Integrand size = 13, antiderivative size = 226 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^{10}} \, dx=-\frac {1}{6 a^{10} x^6}+\frac {5 b}{2 a^{11} x^4}-\frac {55 b^2}{2 a^{12} x^2}-\frac {b^3}{18 a^4 \left (a+b x^2\right )^9}-\frac {b^3}{4 a^5 \left (a+b x^2\right )^8}-\frac {5 b^3}{7 a^6 \left (a+b x^2\right )^7}-\frac {5 b^3}{3 a^7 \left (a+b x^2\right )^6}-\frac {7 b^3}{2 a^8 \left (a+b x^2\right )^5}-\frac {7 b^3}{a^9 \left (a+b x^2\right )^4}-\frac {14 b^3}{a^{10} \left (a+b x^2\right )^3}-\frac {30 b^3}{a^{11} \left (a+b x^2\right )^2}-\frac {165 b^3}{2 a^{12} \left (a+b x^2\right )}-\frac {220 b^3 \log (x)}{a^{13}}+\frac {110 b^3 \log \left (a+b x^2\right )}{a^{13}} \]
-1/6/a^10/x^6+5/2*b/a^11/x^4-55/2*b^2/a^12/x^2-1/18*b^3/a^4/(b*x^2+a)^9-1/ 4*b^3/a^5/(b*x^2+a)^8-5/7*b^3/a^6/(b*x^2+a)^7-5/3*b^3/a^7/(b*x^2+a)^6-7/2* b^3/a^8/(b*x^2+a)^5-7*b^3/a^9/(b*x^2+a)^4-14*b^3/a^10/(b*x^2+a)^3-30*b^3/a ^11/(b*x^2+a)^2-165/2*b^3/a^12/(b*x^2+a)-220*b^3*ln(x)/a^13+110*b^3*ln(b*x ^2+a)/a^13
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^{10}} \, dx=-\frac {\frac {a \left (42 a^{11}-252 a^{10} b x^2+2772 a^9 b^2 x^4+78419 a^8 b^3 x^6+456291 a^7 b^4 x^8+1326204 a^6 b^5 x^{10}+2318316 a^5 b^6 x^{12}+2604294 a^4 b^7 x^{14}+1905750 a^3 b^8 x^{16}+882420 a^2 b^9 x^{18}+235620 a b^{10} x^{20}+27720 b^{11} x^{22}\right )}{x^6 \left (a+b x^2\right )^9}+55440 b^3 \log (x)-27720 b^3 \log \left (a+b x^2\right )}{252 a^{13}} \]
-1/252*((a*(42*a^11 - 252*a^10*b*x^2 + 2772*a^9*b^2*x^4 + 78419*a^8*b^3*x^ 6 + 456291*a^7*b^4*x^8 + 1326204*a^6*b^5*x^10 + 2318316*a^5*b^6*x^12 + 260 4294*a^4*b^7*x^14 + 1905750*a^3*b^8*x^16 + 882420*a^2*b^9*x^18 + 235620*a* b^10*x^20 + 27720*b^11*x^22))/(x^6*(a + b*x^2)^9) + 55440*b^3*Log[x] - 277 20*b^3*Log[a + b*x^2])/a^13
Time = 0.41 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {243, 54, 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.
| \(\displaystyle \int \frac {1}{x^7 \left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^8 \left (b x^2+a\right )^{10}}dx^2\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {220 b^4}{a^{13} \left (b x^2+a\right )}+\frac {165 b^4}{a^{12} \left (b x^2+a\right )^2}+\frac {120 b^4}{a^{11} \left (b x^2+a\right )^3}+\frac {84 b^4}{a^{10} \left (b x^2+a\right )^4}+\frac {56 b^4}{a^9 \left (b x^2+a\right )^5}+\frac {35 b^4}{a^8 \left (b x^2+a\right )^6}+\frac {20 b^4}{a^7 \left (b x^2+a\right )^7}+\frac {10 b^4}{a^6 \left (b x^2+a\right )^8}+\frac {4 b^4}{a^5 \left (b x^2+a\right )^9}+\frac {b^4}{a^4 \left (b x^2+a\right )^{10}}-\frac {220 b^3}{a^{13} x^2}+\frac {55 b^2}{a^{12} x^4}-\frac {10 b}{a^{11} x^6}+\frac {1}{a^{10} x^8}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {220 b^3 \log \left (x^2\right )}{a^{13}}+\frac {220 b^3 \log \left (a+b x^2\right )}{a^{13}}-\frac {165 b^3}{a^{12} \left (a+b x^2\right )}-\frac {55 b^2}{a^{12} x^2}-\frac {60 b^3}{a^{11} \left (a+b x^2\right )^2}+\frac {5 b}{a^{11} x^4}-\frac {28 b^3}{a^{10} \left (a+b x^2\right )^3}-\frac {1}{3 a^{10} x^6}-\frac {14 b^3}{a^9 \left (a+b x^2\right )^4}-\frac {7 b^3}{a^8 \left (a+b x^2\right )^5}-\frac {10 b^3}{3 a^7 \left (a+b x^2\right )^6}-\frac {10 b^3}{7 a^6 \left (a+b x^2\right )^7}-\frac {b^3}{2 a^5 \left (a+b x^2\right )^8}-\frac {b^3}{9 a^4 \left (a+b x^2\right )^9}\right )\) |
(-1/3*1/(a^10*x^6) + (5*b)/(a^11*x^4) - (55*b^2)/(a^12*x^2) - b^3/(9*a^4*( a + b*x^2)^9) - b^3/(2*a^5*(a + b*x^2)^8) - (10*b^3)/(7*a^6*(a + b*x^2)^7) - (10*b^3)/(3*a^7*(a + b*x^2)^6) - (7*b^3)/(a^8*(a + b*x^2)^5) - (14*b^3) /(a^9*(a + b*x^2)^4) - (28*b^3)/(a^10*(a + b*x^2)^3) - (60*b^3)/(a^11*(a + b*x^2)^2) - (165*b^3)/(a^12*(a + b*x^2)) - (220*b^3*Log[x^2])/a^13 + (220 *b^3*Log[a + b*x^2])/a^13)/2
3.3.8.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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 = 1.80 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.73
| method | result | size |
| norman | \(\frac {\frac {b \,x^{2}}{a^{2}}-\frac {1}{6 a}-\frac {11 b^{2} x^{4}}{a^{3}}+\frac {990 b^{4} x^{8}}{a^{5}}+\frac {5940 b^{5} x^{10}}{a^{6}}+\frac {16940 b^{6} x^{12}}{a^{7}}+\frac {28875 b^{7} x^{14}}{a^{8}}+\frac {31647 b^{8} x^{16}}{a^{9}}+\frac {22638 b^{9} x^{18}}{a^{10}}+\frac {71874 b^{10} x^{20}}{7 a^{11}}+\frac {75339 b^{11} x^{22}}{28 a^{12}}+\frac {78419 b^{12} x^{24}}{252 a^{13}}}{x^{6} \left (b \,x^{2}+a \right )^{9}}-\frac {220 b^{3} \ln \left (x \right )}{a^{13}}+\frac {110 b^{3} \ln \left (b \,x^{2}+a \right )}{a^{13}}\) | \(165\) |
| risch | \(\frac {-\frac {1}{6 a}+\frac {b \,x^{2}}{a^{2}}-\frac {11 b^{2} x^{4}}{a^{3}}-\frac {78419 b^{3} x^{6}}{252 a^{4}}-\frac {50699 b^{4} x^{8}}{28 a^{5}}-\frac {36839 b^{5} x^{10}}{7 a^{6}}-\frac {27599 b^{6} x^{12}}{3 a^{7}}-\frac {20669 b^{7} x^{14}}{2 a^{8}}-\frac {15125 b^{8} x^{16}}{2 a^{9}}-\frac {10505 b^{9} x^{18}}{3 a^{10}}-\frac {935 b^{10} x^{20}}{a^{11}}-\frac {110 b^{11} x^{22}}{a^{12}}}{x^{6} \left (b \,x^{2}+a \right )^{9}}-\frac {220 b^{3} \ln \left (x \right )}{a^{13}}+\frac {110 b^{3} \ln \left (-b \,x^{2}-a \right )}{a^{13}}\) | \(168\) |
| default | \(-\frac {1}{6 a^{10} x^{6}}-\frac {220 b^{3} \ln \left (x \right )}{a^{13}}-\frac {55 b^{2}}{2 a^{12} x^{2}}+\frac {5 b}{2 a^{11} x^{4}}+\frac {b^{4} \left (-\frac {28 a^{3}}{b \left (b \,x^{2}+a \right )^{3}}-\frac {a^{8}}{2 b \left (b \,x^{2}+a \right )^{8}}-\frac {7 a^{5}}{b \left (b \,x^{2}+a \right )^{5}}-\frac {a^{9}}{9 b \left (b \,x^{2}+a \right )^{9}}+\frac {220 \ln \left (b \,x^{2}+a \right )}{b}-\frac {10 a^{6}}{3 b \left (b \,x^{2}+a \right )^{6}}-\frac {14 a^{4}}{b \left (b \,x^{2}+a \right )^{4}}-\frac {10 a^{7}}{7 b \left (b \,x^{2}+a \right )^{7}}-\frac {60 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {165 a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{13}}\) | \(213\) |
| parallelrisch | \(-\frac {42 a^{12}-27720 \ln \left (b \,x^{2}+a \right ) x^{24} b^{12}-5704776 x^{18} a^{3} b^{9}-2328480 \ln \left (b \,x^{2}+a \right ) x^{18} a^{3} b^{9}-3492720 \ln \left (b \,x^{2}+a \right ) x^{16} a^{4} b^{8}-3492720 \ln \left (b \,x^{2}+a \right ) x^{14} a^{5} b^{7}-2328480 \ln \left (b \,x^{2}+a \right ) x^{12} a^{6} b^{6}-997920 \ln \left (b \,x^{2}+a \right ) x^{10} a^{7} b^{5}-249480 \ln \left (b \,x^{2}+a \right ) x^{8} a^{8} b^{4}-27720 \ln \left (b \,x^{2}+a \right ) x^{6} a^{9} b^{3}+55440 \ln \left (x \right ) x^{24} b^{12}-78419 b^{12} x^{24}+498960 \ln \left (x \right ) x^{22} a \,b^{11}-249480 \ln \left (b \,x^{2}+a \right ) x^{22} a \,b^{11}+1995840 \ln \left (x \right ) x^{20} a^{2} b^{10}-997920 \ln \left (b \,x^{2}+a \right ) x^{20} a^{2} b^{10}+4656960 \ln \left (x \right ) x^{18} a^{3} b^{9}+6985440 \ln \left (x \right ) x^{16} a^{4} b^{8}+6985440 \ln \left (x \right ) x^{14} a^{5} b^{7}+4656960 \ln \left (x \right ) x^{12} a^{6} b^{6}+1995840 \ln \left (x \right ) x^{10} a^{7} b^{5}+498960 \ln \left (x \right ) x^{8} a^{8} b^{4}+55440 \ln \left (x \right ) x^{6} a^{9} b^{3}-2587464 a^{2} x^{20} b^{10}-678051 a \,x^{22} b^{11}-7276500 x^{14} a^{5} b^{7}-249480 x^{8} a^{8} b^{4}-1496880 x^{10} a^{7} b^{5}-4268880 x^{12} a^{6} b^{6}+2772 x^{4} a^{10} b^{2}-252 x^{2} a^{11} b -7975044 x^{16} a^{4} b^{8}}{252 a^{13} x^{6} \left (b \,x^{2}+a \right )^{9}}\) | \(448\) |
(b/a^2*x^2-1/6/a-11*b^2/a^3*x^4+990*b^4/a^5*x^8+5940*b^5/a^6*x^10+16940*b^ 6/a^7*x^12+28875*b^7/a^8*x^14+31647*b^8/a^9*x^16+22638*b^9/a^10*x^18+71874 /7*b^10/a^11*x^20+75339/28*b^11/a^12*x^22+78419/252*b^12/a^13*x^24)/x^6/(b *x^2+a)^9-220*b^3*ln(x)/a^13+110*b^3*ln(b*x^2+a)/a^13
Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (208) = 416\).
Time = 0.28 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^{10}} \, dx=-\frac {27720 \, a b^{11} x^{22} + 235620 \, a^{2} b^{10} x^{20} + 882420 \, a^{3} b^{9} x^{18} + 1905750 \, a^{4} b^{8} x^{16} + 2604294 \, a^{5} b^{7} x^{14} + 2318316 \, a^{6} b^{6} x^{12} + 1326204 \, a^{7} b^{5} x^{10} + 456291 \, a^{8} b^{4} x^{8} + 78419 \, a^{9} b^{3} x^{6} + 2772 \, a^{10} b^{2} x^{4} - 252 \, a^{11} b x^{2} + 42 \, a^{12} - 27720 \, {\left (b^{12} x^{24} + 9 \, a b^{11} x^{22} + 36 \, a^{2} b^{10} x^{20} + 84 \, a^{3} b^{9} x^{18} + 126 \, a^{4} b^{8} x^{16} + 126 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 36 \, a^{7} b^{5} x^{10} + 9 \, a^{8} b^{4} x^{8} + a^{9} b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 55440 \, {\left (b^{12} x^{24} + 9 \, a b^{11} x^{22} + 36 \, a^{2} b^{10} x^{20} + 84 \, a^{3} b^{9} x^{18} + 126 \, a^{4} b^{8} x^{16} + 126 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 36 \, a^{7} b^{5} x^{10} + 9 \, a^{8} b^{4} x^{8} + a^{9} b^{3} x^{6}\right )} \log \left (x\right )}{252 \, {\left (a^{13} b^{9} x^{24} + 9 \, a^{14} b^{8} x^{22} + 36 \, a^{15} b^{7} x^{20} + 84 \, a^{16} b^{6} x^{18} + 126 \, a^{17} b^{5} x^{16} + 126 \, a^{18} b^{4} x^{14} + 84 \, a^{19} b^{3} x^{12} + 36 \, a^{20} b^{2} x^{10} + 9 \, a^{21} b x^{8} + a^{22} x^{6}\right )}} \]
-1/252*(27720*a*b^11*x^22 + 235620*a^2*b^10*x^20 + 882420*a^3*b^9*x^18 + 1 905750*a^4*b^8*x^16 + 2604294*a^5*b^7*x^14 + 2318316*a^6*b^6*x^12 + 132620 4*a^7*b^5*x^10 + 456291*a^8*b^4*x^8 + 78419*a^9*b^3*x^6 + 2772*a^10*b^2*x^ 4 - 252*a^11*b*x^2 + 42*a^12 - 27720*(b^12*x^24 + 9*a*b^11*x^22 + 36*a^2*b ^10*x^20 + 84*a^3*b^9*x^18 + 126*a^4*b^8*x^16 + 126*a^5*b^7*x^14 + 84*a^6* b^6*x^12 + 36*a^7*b^5*x^10 + 9*a^8*b^4*x^8 + a^9*b^3*x^6)*log(b*x^2 + a) + 55440*(b^12*x^24 + 9*a*b^11*x^22 + 36*a^2*b^10*x^20 + 84*a^3*b^9*x^18 + 1 26*a^4*b^8*x^16 + 126*a^5*b^7*x^14 + 84*a^6*b^6*x^12 + 36*a^7*b^5*x^10 + 9 *a^8*b^4*x^8 + a^9*b^3*x^6)*log(x))/(a^13*b^9*x^24 + 9*a^14*b^8*x^22 + 36* a^15*b^7*x^20 + 84*a^16*b^6*x^18 + 126*a^17*b^5*x^16 + 126*a^18*b^4*x^14 + 84*a^19*b^3*x^12 + 36*a^20*b^2*x^10 + 9*a^21*b*x^8 + a^22*x^6)
Time = 0.78 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^{10}} \, dx=\frac {- 42 a^{11} + 252 a^{10} b x^{2} - 2772 a^{9} b^{2} x^{4} - 78419 a^{8} b^{3} x^{6} - 456291 a^{7} b^{4} x^{8} - 1326204 a^{6} b^{5} x^{10} - 2318316 a^{5} b^{6} x^{12} - 2604294 a^{4} b^{7} x^{14} - 1905750 a^{3} b^{8} x^{16} - 882420 a^{2} b^{9} x^{18} - 235620 a b^{10} x^{20} - 27720 b^{11} x^{22}}{252 a^{21} x^{6} + 2268 a^{20} b x^{8} + 9072 a^{19} b^{2} x^{10} + 21168 a^{18} b^{3} x^{12} + 31752 a^{17} b^{4} x^{14} + 31752 a^{16} b^{5} x^{16} + 21168 a^{15} b^{6} x^{18} + 9072 a^{14} b^{7} x^{20} + 2268 a^{13} b^{8} x^{22} + 252 a^{12} b^{9} x^{24}} - \frac {220 b^{3} \log {\left (x \right )}}{a^{13}} + \frac {110 b^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{13}} \]
(-42*a**11 + 252*a**10*b*x**2 - 2772*a**9*b**2*x**4 - 78419*a**8*b**3*x**6 - 456291*a**7*b**4*x**8 - 1326204*a**6*b**5*x**10 - 2318316*a**5*b**6*x** 12 - 2604294*a**4*b**7*x**14 - 1905750*a**3*b**8*x**16 - 882420*a**2*b**9* x**18 - 235620*a*b**10*x**20 - 27720*b**11*x**22)/(252*a**21*x**6 + 2268*a **20*b*x**8 + 9072*a**19*b**2*x**10 + 21168*a**18*b**3*x**12 + 31752*a**17 *b**4*x**14 + 31752*a**16*b**5*x**16 + 21168*a**15*b**6*x**18 + 9072*a**14 *b**7*x**20 + 2268*a**13*b**8*x**22 + 252*a**12*b**9*x**24) - 220*b**3*log (x)/a**13 + 110*b**3*log(a/b + x**2)/a**13
Time = 0.22 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^{10}} \, dx=-\frac {27720 \, b^{11} x^{22} + 235620 \, a b^{10} x^{20} + 882420 \, a^{2} b^{9} x^{18} + 1905750 \, a^{3} b^{8} x^{16} + 2604294 \, a^{4} b^{7} x^{14} + 2318316 \, a^{5} b^{6} x^{12} + 1326204 \, a^{6} b^{5} x^{10} + 456291 \, a^{7} b^{4} x^{8} + 78419 \, a^{8} b^{3} x^{6} + 2772 \, a^{9} b^{2} x^{4} - 252 \, a^{10} b x^{2} + 42 \, a^{11}}{252 \, {\left (a^{12} b^{9} x^{24} + 9 \, a^{13} b^{8} x^{22} + 36 \, a^{14} b^{7} x^{20} + 84 \, a^{15} b^{6} x^{18} + 126 \, a^{16} b^{5} x^{16} + 126 \, a^{17} b^{4} x^{14} + 84 \, a^{18} b^{3} x^{12} + 36 \, a^{19} b^{2} x^{10} + 9 \, a^{20} b x^{8} + a^{21} x^{6}\right )}} + \frac {110 \, b^{3} \log \left (b x^{2} + a\right )}{a^{13}} - \frac {110 \, b^{3} \log \left (x^{2}\right )}{a^{13}} \]
-1/252*(27720*b^11*x^22 + 235620*a*b^10*x^20 + 882420*a^2*b^9*x^18 + 19057 50*a^3*b^8*x^16 + 2604294*a^4*b^7*x^14 + 2318316*a^5*b^6*x^12 + 1326204*a^ 6*b^5*x^10 + 456291*a^7*b^4*x^8 + 78419*a^8*b^3*x^6 + 2772*a^9*b^2*x^4 - 2 52*a^10*b*x^2 + 42*a^11)/(a^12*b^9*x^24 + 9*a^13*b^8*x^22 + 36*a^14*b^7*x^ 20 + 84*a^15*b^6*x^18 + 126*a^16*b^5*x^16 + 126*a^17*b^4*x^14 + 84*a^18*b^ 3*x^12 + 36*a^19*b^2*x^10 + 9*a^20*b*x^8 + a^21*x^6) + 110*b^3*log(b*x^2 + a)/a^13 - 110*b^3*log(x^2)/a^13
Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^{10}} \, dx=-\frac {110 \, b^{3} \log \left (x^{2}\right )}{a^{13}} + \frac {110 \, b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{13}} + \frac {1210 \, b^{3} x^{6} - 165 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} - a^{3}}{6 \, a^{13} x^{6}} - \frac {78419 \, b^{12} x^{18} + 726561 \, a b^{11} x^{16} + 2996964 \, a^{2} b^{10} x^{14} + 7225764 \, a^{3} b^{9} x^{12} + 11226726 \, a^{4} b^{8} x^{10} + 11663316 \, a^{5} b^{7} x^{8} + 8108184 \, a^{6} b^{6} x^{6} + 3641256 \, a^{7} b^{5} x^{4} + 960210 \, a^{8} b^{4} x^{2} + 113620 \, a^{9} b^{3}}{252 \, {\left (b x^{2} + a\right )}^{9} a^{13}} \]
-110*b^3*log(x^2)/a^13 + 110*b^3*log(abs(b*x^2 + a))/a^13 + 1/6*(1210*b^3* x^6 - 165*a*b^2*x^4 + 15*a^2*b*x^2 - a^3)/(a^13*x^6) - 1/252*(78419*b^12*x ^18 + 726561*a*b^11*x^16 + 2996964*a^2*b^10*x^14 + 7225764*a^3*b^9*x^12 + 11226726*a^4*b^8*x^10 + 11663316*a^5*b^7*x^8 + 8108184*a^6*b^6*x^6 + 36412 56*a^7*b^5*x^4 + 960210*a^8*b^4*x^2 + 113620*a^9*b^3)/((b*x^2 + a)^9*a^13)
Time = 5.53 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^{10}} \, dx=\frac {110\,b^3\,\ln \left (b\,x^2+a\right )}{a^{13}}-\frac {\frac {1}{6\,a}-\frac {b\,x^2}{a^2}+\frac {11\,b^2\,x^4}{a^3}+\frac {78419\,b^3\,x^6}{252\,a^4}+\frac {50699\,b^4\,x^8}{28\,a^5}+\frac {36839\,b^5\,x^{10}}{7\,a^6}+\frac {27599\,b^6\,x^{12}}{3\,a^7}+\frac {20669\,b^7\,x^{14}}{2\,a^8}+\frac {15125\,b^8\,x^{16}}{2\,a^9}+\frac {10505\,b^9\,x^{18}}{3\,a^{10}}+\frac {935\,b^{10}\,x^{20}}{a^{11}}+\frac {110\,b^{11}\,x^{22}}{a^{12}}}{a^9\,x^6+9\,a^8\,b\,x^8+36\,a^7\,b^2\,x^{10}+84\,a^6\,b^3\,x^{12}+126\,a^5\,b^4\,x^{14}+126\,a^4\,b^5\,x^{16}+84\,a^3\,b^6\,x^{18}+36\,a^2\,b^7\,x^{20}+9\,a\,b^8\,x^{22}+b^9\,x^{24}}-\frac {220\,b^3\,\ln \left (x\right )}{a^{13}} \]
(110*b^3*log(a + b*x^2))/a^13 - (1/(6*a) - (b*x^2)/a^2 + (11*b^2*x^4)/a^3 + (78419*b^3*x^6)/(252*a^4) + (50699*b^4*x^8)/(28*a^5) + (36839*b^5*x^10)/ (7*a^6) + (27599*b^6*x^12)/(3*a^7) + (20669*b^7*x^14)/(2*a^8) + (15125*b^8 *x^16)/(2*a^9) + (10505*b^9*x^18)/(3*a^10) + (935*b^10*x^20)/a^11 + (110*b ^11*x^22)/a^12)/(a^9*x^6 + b^9*x^24 + 9*a^8*b*x^8 + 9*a*b^8*x^22 + 36*a^7* b^2*x^10 + 84*a^6*b^3*x^12 + 126*a^5*b^4*x^14 + 126*a^4*b^5*x^16 + 84*a^3* b^6*x^18 + 36*a^2*b^7*x^20) - (220*b^3*log(x))/a^13