Integrand size = 13, antiderivative size = 216 \[ \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx=\frac {55 a^2 x^2}{2 b^{12}}-\frac {5 a x^4}{2 b^{11}}+\frac {x^6}{6 b^{10}}-\frac {a^{12}}{18 b^{13} \left (a+b x^2\right )^9}+\frac {3 a^{11}}{4 b^{13} \left (a+b x^2\right )^8}-\frac {33 a^{10}}{7 b^{13} \left (a+b x^2\right )^7}+\frac {55 a^9}{3 b^{13} \left (a+b x^2\right )^6}-\frac {99 a^8}{2 b^{13} \left (a+b x^2\right )^5}+\frac {99 a^7}{b^{13} \left (a+b x^2\right )^4}-\frac {154 a^6}{b^{13} \left (a+b x^2\right )^3}+\frac {198 a^5}{b^{13} \left (a+b x^2\right )^2}-\frac {495 a^4}{2 b^{13} \left (a+b x^2\right )}-\frac {110 a^3 \log \left (a+b x^2\right )}{b^{13}} \] Output:
55/2*a^2*x^2/b^12-5/2*a*x^4/b^11+1/6*x^6/b^10-1/18*a^12/b^13/(b*x^2+a)^9+3 /4*a^11/b^13/(b*x^2+a)^8-33/7*a^10/b^13/(b*x^2+a)^7+55/3*a^9/b^13/(b*x^2+a )^6-99/2*a^8/b^13/(b*x^2+a)^5+99*a^7/b^13/(b*x^2+a)^4-154*a^6/b^13/(b*x^2+ a)^3+198*a^5/b^13/(b*x^2+a)^2-495/2*a^4/b^13/(b*x^2+a)-110*a^3*ln(b*x^2+a) /b^13
Time = 0.03 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.78 \[ \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {35201 a^{12}+289089 a^{11} b x^2+1031616 a^{10} b^2 x^4+2074464 a^9 b^3 x^6+2529576 a^8 b^4 x^8+1831032 a^7 b^5 x^{10}+638568 a^6 b^6 x^{12}-58968 a^5 b^7 x^{14}-139482 a^4 b^8 x^{16}-43218 a^3 b^9 x^{18}-2772 a^2 b^{10} x^{20}+252 a b^{11} x^{22}-42 b^{12} x^{24}+27720 a^3 \left (a+b x^2\right )^9 \log \left (a+b x^2\right )}{252 b^{13} \left (a+b x^2\right )^9} \] Input:
Integrate[x^25/(a + b*x^2)^10,x]
Output:
-1/252*(35201*a^12 + 289089*a^11*b*x^2 + 1031616*a^10*b^2*x^4 + 2074464*a^ 9*b^3*x^6 + 2529576*a^8*b^4*x^8 + 1831032*a^7*b^5*x^10 + 638568*a^6*b^6*x^ 12 - 58968*a^5*b^7*x^14 - 139482*a^4*b^8*x^16 - 43218*a^3*b^9*x^18 - 2772* a^2*b^10*x^20 + 252*a*b^11*x^22 - 42*b^12*x^24 + 27720*a^3*(a + b*x^2)^9*L og[a + b*x^2])/(b^13*(a + b*x^2)^9)
Time = 0.40 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {243, 49, 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.
| \(\displaystyle \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {x^{24}}{\left (b x^2+a\right )^{10}}dx^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {a^{12}}{b^{12} \left (b x^2+a\right )^{10}}-\frac {12 a^{11}}{b^{12} \left (b x^2+a\right )^9}+\frac {66 a^{10}}{b^{12} \left (b x^2+a\right )^8}-\frac {220 a^9}{b^{12} \left (b x^2+a\right )^7}+\frac {495 a^8}{b^{12} \left (b x^2+a\right )^6}-\frac {792 a^7}{b^{12} \left (b x^2+a\right )^5}+\frac {924 a^6}{b^{12} \left (b x^2+a\right )^4}-\frac {792 a^5}{b^{12} \left (b x^2+a\right )^3}+\frac {495 a^4}{b^{12} \left (b x^2+a\right )^2}-\frac {220 a^3}{b^{12} \left (b x^2+a\right )}+\frac {55 a^2}{b^{12}}-\frac {10 x^2 a}{b^{11}}+\frac {x^4}{b^{10}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a^{12}}{9 b^{13} \left (a+b x^2\right )^9}+\frac {3 a^{11}}{2 b^{13} \left (a+b x^2\right )^8}-\frac {66 a^{10}}{7 b^{13} \left (a+b x^2\right )^7}+\frac {110 a^9}{3 b^{13} \left (a+b x^2\right )^6}-\frac {99 a^8}{b^{13} \left (a+b x^2\right )^5}+\frac {198 a^7}{b^{13} \left (a+b x^2\right )^4}-\frac {308 a^6}{b^{13} \left (a+b x^2\right )^3}+\frac {396 a^5}{b^{13} \left (a+b x^2\right )^2}-\frac {495 a^4}{b^{13} \left (a+b x^2\right )}-\frac {220 a^3 \log \left (a+b x^2\right )}{b^{13}}+\frac {55 a^2 x^2}{b^{12}}-\frac {5 a x^4}{b^{11}}+\frac {x^6}{3 b^{10}}\right )\) |
Input:
Int[x^25/(a + b*x^2)^10,x]
Output:
((55*a^2*x^2)/b^12 - (5*a*x^4)/b^11 + x^6/(3*b^10) - a^12/(9*b^13*(a + b*x ^2)^9) + (3*a^11)/(2*b^13*(a + b*x^2)^8) - (66*a^10)/(7*b^13*(a + b*x^2)^7 ) + (110*a^9)/(3*b^13*(a + b*x^2)^6) - (99*a^8)/(b^13*(a + b*x^2)^5) + (19 8*a^7)/(b^13*(a + b*x^2)^4) - (308*a^6)/(b^13*(a + b*x^2)^3) + (396*a^5)/( b^13*(a + b*x^2)^2) - (495*a^4)/(b^13*(a + b*x^2)) - (220*a^3*Log[a + b*x^ 2])/b^13)/2
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {x^{6}}{6 b^{10}}-\frac {5 a \,x^{4}}{2 b^{11}}+\frac {55 a^{2} x^{2}}{2 b^{12}}+\frac {-\frac {35201 a^{12}}{252 b}-\frac {32891 a^{11} x^{2}}{28}-\frac {30371 a^{10} b \,x^{4}}{7}-\frac {27599 b^{2} a^{9} x^{6}}{3}-\frac {24519 b^{3} a^{8} x^{8}}{2}-10527 a^{7} b^{4} x^{10}-5698 a^{6} b^{5} x^{12}-1782 a^{5} b^{6} x^{14}-\frac {495 a^{4} b^{7} x^{16}}{2}}{b^{12} \left (b \,x^{2}+a \right )^{9}}-\frac {110 a^{3} \ln \left (b \,x^{2}+a \right )}{b^{13}}\) | \(151\) |
| norman | \(\frac {\frac {x^{24}}{6 b}-\frac {a \,x^{22}}{b^{2}}+\frac {11 a^{2} x^{20}}{b^{3}}-\frac {78419 a^{12}}{252 b^{13}}-\frac {990 a^{4} x^{16}}{b^{5}}-\frac {5940 a^{5} x^{14}}{b^{6}}-\frac {16940 a^{6} x^{12}}{b^{7}}-\frac {28875 a^{7} x^{10}}{b^{8}}-\frac {31647 a^{8} x^{8}}{b^{9}}-\frac {22638 a^{9} x^{6}}{b^{10}}-\frac {71874 a^{10} x^{4}}{7 b^{11}}-\frac {75339 a^{11} x^{2}}{28 b^{12}}}{\left (b \,x^{2}+a \right )^{9}}-\frac {110 a^{3} \ln \left (b \,x^{2}+a \right )}{b^{13}}\) | \(153\) |
| default | \(\frac {\frac {1}{6} b^{2} x^{6}-\frac {5}{2} a b \,x^{4}+\frac {55}{2} a^{2} x^{2}}{b^{12}}-\frac {a^{3} \left (\frac {a^{9}}{9 b \left (b \,x^{2}+a \right )^{9}}+\frac {99 a^{5}}{b \left (b \,x^{2}+a \right )^{5}}-\frac {198 a^{4}}{b \left (b \,x^{2}+a \right )^{4}}+\frac {66 a^{7}}{7 b \left (b \,x^{2}+a \right )^{7}}-\frac {3 a^{8}}{2 b \left (b \,x^{2}+a \right )^{8}}-\frac {396 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}+\frac {495 a}{b \left (b \,x^{2}+a \right )}+\frac {220 \ln \left (b \,x^{2}+a \right )}{b}+\frac {308 a^{3}}{b \left (b \,x^{2}+a \right )^{3}}-\frac {110 a^{6}}{3 b \left (b \,x^{2}+a \right )^{6}}\right )}{2 b^{12}}\) | \(203\) |
| parallelrisch | \(-\frac {78419 a^{12}+27720 \ln \left (b \,x^{2}+a \right ) x^{18} a^{3} b^{9}+249480 \ln \left (b \,x^{2}+a \right ) x^{16} a^{4} b^{8}+997920 \ln \left (b \,x^{2}+a \right ) x^{14} a^{5} b^{7}+2328480 \ln \left (b \,x^{2}+a \right ) x^{12} a^{6} b^{6}+3492720 \ln \left (b \,x^{2}+a \right ) x^{10} a^{7} b^{5}+3492720 \ln \left (b \,x^{2}+a \right ) x^{8} a^{8} b^{4}+2328480 \ln \left (b \,x^{2}+a \right ) x^{6} a^{9} b^{3}+997920 \ln \left (b \,x^{2}+a \right ) x^{4} a^{10} b^{2}+249480 \ln \left (b \,x^{2}+a \right ) x^{2} a^{11} b -42 b^{12} x^{24}+27720 \ln \left (b \,x^{2}+a \right ) a^{12}-2772 a^{2} x^{20} b^{10}+252 a \,x^{22} b^{11}+678051 x^{2} a^{11} b +5704776 x^{6} a^{9} b^{3}+7975044 x^{8} a^{8} b^{4}+7276500 x^{10} a^{7} b^{5}+4268880 x^{12} a^{6} b^{6}+1496880 x^{14} a^{5} b^{7}+249480 x^{16} a^{4} b^{8}+2587464 x^{4} a^{10} b^{2}}{252 b^{13} \left (b \,x^{2}+a \right )^{9}}\) | \(317\) |
Input:
int(x^25/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
Output:
1/6*x^6/b^10-5/2*a*x^4/b^11+55/2*a^2*x^2/b^12+(-35201/252*a^12/b-32891/28* a^11*x^2-30371/7*a^10*b*x^4-27599/3*b^2*a^9*x^6-24519/2*b^3*a^8*x^8-10527* a^7*b^4*x^10-5698*a^6*b^5*x^12-1782*a^5*b^6*x^14-495/2*a^4*b^7*x^16)/b^12/ (b*x^2+a)^9-110*a^3*ln(b*x^2+a)/b^13
Time = 0.06 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.60 \[ \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx=\frac {42 \, b^{12} x^{24} - 252 \, a b^{11} x^{22} + 2772 \, a^{2} b^{10} x^{20} + 43218 \, a^{3} b^{9} x^{18} + 139482 \, a^{4} b^{8} x^{16} + 58968 \, a^{5} b^{7} x^{14} - 638568 \, a^{6} b^{6} x^{12} - 1831032 \, a^{7} b^{5} x^{10} - 2529576 \, a^{8} b^{4} x^{8} - 2074464 \, a^{9} b^{3} x^{6} - 1031616 \, a^{10} b^{2} x^{4} - 289089 \, a^{11} b x^{2} - 35201 \, a^{12} - 27720 \, {\left (a^{3} b^{9} x^{18} + 9 \, a^{4} b^{8} x^{16} + 36 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 126 \, a^{7} b^{5} x^{10} + 126 \, a^{8} b^{4} x^{8} + 84 \, a^{9} b^{3} x^{6} + 36 \, a^{10} b^{2} x^{4} + 9 \, a^{11} b x^{2} + a^{12}\right )} \log \left (b x^{2} + a\right )}{252 \, {\left (b^{22} x^{18} + 9 \, a b^{21} x^{16} + 36 \, a^{2} b^{20} x^{14} + 84 \, a^{3} b^{19} x^{12} + 126 \, a^{4} b^{18} x^{10} + 126 \, a^{5} b^{17} x^{8} + 84 \, a^{6} b^{16} x^{6} + 36 \, a^{7} b^{15} x^{4} + 9 \, a^{8} b^{14} x^{2} + a^{9} b^{13}\right )}} \] Input:
integrate(x^25/(b*x^2+a)^10,x, algorithm="fricas")
Output:
1/252*(42*b^12*x^24 - 252*a*b^11*x^22 + 2772*a^2*b^10*x^20 + 43218*a^3*b^9 *x^18 + 139482*a^4*b^8*x^16 + 58968*a^5*b^7*x^14 - 638568*a^6*b^6*x^12 - 1 831032*a^7*b^5*x^10 - 2529576*a^8*b^4*x^8 - 2074464*a^9*b^3*x^6 - 1031616* a^10*b^2*x^4 - 289089*a^11*b*x^2 - 35201*a^12 - 27720*(a^3*b^9*x^18 + 9*a^ 4*b^8*x^16 + 36*a^5*b^7*x^14 + 84*a^6*b^6*x^12 + 126*a^7*b^5*x^10 + 126*a^ 8*b^4*x^8 + 84*a^9*b^3*x^6 + 36*a^10*b^2*x^4 + 9*a^11*b*x^2 + a^12)*log(b* x^2 + a))/(b^22*x^18 + 9*a*b^21*x^16 + 36*a^2*b^20*x^14 + 84*a^3*b^19*x^12 + 126*a^4*b^18*x^10 + 126*a^5*b^17*x^8 + 84*a^6*b^16*x^6 + 36*a^7*b^15*x^ 4 + 9*a^8*b^14*x^2 + a^9*b^13)
Time = 1.16 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.20 \[ \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx=- \frac {110 a^{3} \log {\left (a + b x^{2} \right )}}{b^{13}} + \frac {55 a^{2} x^{2}}{2 b^{12}} - \frac {5 a x^{4}}{2 b^{11}} + \frac {- 35201 a^{12} - 296019 a^{11} b x^{2} - 1093356 a^{10} b^{2} x^{4} - 2318316 a^{9} b^{3} x^{6} - 3089394 a^{8} b^{4} x^{8} - 2652804 a^{7} b^{5} x^{10} - 1435896 a^{6} b^{6} x^{12} - 449064 a^{5} b^{7} x^{14} - 62370 a^{4} b^{8} x^{16}}{252 a^{9} b^{13} + 2268 a^{8} b^{14} x^{2} + 9072 a^{7} b^{15} x^{4} + 21168 a^{6} b^{16} x^{6} + 31752 a^{5} b^{17} x^{8} + 31752 a^{4} b^{18} x^{10} + 21168 a^{3} b^{19} x^{12} + 9072 a^{2} b^{20} x^{14} + 2268 a b^{21} x^{16} + 252 b^{22} x^{18}} + \frac {x^{6}}{6 b^{10}} \] Input:
integrate(x**25/(b*x**2+a)**10,x)
Output:
-110*a**3*log(a + b*x**2)/b**13 + 55*a**2*x**2/(2*b**12) - 5*a*x**4/(2*b** 11) + (-35201*a**12 - 296019*a**11*b*x**2 - 1093356*a**10*b**2*x**4 - 2318 316*a**9*b**3*x**6 - 3089394*a**8*b**4*x**8 - 2652804*a**7*b**5*x**10 - 14 35896*a**6*b**6*x**12 - 449064*a**5*b**7*x**14 - 62370*a**4*b**8*x**16)/(2 52*a**9*b**13 + 2268*a**8*b**14*x**2 + 9072*a**7*b**15*x**4 + 21168*a**6*b **16*x**6 + 31752*a**5*b**17*x**8 + 31752*a**4*b**18*x**10 + 21168*a**3*b* *19*x**12 + 9072*a**2*b**20*x**14 + 2268*a*b**21*x**16 + 252*b**22*x**18) + x**6/(6*b**10)
Time = 0.05 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.12 \[ \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {62370 \, a^{4} b^{8} x^{16} + 449064 \, a^{5} b^{7} x^{14} + 1435896 \, a^{6} b^{6} x^{12} + 2652804 \, a^{7} b^{5} x^{10} + 3089394 \, a^{8} b^{4} x^{8} + 2318316 \, a^{9} b^{3} x^{6} + 1093356 \, a^{10} b^{2} x^{4} + 296019 \, a^{11} b x^{2} + 35201 \, a^{12}}{252 \, {\left (b^{22} x^{18} + 9 \, a b^{21} x^{16} + 36 \, a^{2} b^{20} x^{14} + 84 \, a^{3} b^{19} x^{12} + 126 \, a^{4} b^{18} x^{10} + 126 \, a^{5} b^{17} x^{8} + 84 \, a^{6} b^{16} x^{6} + 36 \, a^{7} b^{15} x^{4} + 9 \, a^{8} b^{14} x^{2} + a^{9} b^{13}\right )}} - \frac {110 \, a^{3} \log \left (b x^{2} + a\right )}{b^{13}} + \frac {b^{2} x^{6} - 15 \, a b x^{4} + 165 \, a^{2} x^{2}}{6 \, b^{12}} \] Input:
integrate(x^25/(b*x^2+a)^10,x, algorithm="maxima")
Output:
-1/252*(62370*a^4*b^8*x^16 + 449064*a^5*b^7*x^14 + 1435896*a^6*b^6*x^12 + 2652804*a^7*b^5*x^10 + 3089394*a^8*b^4*x^8 + 2318316*a^9*b^3*x^6 + 1093356 *a^10*b^2*x^4 + 296019*a^11*b*x^2 + 35201*a^12)/(b^22*x^18 + 9*a*b^21*x^16 + 36*a^2*b^20*x^14 + 84*a^3*b^19*x^12 + 126*a^4*b^18*x^10 + 126*a^5*b^17* x^8 + 84*a^6*b^16*x^6 + 36*a^7*b^15*x^4 + 9*a^8*b^14*x^2 + a^9*b^13) - 110 *a^3*log(b*x^2 + a)/b^13 + 1/6*(b^2*x^6 - 15*a*b*x^4 + 165*a^2*x^2)/b^12
Time = 0.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.78 \[ \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx=-\frac {110 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{13}} + \frac {78419 \, a^{3} b^{9} x^{18} + 643401 \, a^{4} b^{8} x^{16} + 2374020 \, a^{5} b^{7} x^{14} + 5151300 \, a^{6} b^{6} x^{12} + 7227990 \, a^{7} b^{5} x^{10} + 6791400 \, a^{8} b^{4} x^{8} + 4268880 \, a^{9} b^{3} x^{6} + 1729728 \, a^{10} b^{2} x^{4} + 409752 \, a^{11} b x^{2} + 43218 \, a^{12}}{252 \, {\left (b x^{2} + a\right )}^{9} b^{13}} + \frac {b^{20} x^{6} - 15 \, a b^{19} x^{4} + 165 \, a^{2} b^{18} x^{2}}{6 \, b^{30}} \] Input:
integrate(x^25/(b*x^2+a)^10,x, algorithm="giac")
Output:
-110*a^3*log(abs(b*x^2 + a))/b^13 + 1/252*(78419*a^3*b^9*x^18 + 643401*a^4 *b^8*x^16 + 2374020*a^5*b^7*x^14 + 5151300*a^6*b^6*x^12 + 7227990*a^7*b^5* x^10 + 6791400*a^8*b^4*x^8 + 4268880*a^9*b^3*x^6 + 1729728*a^10*b^2*x^4 + 409752*a^11*b*x^2 + 43218*a^12)/((b*x^2 + a)^9*b^13) + 1/6*(b^20*x^6 - 15* a*b^19*x^4 + 165*a^2*b^18*x^2)/b^30
Time = 0.48 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.12 \[ \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx=\frac {x^6}{6\,b^{10}}-\frac {\frac {35201\,a^{12}}{252\,b}+\frac {32891\,a^{11}\,x^2}{28}+\frac {30371\,a^{10}\,b\,x^4}{7}+\frac {27599\,a^9\,b^2\,x^6}{3}+\frac {24519\,a^8\,b^3\,x^8}{2}+10527\,a^7\,b^4\,x^{10}+5698\,a^6\,b^5\,x^{12}+1782\,a^5\,b^6\,x^{14}+\frac {495\,a^4\,b^7\,x^{16}}{2}}{a^9\,b^{12}+9\,a^8\,b^{13}\,x^2+36\,a^7\,b^{14}\,x^4+84\,a^6\,b^{15}\,x^6+126\,a^5\,b^{16}\,x^8+126\,a^4\,b^{17}\,x^{10}+84\,a^3\,b^{18}\,x^{12}+36\,a^2\,b^{19}\,x^{14}+9\,a\,b^{20}\,x^{16}+b^{21}\,x^{18}}-\frac {5\,a\,x^4}{2\,b^{11}}-\frac {110\,a^3\,\ln \left (b\,x^2+a\right )}{b^{13}}+\frac {55\,a^2\,x^2}{2\,b^{12}} \] Input:
int(x^25/(a + b*x^2)^10,x)
Output:
x^6/(6*b^10) - ((35201*a^12)/(252*b) + (32891*a^11*x^2)/28 + (30371*a^10*b *x^4)/7 + (27599*a^9*b^2*x^6)/3 + (24519*a^8*b^3*x^8)/2 + 10527*a^7*b^4*x^ 10 + 5698*a^6*b^5*x^12 + 1782*a^5*b^6*x^14 + (495*a^4*b^7*x^16)/2)/(a^9*b^ 12 + b^21*x^18 + 9*a*b^20*x^16 + 9*a^8*b^13*x^2 + 36*a^7*b^14*x^4 + 84*a^6 *b^15*x^6 + 126*a^5*b^16*x^8 + 126*a^4*b^17*x^10 + 84*a^3*b^18*x^12 + 36*a ^2*b^19*x^14) - (5*a*x^4)/(2*b^11) - (110*a^3*log(a + b*x^2))/b^13 + (55*a ^2*x^2)/(2*b^12)
Time = 0.25 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.87 \[ \int \frac {x^{25}}{\left (a+b x^2\right )^{10}} \, dx=\frac {-27720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{12}-249480 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{11} b \,x^{2}-997920 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{10} b^{2} x^{4}-2328480 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{9} b^{3} x^{6}-3492720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{8} b^{4} x^{8}-3492720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{7} b^{5} x^{10}-2328480 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{6} b^{6} x^{12}-997920 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{5} b^{7} x^{14}-249480 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} b^{8} x^{16}-27720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b^{9} x^{18}-50699 a^{12}-428571 a^{11} b \,x^{2}-1589544 a^{10} b^{2} x^{4}-3376296 a^{9} b^{3} x^{6}-4482324 a^{8} b^{4} x^{8}-3783780 a^{7} b^{5} x^{10}-1940400 a^{6} b^{6} x^{12}-498960 a^{5} b^{7} x^{14}+27720 a^{3} b^{9} x^{18}+2772 a^{2} b^{10} x^{20}-252 a \,b^{11} x^{22}+42 b^{12} x^{24}}{252 b^{13} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:
int(x^25/(b*x^2+a)^10,x)
Output:
( - 27720*log(a + b*x**2)*a**12 - 249480*log(a + b*x**2)*a**11*b*x**2 - 99 7920*log(a + b*x**2)*a**10*b**2*x**4 - 2328480*log(a + b*x**2)*a**9*b**3*x **6 - 3492720*log(a + b*x**2)*a**8*b**4*x**8 - 3492720*log(a + b*x**2)*a** 7*b**5*x**10 - 2328480*log(a + b*x**2)*a**6*b**6*x**12 - 997920*log(a + b* x**2)*a**5*b**7*x**14 - 249480*log(a + b*x**2)*a**4*b**8*x**16 - 27720*log (a + b*x**2)*a**3*b**9*x**18 - 50699*a**12 - 428571*a**11*b*x**2 - 1589544 *a**10*b**2*x**4 - 3376296*a**9*b**3*x**6 - 4482324*a**8*b**4*x**8 - 37837 80*a**7*b**5*x**10 - 1940400*a**6*b**6*x**12 - 498960*a**5*b**7*x**14 + 27 720*a**3*b**9*x**18 + 2772*a**2*b**10*x**20 - 252*a*b**11*x**22 + 42*b**12 *x**24)/(252*b**13*(a**9 + 9*a**8*b*x**2 + 36*a**7*b**2*x**4 + 84*a**6*b** 3*x**6 + 126*a**5*b**4*x**8 + 126*a**4*b**5*x**10 + 84*a**3*b**6*x**12 + 3 6*a**2*b**7*x**14 + 9*a*b**8*x**16 + b**9*x**18))