/2))/(a*x^15) - ((8*A*b - 15*a*B)*(-1/13*(a + b*x^2 )^(7/2)/(a*x^13) - (6*b*(-1/11*(a + b*x^2)^(7/2)/(a*x^11) - (4*b*(-1/9*(a + b*x^2)^(7/2)/(a*x^9) + (2*b*(a + b*x^2)^(7/2))/(63*a^2*x^7)))/(11*a)))/( 13*a)))/(15*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 = 0.79 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (\frac {15 x^{2} B}{13}+A \right ) a^{4}-\frac {8 b \left (\frac {45 x^{2} B}{44}+A \right ) x^{2} a^{3}}{13}+\frac {48 \left (\frac {5 x^{2} B}{6}+A \right ) b^{2} x^{4} a^{2}}{143}-\frac {64 \left (\frac {15 x^{2} B}{28}+A \right ) b^{3} x^{6} a}{429}+\frac {128 A \,x^{8} b^{4}}{3003}\right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{15 x^{15} a^{5}}\) | \(93\) |
| gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (128 A \,x^{8} b^{4}-240 B \,x^{8} a \,b^{3}-448 A \,x^{6} a \,b^{3}+840 B \,x^{6} a^{2} b^{2}+1008 A \,x^{4} a^{2} b^{2}-1890 B \,x^{4} a^{3} b -1848 A \,x^{2} a^{3} b +3465 B \,x^{2} a^{4}+3003 A \,a^{4}\right )}{45045 x^{15} a^{5}}\) | \(107\) |
| orering | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (128 A \,x^{8} b^{4}-240 B \,x^{8} a \,b^{3}-448 A \,x^{6} a \,b^{3}+840 B \,x^{6} a^{2} b^{2}+1008 A \,x^{4} a^{2} b^{2}-1890 B \,x^{4} a^{3} b -1848 A \,x^{2} a^{3} b +3465 B \,x^{2} a^{4}+3003 A \,a^{4}\right )}{45045 x^{15} a^{5}}\) | \(107\) |
| trager | \(-\frac {\left (128 A \,b^{7} x^{14}-240 B a \,b^{6} x^{14}-64 A a \,b^{6} x^{12}+120 B \,a^{2} b^{5} x^{12}+48 A \,a^{2} b^{5} x^{10}-90 B \,a^{3} b^{4} x^{10}-40 A \,a^{3} b^{4} x^{8}+75 B \,a^{4} b^{3} x^{8}+35 A \,a^{4} b^{3} x^{6}+5565 B \,a^{5} b^{2} x^{6}+4473 A \,a^{5} b^{2} x^{4}+8505 B \,a^{6} b \,x^{4}+7161 A \,a^{6} b \,x^{2}+3465 B \,a^{7} x^{2}+3003 A \,a^{7}\right ) \sqrt {b \,x^{2}+a}}{45045 x^{15} a^{5}}\) | \(179\) |
| risch | \(-\frac {\left (128 A \,b^{7} x^{14}-240 B a \,b^{6} x^{14}-64 A a \,b^{6} x^{12}+120 B \,a^{2} b^{5} x^{12}+48 A \,a^{2} b^{5} x^{10}-90 B \,a^{3} b^{4} x^{10}-40 A \,a^{3} b^{4} x^{8}+75 B \,a^{4} b^{3} x^{8}+35 A \,a^{4} b^{3} x^{6}+5565 B \,a^{5} b^{2} x^{6}+4473 A \,a^{5} b^{2} x^{4}+8505 B \,a^{6} b \,x^{4}+7161 A \,a^{6} b \,x^{2}+3465 B \,a^{7} x^{2}+3003 A \,a^{7}\right ) \sqrt {b \,x^{2}+a}}{45045 x^{15} a^{5}}\) | \(179\) |
| default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{15 a \,x^{15}}-\frac {8 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 a \,x^{13}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 a \,x^{11}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{13 a \,x^{13}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 a \,x^{11}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\right )\) | \(198\) |
Input:
int((b*x^2+a)^(5/2)*(B*x^2+A)/x^16,x,method=_RETURNVERBOSE)
Output:
-1/15*((15/13*x^2*B+A)*a^4-8/13*b*(45/44*x^2*B+A)*x^2*a^3+48/143*(5/6*x^2* B+A)*b^2*x^4*a^2-64/429*(15/28*x^2*B+A)*b^3*x^6*a+128/3003*A*x^8*b^4)*(b*x ^2+a)^(7/2)/x^15/a^5
Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx=\frac {{\left (16 \, {\left (15 \, B a b^{6} - 8 \, A b^{7}\right )} x^{14} - 8 \, {\left (15 \, B a^{2} b^{5} - 8 \, A a b^{6}\right )} x^{12} + 6 \, {\left (15 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )} x^{10} - 5 \, {\left (15 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{8} - 3003 \, A a^{7} - 35 \, {\left (159 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{6} - 63 \, {\left (135 \, B a^{6} b + 71 \, A a^{5} b^{2}\right )} x^{4} - 231 \, {\left (15 \, B a^{7} + 31 \, A a^{6} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{45045 \, a^{5} x^{15}} \] Input:
integrate((b*x^2+a)^(5/2)*(B*x^2+A)/x^16,x, algorithm="fricas")
Output:
1/45045*(16*(15*B*a*b^6 - 8*A*b^7)*x^14 - 8*(15*B*a^2*b^5 - 8*A*a*b^6)*x^1 2 + 6*(15*B*a^3*b^4 - 8*A*a^2*b^5)*x^10 - 5*(15*B*a^4*b^3 - 8*A*a^3*b^4)*x ^8 - 3003*A*a^7 - 35*(159*B*a^5*b^2 + A*a^4*b^3)*x^6 - 63*(135*B*a^6*b + 7 1*A*a^5*b^2)*x^4 - 231*(15*B*a^7 + 31*A*a^6*b)*x^2)*sqrt(b*x^2 + a)/(a^5*x ^15)
Leaf count of result is larger than twice the leaf count of optimal. 6210 vs. \(2 (146) = 292\).
Time = 8.07 (sec) , antiderivative size = 6210, normalized size of antiderivative = 41.40 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**16,x)
Output:
-3003*A*a**15*b**(73/2)*sqrt(a/(b*x**2) + 1)/(45045*a**13*b**36*x**14 + 27 0270*a**12*b**37*x**16 + 675675*a**11*b**38*x**18 + 900900*a**10*b**39*x** 20 + 675675*a**9*b**40*x**22 + 270270*a**8*b**41*x**24 + 45045*a**7*b**42* x**26) - 18249*A*a**14*b**(75/2)*x**2*sqrt(a/(b*x**2) + 1)/(45045*a**13*b* *36*x**14 + 270270*a**12*b**37*x**16 + 675675*a**11*b**38*x**18 + 900900*a **10*b**39*x**20 + 675675*a**9*b**40*x**22 + 270270*a**8*b**41*x**24 + 450 45*a**7*b**42*x**26) - 46179*A*a**13*b**(77/2)*x**4*sqrt(a/(b*x**2) + 1)/( 45045*a**13*b**36*x**14 + 270270*a**12*b**37*x**16 + 675675*a**11*b**38*x* *18 + 900900*a**10*b**39*x**20 + 675675*a**9*b**40*x**22 + 270270*a**8*b** 41*x**24 + 45045*a**7*b**42*x**26) - 62293*A*a**12*b**(79/2)*x**6*sqrt(a/( b*x**2) + 1)/(45045*a**13*b**36*x**14 + 270270*a**12*b**37*x**16 + 675675* a**11*b**38*x**18 + 900900*a**10*b**39*x**20 + 675675*a**9*b**40*x**22 + 2 70270*a**8*b**41*x**24 + 45045*a**7*b**42*x**26) - 1386*A*a**12*b**(53/2)* sqrt(a/(b*x**2) + 1)/(9009*a**11*b**25*x**12 + 45045*a**10*b**26*x**14 + 9 0090*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) - 47245*A*a**11*b**(81/2)*x**8*sqrt(a/(b*x**2) + 1) /(45045*a**13*b**36*x**14 + 270270*a**12*b**37*x**16 + 675675*a**11*b**38* x**18 + 900900*a**10*b**39*x**20 + 675675*a**9*b**40*x**22 + 270270*a**8*b **41*x**24 + 45045*a**7*b**42*x**26) - 7056*A*a**11*b**(55/2)*x**2*sqrt(a/ (b*x**2) + 1)/(9009*a**11*b**25*x**12 + 45045*a**10*b**26*x**14 + 90090...
Time = 0.05 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx=\frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B b^{3}}{3003 \, a^{4} x^{7}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{4}}{45045 \, a^{5} x^{7}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B b^{2}}{429 \, a^{3} x^{9}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{3}}{6435 \, a^{4} x^{9}} + \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B b}{143 \, a^{2} x^{11}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{2}}{715 \, a^{3} x^{11}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{13 \, a x^{13}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{195 \, a^{2} x^{13}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{15 \, a x^{15}} \] Input:
integrate((b*x^2+a)^(5/2)*(B*x^2+A)/x^16,x, algorithm="maxima")
Output:
16/3003*(b*x^2 + a)^(7/2)*B*b^3/(a^4*x^7) - 128/45045*(b*x^2 + a)^(7/2)*A* b^4/(a^5*x^7) - 8/429*(b*x^2 + a)^(7/2)*B*b^2/(a^3*x^9) + 64/6435*(b*x^2 + a)^(7/2)*A*b^3/(a^4*x^9) + 6/143*(b*x^2 + a)^(7/2)*B*b/(a^2*x^11) - 16/71 5*(b*x^2 + a)^(7/2)*A*b^2/(a^3*x^11) - 1/13*(b*x^2 + a)^(7/2)*B/(a*x^13) + 8/195*(b*x^2 + a)^(7/2)*A*b/(a^2*x^13) - 1/15*(b*x^2 + a)^(7/2)*A/(a*x^15 )
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (130) = 260\).
Time = 0.14 (sec) , antiderivative size = 624, normalized size of antiderivative = 4.16 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx=\frac {32 \, {\left (45045 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} B b^{\frac {13}{2}} + 45045 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} B a b^{\frac {13}{2}} + 144144 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} A b^{\frac {15}{2}} + 45045 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} B a^{2} b^{\frac {13}{2}} + 480480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} A a b^{\frac {15}{2}} - 160875 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} B a^{3} b^{\frac {13}{2}} + 926640 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} A a^{2} b^{\frac {15}{2}} - 64350 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} B a^{4} b^{\frac {13}{2}} + 875160 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} A a^{3} b^{\frac {15}{2}} - 30030 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a^{5} b^{\frac {13}{2}} + 520520 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A a^{4} b^{\frac {15}{2}} + 90090 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{6} b^{\frac {13}{2}} + 120120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a^{5} b^{\frac {15}{2}} + 24570 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{7} b^{\frac {13}{2}} + 10920 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{6} b^{\frac {15}{2}} + 6825 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{8} b^{\frac {13}{2}} - 3640 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{7} b^{\frac {15}{2}} - 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{9} b^{\frac {13}{2}} + 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{8} b^{\frac {15}{2}} + 225 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{10} b^{\frac {13}{2}} - 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{9} b^{\frac {15}{2}} - 15 \, B a^{11} b^{\frac {13}{2}} + 8 \, A a^{10} b^{\frac {15}{2}}\right )}}{45045 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{15}} \] Input:
integrate((b*x^2+a)^(5/2)*(B*x^2+A)/x^16,x, algorithm="giac")
Output:
32/45045*(45045*(sqrt(b)*x - sqrt(b*x^2 + a))^22*B*b^(13/2) + 45045*(sqrt( b)*x - sqrt(b*x^2 + a))^20*B*a*b^(13/2) + 144144*(sqrt(b)*x - sqrt(b*x^2 + a))^20*A*b^(15/2) + 45045*(sqrt(b)*x - sqrt(b*x^2 + a))^18*B*a^2*b^(13/2) + 480480*(sqrt(b)*x - sqrt(b*x^2 + a))^18*A*a*b^(15/2) - 160875*(sqrt(b)* x - sqrt(b*x^2 + a))^16*B*a^3*b^(13/2) + 926640*(sqrt(b)*x - sqrt(b*x^2 + a))^16*A*a^2*b^(15/2) - 64350*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^4*b^(13 /2) + 875160*(sqrt(b)*x - sqrt(b*x^2 + a))^14*A*a^3*b^(15/2) - 30030*(sqrt (b)*x - sqrt(b*x^2 + a))^12*B*a^5*b^(13/2) + 520520*(sqrt(b)*x - sqrt(b*x^ 2 + a))^12*A*a^4*b^(15/2) + 90090*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^6*b ^(13/2) + 120120*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a^5*b^(15/2) + 24570*( sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^7*b^(13/2) + 10920*(sqrt(b)*x - sqrt(b* x^2 + a))^8*A*a^6*b^(15/2) + 6825*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^8*b^ (13/2) - 3640*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^7*b^(15/2) - 1575*(sqrt( b)*x - sqrt(b*x^2 + a))^4*B*a^9*b^(13/2) + 840*(sqrt(b)*x - sqrt(b*x^2 + a ))^4*A*a^8*b^(15/2) + 225*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^10*b^(13/2) - 120*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^9*b^(15/2) - 15*B*a^11*b^(13/2) + 8*A*a^10*b^(15/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^15
Time = 4.91 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx=\frac {8\,A\,b^4\,\sqrt {b\,x^2+a}}{9009\,a^2\,x^7}-\frac {71\,A\,b^2\,\sqrt {b\,x^2+a}}{715\,x^{11}}-\frac {B\,a^2\,\sqrt {b\,x^2+a}}{13\,x^{13}}-\frac {53\,B\,b^2\,\sqrt {b\,x^2+a}}{429\,x^9}-\frac {A\,b^3\,\sqrt {b\,x^2+a}}{1287\,a\,x^9}-\frac {A\,a^2\,\sqrt {b\,x^2+a}}{15\,x^{15}}-\frac {16\,A\,b^5\,\sqrt {b\,x^2+a}}{15015\,a^3\,x^5}+\frac {64\,A\,b^6\,\sqrt {b\,x^2+a}}{45045\,a^4\,x^3}-\frac {128\,A\,b^7\,\sqrt {b\,x^2+a}}{45045\,a^5\,x}-\frac {5\,B\,b^3\,\sqrt {b\,x^2+a}}{3003\,a\,x^7}+\frac {2\,B\,b^4\,\sqrt {b\,x^2+a}}{1001\,a^2\,x^5}-\frac {8\,B\,b^5\,\sqrt {b\,x^2+a}}{3003\,a^3\,x^3}+\frac {16\,B\,b^6\,\sqrt {b\,x^2+a}}{3003\,a^4\,x}-\frac {31\,A\,a\,b\,\sqrt {b\,x^2+a}}{195\,x^{13}}-\frac {27\,B\,a\,b\,\sqrt {b\,x^2+a}}{143\,x^{11}} \] Input:
int(((A + B*x^2)*(a + b*x^2)^(5/2))/x^16,x)
Output:
(8*A*b^4*(a + b*x^2)^(1/2))/(9009*a^2*x^7) - (71*A*b^2*(a + b*x^2)^(1/2))/ (715*x^11) - (B*a^2*(a + b*x^2)^(1/2))/(13*x^13) - (53*B*b^2*(a + b*x^2)^( 1/2))/(429*x^9) - (A*b^3*(a + b*x^2)^(1/2))/(1287*a*x^9) - (A*a^2*(a + b*x ^2)^(1/2))/(15*x^15) - (16*A*b^5*(a + b*x^2)^(1/2))/(15015*a^3*x^5) + (64* A*b^6*(a + b*x^2)^(1/2))/(45045*a^4*x^3) - (128*A*b^7*(a + b*x^2)^(1/2))/( 45045*a^5*x) - (5*B*b^3*(a + b*x^2)^(1/2))/(3003*a*x^7) + (2*B*b^4*(a + b* x^2)^(1/2))/(1001*a^2*x^5) - (8*B*b^5*(a + b*x^2)^(1/2))/(3003*a^3*x^3) + (16*B*b^6*(a + b*x^2)^(1/2))/(3003*a^4*x) - (31*A*a*b*(a + b*x^2)^(1/2))/( 195*x^13) - (27*B*a*b*(a + b*x^2)^(1/2))/(143*x^11)
Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{16}} \, dx=\frac {-429 \sqrt {b \,x^{2}+a}\, a^{7}-1518 \sqrt {b \,x^{2}+a}\, a^{6} b \,x^{2}-1854 \sqrt {b \,x^{2}+a}\, a^{5} b^{2} x^{4}-800 \sqrt {b \,x^{2}+a}\, a^{4} b^{3} x^{6}-5 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} x^{8}+6 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} x^{10}-8 \sqrt {b \,x^{2}+a}\, a \,b^{6} x^{12}+16 \sqrt {b \,x^{2}+a}\, b^{7} x^{14}-16 \sqrt {b}\, b^{7} x^{15}}{6435 a^{4} x^{15}} \] Input:
int((b*x^2+a)^(5/2)*(B*x^2+A)/x^16,x)
Output:
( - 429*sqrt(a + b*x**2)*a**7 - 1518*sqrt(a + b*x**2)*a**6*b*x**2 - 1854*s qrt(a + b*x**2)*a**5*b**2*x**4 - 800*sqrt(a + b*x**2)*a**4*b**3*x**6 - 5*s qrt(a + b*x**2)*a**3*b**4*x**8 + 6*sqrt(a + b*x**2)*a**2*b**5*x**10 - 8*sq rt(a + b*x**2)*a*b**6*x**12 + 16*sqrt(a + b*x**2)*b**7*x**14 - 16*sqrt(b)* b**7*x**15)/(6435*a**4*x**15)