Integrand size = 13, antiderivative size = 217 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx=-\frac {1}{4 a^{10} x^4}+\frac {5 b}{a^{11} x^2}+\frac {b^2}{18 a^3 \left (a+b x^2\right )^9}+\frac {3 b^2}{16 a^4 \left (a+b x^2\right )^8}+\frac {3 b^2}{7 a^5 \left (a+b x^2\right )^7}+\frac {5 b^2}{6 a^6 \left (a+b x^2\right )^6}+\frac {3 b^2}{2 a^7 \left (a+b x^2\right )^5}+\frac {21 b^2}{8 a^8 \left (a+b x^2\right )^4}+\frac {14 b^2}{3 a^9 \left (a+b x^2\right )^3}+\frac {9 b^2}{a^{10} \left (a+b x^2\right )^2}+\frac {45 b^2}{2 a^{11} \left (a+b x^2\right )}+\frac {55 b^2 \log (x)}{a^{12}}-\frac {55 b^2 \log \left (a+b x^2\right )}{2 a^{12}} \] Output:
-1/4/a^10/x^4+5*b/a^11/x^2+1/18*b^2/a^3/(b*x^2+a)^9+3/16*b^2/a^4/(b*x^2+a) ^8+3/7*b^2/a^5/(b*x^2+a)^7+5/6*b^2/a^6/(b*x^2+a)^6+3/2*b^2/a^7/(b*x^2+a)^5 +21/8*b^2/a^8/(b*x^2+a)^4+14/3*b^2/a^9/(b*x^2+a)^3+9*b^2/a^10/(b*x^2+a)^2+ 45/2*b^2/a^11/(b*x^2+a)+55*b^2*ln(x)/a^12-55/2*b^2*ln(b*x^2+a)/a^12
Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx=\frac {\frac {a \left (-252 a^{10}+2772 a^9 b x^2+78419 a^8 b^2 x^4+456291 a^7 b^3 x^6+1326204 a^6 b^4 x^8+2318316 a^5 b^5 x^{10}+2604294 a^4 b^6 x^{12}+1905750 a^3 b^7 x^{14}+882420 a^2 b^8 x^{16}+235620 a b^9 x^{18}+27720 b^{10} x^{20}\right )}{x^4 \left (a+b x^2\right )^9}+55440 b^2 \log (x)-27720 b^2 \log \left (a+b x^2\right )}{1008 a^{12}} \] Input:
Integrate[1/(x^5*(a + b*x^2)^10),x]
Output:
((a*(-252*a^10 + 2772*a^9*b*x^2 + 78419*a^8*b^2*x^4 + 456291*a^7*b^3*x^6 + 1326204*a^6*b^4*x^8 + 2318316*a^5*b^5*x^10 + 2604294*a^4*b^6*x^12 + 19057 50*a^3*b^7*x^14 + 882420*a^2*b^8*x^16 + 235620*a*b^9*x^18 + 27720*b^10*x^2 0))/(x^4*(a + b*x^2)^9) + 55440*b^2*Log[x] - 27720*b^2*Log[a + b*x^2])/(10 08*a^12)
Time = 0.38 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (b x^2+a\right )^{10}}dx^2\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {55 b^3}{a^{12} \left (b x^2+a\right )}-\frac {45 b^3}{a^{11} \left (b x^2+a\right )^2}-\frac {36 b^3}{a^{10} \left (b x^2+a\right )^3}-\frac {28 b^3}{a^9 \left (b x^2+a\right )^4}-\frac {21 b^3}{a^8 \left (b x^2+a\right )^5}-\frac {15 b^3}{a^7 \left (b x^2+a\right )^6}-\frac {10 b^3}{a^6 \left (b x^2+a\right )^7}-\frac {6 b^3}{a^5 \left (b x^2+a\right )^8}-\frac {3 b^3}{a^4 \left (b x^2+a\right )^9}-\frac {b^3}{a^3 \left (b x^2+a\right )^{10}}+\frac {55 b^2}{a^{12} x^2}-\frac {10 b}{a^{11} x^4}+\frac {1}{a^{10} x^6}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {55 b^2 \log \left (x^2\right )}{a^{12}}-\frac {55 b^2 \log \left (a+b x^2\right )}{a^{12}}+\frac {45 b^2}{a^{11} \left (a+b x^2\right )}+\frac {10 b}{a^{11} x^2}+\frac {18 b^2}{a^{10} \left (a+b x^2\right )^2}-\frac {1}{2 a^{10} x^4}+\frac {28 b^2}{3 a^9 \left (a+b x^2\right )^3}+\frac {21 b^2}{4 a^8 \left (a+b x^2\right )^4}+\frac {3 b^2}{a^7 \left (a+b x^2\right )^5}+\frac {5 b^2}{3 a^6 \left (a+b x^2\right )^6}+\frac {6 b^2}{7 a^5 \left (a+b x^2\right )^7}+\frac {3 b^2}{8 a^4 \left (a+b x^2\right )^8}+\frac {b^2}{9 a^3 \left (a+b x^2\right )^9}\right )\) |
Input:
Int[1/(x^5*(a + b*x^2)^10),x]
Output:
(-1/2*1/(a^10*x^4) + (10*b)/(a^11*x^2) + b^2/(9*a^3*(a + b*x^2)^9) + (3*b^ 2)/(8*a^4*(a + b*x^2)^8) + (6*b^2)/(7*a^5*(a + b*x^2)^7) + (5*b^2)/(3*a^6* (a + b*x^2)^6) + (3*b^2)/(a^7*(a + b*x^2)^5) + (21*b^2)/(4*a^8*(a + b*x^2) ^4) + (28*b^2)/(3*a^9*(a + b*x^2)^3) + (18*b^2)/(a^10*(a + b*x^2)^2) + (45 *b^2)/(a^11*(a + b*x^2)) + (55*b^2*Log[x^2])/a^12 - (55*b^2*Log[a + b*x^2] )/a^12)/2
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 = 0.34 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.71
| method | result | size |
| norman | \(\frac {-\frac {1}{4 a}+\frac {11 b \,x^{2}}{4 a^{2}}-\frac {495 b^{3} x^{6}}{2 a^{4}}-\frac {1485 b^{4} x^{8}}{a^{5}}-\frac {4235 b^{5} x^{10}}{a^{6}}-\frac {28875 b^{6} x^{12}}{4 a^{7}}-\frac {31647 b^{7} x^{14}}{4 a^{8}}-\frac {11319 b^{8} x^{16}}{2 a^{9}}-\frac {35937 b^{9} x^{18}}{14 a^{10}}-\frac {75339 b^{10} x^{20}}{112 a^{11}}-\frac {78419 b^{11} x^{22}}{1008 a^{12}}}{x^{4} \left (b \,x^{2}+a \right )^{9}}+\frac {55 b^{2} \ln \left (x \right )}{a^{12}}-\frac {55 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{12}}\) | \(155\) |
| risch | \(\frac {-\frac {1}{4 a}+\frac {11 b \,x^{2}}{4 a^{2}}+\frac {78419 b^{2} x^{4}}{1008 a^{3}}+\frac {50699 b^{3} x^{6}}{112 a^{4}}+\frac {36839 b^{4} x^{8}}{28 a^{5}}+\frac {27599 b^{5} x^{10}}{12 a^{6}}+\frac {20669 b^{6} x^{12}}{8 a^{7}}+\frac {15125 b^{7} x^{14}}{8 a^{8}}+\frac {10505 b^{8} x^{16}}{12 a^{9}}+\frac {935 b^{9} x^{18}}{4 a^{10}}+\frac {55 b^{10} x^{20}}{2 a^{11}}}{x^{4} \left (b \,x^{2}+a \right )^{9}}+\frac {55 b^{2} \ln \left (x \right )}{a^{12}}-\frac {55 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{12}}\) | \(155\) |
| default | \(-\frac {b^{3} \left (-\frac {a^{9}}{9 b \left (b \,x^{2}+a \right )^{9}}-\frac {3 a^{5}}{b \left (b \,x^{2}+a \right )^{5}}-\frac {21 a^{4}}{4 b \left (b \,x^{2}+a \right )^{4}}-\frac {6 a^{7}}{7 b \left (b \,x^{2}+a \right )^{7}}-\frac {3 a^{8}}{8 b \left (b \,x^{2}+a \right )^{8}}-\frac {18 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {45 a}{b \left (b \,x^{2}+a \right )}+\frac {55 \ln \left (b \,x^{2}+a \right )}{b}-\frac {28 a^{3}}{3 b \left (b \,x^{2}+a \right )^{3}}-\frac {5 a^{6}}{3 b \left (b \,x^{2}+a \right )^{6}}\right )}{2 a^{12}}-\frac {1}{4 a^{10} x^{4}}+\frac {5 b}{a^{11} x^{2}}+\frac {55 b^{2} \ln \left (x \right )}{a^{12}}\) | \(202\) |
| parallelrisch | \(\frac {55440 \ln \left (x \right ) x^{4} a^{9} b^{2}+6985440 \ln \left (x \right ) x^{14} a^{4} b^{7}+6985440 \ln \left (x \right ) x^{12} a^{5} b^{6}+4656960 \ln \left (x \right ) x^{10} a^{6} b^{5}-252 a^{11}+2772 a^{10} b \,x^{2}+4656960 \ln \left (x \right ) x^{16} a^{3} b^{8}-249480 \ln \left (b \,x^{2}+a \right ) x^{20} a \,b^{10}+1995840 \ln \left (x \right ) x^{18} a^{2} b^{9}-997920 \ln \left (b \,x^{2}+a \right ) x^{18} a^{2} b^{9}-2328480 \ln \left (b \,x^{2}+a \right ) x^{16} a^{3} b^{8}-3492720 \ln \left (b \,x^{2}+a \right ) x^{14} a^{4} b^{7}-3492720 \ln \left (b \,x^{2}+a \right ) x^{12} a^{5} b^{6}-2328480 \ln \left (b \,x^{2}+a \right ) x^{10} a^{6} b^{5}-997920 \ln \left (b \,x^{2}+a \right ) x^{8} a^{7} b^{4}-249480 \ln \left (b \,x^{2}+a \right ) x^{6} a^{8} b^{3}-27720 \ln \left (b \,x^{2}+a \right ) x^{4} a^{9} b^{2}+498960 \ln \left (x \right ) x^{20} a \,b^{10}+498960 \ln \left (x \right ) x^{6} a^{8} b^{3}+1995840 \ln \left (x \right ) x^{8} a^{7} b^{4}-78419 b^{11} x^{22}-678051 a \,x^{20} b^{10}+55440 \ln \left (x \right ) x^{22} b^{11}-27720 \ln \left (b \,x^{2}+a \right ) x^{22} b^{11}-249480 x^{6} a^{8} b^{3}-1496880 x^{8} a^{7} b^{4}-4268880 x^{10} a^{6} b^{5}-7276500 x^{12} a^{5} b^{6}-7975044 x^{14} a^{4} b^{7}-5704776 x^{16} a^{3} b^{8}-2587464 x^{18} a^{2} b^{9}}{1008 a^{12} x^{4} \left (b \,x^{2}+a \right )^{9}}\) | \(437\) |
Input:
int(1/x^5/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
Output:
(-1/4/a+11/4*b/a^2*x^2-495/2*b^3/a^4*x^6-1485*b^4/a^5*x^8-4235*b^5/a^6*x^1 0-28875/4*b^6/a^7*x^12-31647/4*b^7/a^8*x^14-11319/2*b^8/a^9*x^16-35937/14* b^9/a^10*x^18-75339/112*b^10/a^11*x^20-78419/1008*b^11/a^12*x^22)/x^4/(b*x ^2+a)^9+55*b^2*ln(x)/a^12-55/2*b^2*ln(b*x^2+a)/a^12
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (197) = 394\).
Time = 0.08 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.04 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx=\frac {27720 \, a b^{10} x^{20} + 235620 \, a^{2} b^{9} x^{18} + 882420 \, a^{3} b^{8} x^{16} + 1905750 \, a^{4} b^{7} x^{14} + 2604294 \, a^{5} b^{6} x^{12} + 2318316 \, a^{6} b^{5} x^{10} + 1326204 \, a^{7} b^{4} x^{8} + 456291 \, a^{8} b^{3} x^{6} + 78419 \, a^{9} b^{2} x^{4} + 2772 \, a^{10} b x^{2} - 252 \, a^{11} - 27720 \, {\left (b^{11} x^{22} + 9 \, a b^{10} x^{20} + 36 \, a^{2} b^{9} x^{18} + 84 \, a^{3} b^{8} x^{16} + 126 \, a^{4} b^{7} x^{14} + 126 \, a^{5} b^{6} x^{12} + 84 \, a^{6} b^{5} x^{10} + 36 \, a^{7} b^{4} x^{8} + 9 \, a^{8} b^{3} x^{6} + a^{9} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 55440 \, {\left (b^{11} x^{22} + 9 \, a b^{10} x^{20} + 36 \, a^{2} b^{9} x^{18} + 84 \, a^{3} b^{8} x^{16} + 126 \, a^{4} b^{7} x^{14} + 126 \, a^{5} b^{6} x^{12} + 84 \, a^{6} b^{5} x^{10} + 36 \, a^{7} b^{4} x^{8} + 9 \, a^{8} b^{3} x^{6} + a^{9} b^{2} x^{4}\right )} \log \left (x\right )}{1008 \, {\left (a^{12} b^{9} x^{22} + 9 \, a^{13} b^{8} x^{20} + 36 \, a^{14} b^{7} x^{18} + 84 \, a^{15} b^{6} x^{16} + 126 \, a^{16} b^{5} x^{14} + 126 \, a^{17} b^{4} x^{12} + 84 \, a^{18} b^{3} x^{10} + 36 \, a^{19} b^{2} x^{8} + 9 \, a^{20} b x^{6} + a^{21} x^{4}\right )}} \] Input:
integrate(1/x^5/(b*x^2+a)^10,x, algorithm="fricas")
Output:
1/1008*(27720*a*b^10*x^20 + 235620*a^2*b^9*x^18 + 882420*a^3*b^8*x^16 + 19 05750*a^4*b^7*x^14 + 2604294*a^5*b^6*x^12 + 2318316*a^6*b^5*x^10 + 1326204 *a^7*b^4*x^8 + 456291*a^8*b^3*x^6 + 78419*a^9*b^2*x^4 + 2772*a^10*b*x^2 - 252*a^11 - 27720*(b^11*x^22 + 9*a*b^10*x^20 + 36*a^2*b^9*x^18 + 84*a^3*b^8 *x^16 + 126*a^4*b^7*x^14 + 126*a^5*b^6*x^12 + 84*a^6*b^5*x^10 + 36*a^7*b^4 *x^8 + 9*a^8*b^3*x^6 + a^9*b^2*x^4)*log(b*x^2 + a) + 55440*(b^11*x^22 + 9* a*b^10*x^20 + 36*a^2*b^9*x^18 + 84*a^3*b^8*x^16 + 126*a^4*b^7*x^14 + 126*a ^5*b^6*x^12 + 84*a^6*b^5*x^10 + 36*a^7*b^4*x^8 + 9*a^8*b^3*x^6 + a^9*b^2*x ^4)*log(x))/(a^12*b^9*x^22 + 9*a^13*b^8*x^20 + 36*a^14*b^7*x^18 + 84*a^15* b^6*x^16 + 126*a^16*b^5*x^14 + 126*a^17*b^4*x^12 + 84*a^18*b^3*x^10 + 36*a ^19*b^2*x^8 + 9*a^20*b*x^6 + a^21*x^4)
Time = 1.01 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx=\frac {- 252 a^{10} + 2772 a^{9} b x^{2} + 78419 a^{8} b^{2} x^{4} + 456291 a^{7} b^{3} x^{6} + 1326204 a^{6} b^{4} x^{8} + 2318316 a^{5} b^{5} x^{10} + 2604294 a^{4} b^{6} x^{12} + 1905750 a^{3} b^{7} x^{14} + 882420 a^{2} b^{8} x^{16} + 235620 a b^{9} x^{18} + 27720 b^{10} x^{20}}{1008 a^{20} x^{4} + 9072 a^{19} b x^{6} + 36288 a^{18} b^{2} x^{8} + 84672 a^{17} b^{3} x^{10} + 127008 a^{16} b^{4} x^{12} + 127008 a^{15} b^{5} x^{14} + 84672 a^{14} b^{6} x^{16} + 36288 a^{13} b^{7} x^{18} + 9072 a^{12} b^{8} x^{20} + 1008 a^{11} b^{9} x^{22}} + \frac {55 b^{2} \log {\left (x \right )}}{a^{12}} - \frac {55 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{12}} \] Input:
integrate(1/x**5/(b*x**2+a)**10,x)
Output:
(-252*a**10 + 2772*a**9*b*x**2 + 78419*a**8*b**2*x**4 + 456291*a**7*b**3*x **6 + 1326204*a**6*b**4*x**8 + 2318316*a**5*b**5*x**10 + 2604294*a**4*b**6 *x**12 + 1905750*a**3*b**7*x**14 + 882420*a**2*b**8*x**16 + 235620*a*b**9* x**18 + 27720*b**10*x**20)/(1008*a**20*x**4 + 9072*a**19*b*x**6 + 36288*a* *18*b**2*x**8 + 84672*a**17*b**3*x**10 + 127008*a**16*b**4*x**12 + 127008* a**15*b**5*x**14 + 84672*a**14*b**6*x**16 + 36288*a**13*b**7*x**18 + 9072* a**12*b**8*x**20 + 1008*a**11*b**9*x**22) + 55*b**2*log(x)/a**12 - 55*b**2 *log(a/b + x**2)/(2*a**12)
Time = 0.06 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx=\frac {27720 \, b^{10} x^{20} + 235620 \, a b^{9} x^{18} + 882420 \, a^{2} b^{8} x^{16} + 1905750 \, a^{3} b^{7} x^{14} + 2604294 \, a^{4} b^{6} x^{12} + 2318316 \, a^{5} b^{5} x^{10} + 1326204 \, a^{6} b^{4} x^{8} + 456291 \, a^{7} b^{3} x^{6} + 78419 \, a^{8} b^{2} x^{4} + 2772 \, a^{9} b x^{2} - 252 \, a^{10}}{1008 \, {\left (a^{11} b^{9} x^{22} + 9 \, a^{12} b^{8} x^{20} + 36 \, a^{13} b^{7} x^{18} + 84 \, a^{14} b^{6} x^{16} + 126 \, a^{15} b^{5} x^{14} + 126 \, a^{16} b^{4} x^{12} + 84 \, a^{17} b^{3} x^{10} + 36 \, a^{18} b^{2} x^{8} + 9 \, a^{19} b x^{6} + a^{20} x^{4}\right )}} - \frac {55 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{12}} + \frac {55 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{12}} \] Input:
integrate(1/x^5/(b*x^2+a)^10,x, algorithm="maxima")
Output:
1/1008*(27720*b^10*x^20 + 235620*a*b^9*x^18 + 882420*a^2*b^8*x^16 + 190575 0*a^3*b^7*x^14 + 2604294*a^4*b^6*x^12 + 2318316*a^5*b^5*x^10 + 1326204*a^6 *b^4*x^8 + 456291*a^7*b^3*x^6 + 78419*a^8*b^2*x^4 + 2772*a^9*b*x^2 - 252*a ^10)/(a^11*b^9*x^22 + 9*a^12*b^8*x^20 + 36*a^13*b^7*x^18 + 84*a^14*b^6*x^1 6 + 126*a^15*b^5*x^14 + 126*a^16*b^4*x^12 + 84*a^17*b^3*x^10 + 36*a^18*b^2 *x^8 + 9*a^19*b*x^6 + a^20*x^4) - 55/2*b^2*log(b*x^2 + a)/a^12 + 55/2*b^2* log(x^2)/a^12
Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx=\frac {55 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{12}} - \frac {55 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{12}} - \frac {165 \, b^{2} x^{4} - 20 \, a b x^{2} + a^{2}}{4 \, a^{12} x^{4}} + \frac {78419 \, b^{11} x^{18} + 728451 \, a b^{10} x^{16} + 3013596 \, a^{2} b^{9} x^{14} + 7290444 \, a^{3} b^{8} x^{12} + 11372256 \, a^{4} b^{7} x^{10} + 11871216 \, a^{5} b^{6} x^{8} + 8302224 \, a^{6} b^{5} x^{6} + 3757680 \, a^{7} b^{4} x^{4} + 1001790 \, a^{8} b^{3} x^{2} + 120550 \, a^{9} b^{2}}{1008 \, {\left (b x^{2} + a\right )}^{9} a^{12}} \] Input:
integrate(1/x^5/(b*x^2+a)^10,x, algorithm="giac")
Output:
55/2*b^2*log(x^2)/a^12 - 55/2*b^2*log(abs(b*x^2 + a))/a^12 - 1/4*(165*b^2* x^4 - 20*a*b*x^2 + a^2)/(a^12*x^4) + 1/1008*(78419*b^11*x^18 + 728451*a*b^ 10*x^16 + 3013596*a^2*b^9*x^14 + 7290444*a^3*b^8*x^12 + 11372256*a^4*b^7*x ^10 + 11871216*a^5*b^6*x^8 + 8302224*a^6*b^5*x^6 + 3757680*a^7*b^4*x^4 + 1 001790*a^8*b^3*x^2 + 120550*a^9*b^2)/((b*x^2 + a)^9*a^12)
Time = 1.05 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx=\frac {\frac {11\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {78419\,b^2\,x^4}{1008\,a^3}+\frac {50699\,b^3\,x^6}{112\,a^4}+\frac {36839\,b^4\,x^8}{28\,a^5}+\frac {27599\,b^5\,x^{10}}{12\,a^6}+\frac {20669\,b^6\,x^{12}}{8\,a^7}+\frac {15125\,b^7\,x^{14}}{8\,a^8}+\frac {10505\,b^8\,x^{16}}{12\,a^9}+\frac {935\,b^9\,x^{18}}{4\,a^{10}}+\frac {55\,b^{10}\,x^{20}}{2\,a^{11}}}{a^9\,x^4+9\,a^8\,b\,x^6+36\,a^7\,b^2\,x^8+84\,a^6\,b^3\,x^{10}+126\,a^5\,b^4\,x^{12}+126\,a^4\,b^5\,x^{14}+84\,a^3\,b^6\,x^{16}+36\,a^2\,b^7\,x^{18}+9\,a\,b^8\,x^{20}+b^9\,x^{22}}-\frac {55\,b^2\,\ln \left (b\,x^2+a\right )}{2\,a^{12}}+\frac {55\,b^2\,\ln \left (x\right )}{a^{12}} \] Input:
int(1/(x^5*(a + b*x^2)^10),x)
Output:
((11*b*x^2)/(4*a^2) - 1/(4*a) + (78419*b^2*x^4)/(1008*a^3) + (50699*b^3*x^ 6)/(112*a^4) + (36839*b^4*x^8)/(28*a^5) + (27599*b^5*x^10)/(12*a^6) + (206 69*b^6*x^12)/(8*a^7) + (15125*b^7*x^14)/(8*a^8) + (10505*b^8*x^16)/(12*a^9 ) + (935*b^9*x^18)/(4*a^10) + (55*b^10*x^20)/(2*a^11))/(a^9*x^4 + b^9*x^22 + 9*a^8*b*x^6 + 9*a*b^8*x^20 + 36*a^7*b^2*x^8 + 84*a^6*b^3*x^10 + 126*a^5 *b^4*x^12 + 126*a^4*b^5*x^14 + 84*a^3*b^6*x^16 + 36*a^2*b^7*x^18) - (55*b^ 2*log(a + b*x^2))/(2*a^12) + (55*b^2*log(x))/a^12
Time = 0.20 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.42 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^{10}} \, dx=\frac {-27720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{9} b^{2} x^{4}-249480 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{8} b^{3} x^{6}-997920 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{7} b^{4} x^{8}-2328480 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{6} b^{5} x^{10}-3492720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{5} b^{6} x^{12}-3492720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} b^{7} x^{14}-2328480 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b^{8} x^{16}-997920 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{9} x^{18}+2772 a^{10} b \,x^{2}+75339 a^{9} b^{2} x^{4}+428571 a^{8} b^{3} x^{6}+1215324 a^{7} b^{4} x^{8}+2059596 a^{6} b^{5} x^{10}+2216214 a^{5} b^{6} x^{12}+1517670 a^{4} b^{7} x^{14}+124740 a^{2} b^{9} x^{18}-27720 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{11} x^{22}+55440 \,\mathrm {log}\left (x \right ) b^{11} x^{22}+623700 a^{3} b^{8} x^{16}-252 a^{11}-249480 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{10} x^{20}+6985440 \,\mathrm {log}\left (x \right ) a^{5} b^{6} x^{12}+6985440 \,\mathrm {log}\left (x \right ) a^{4} b^{7} x^{14}+4656960 \,\mathrm {log}\left (x \right ) a^{3} b^{8} x^{16}+1995840 \,\mathrm {log}\left (x \right ) a^{2} b^{9} x^{18}+498960 \,\mathrm {log}\left (x \right ) a \,b^{10} x^{20}-3080 b^{11} x^{22}+55440 \,\mathrm {log}\left (x \right ) a^{9} b^{2} x^{4}+498960 \,\mathrm {log}\left (x \right ) a^{8} b^{3} x^{6}+1995840 \,\mathrm {log}\left (x \right ) a^{7} b^{4} x^{8}+4656960 \,\mathrm {log}\left (x \right ) a^{6} b^{5} x^{10}}{1008 a^{12} x^{4} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:
int(1/x^5/(b*x^2+a)^10,x)
Output:
( - 27720*log(a + b*x**2)*a**9*b**2*x**4 - 249480*log(a + b*x**2)*a**8*b** 3*x**6 - 997920*log(a + b*x**2)*a**7*b**4*x**8 - 2328480*log(a + b*x**2)*a **6*b**5*x**10 - 3492720*log(a + b*x**2)*a**5*b**6*x**12 - 3492720*log(a + b*x**2)*a**4*b**7*x**14 - 2328480*log(a + b*x**2)*a**3*b**8*x**16 - 99792 0*log(a + b*x**2)*a**2*b**9*x**18 - 249480*log(a + b*x**2)*a*b**10*x**20 - 27720*log(a + b*x**2)*b**11*x**22 + 55440*log(x)*a**9*b**2*x**4 + 498960* log(x)*a**8*b**3*x**6 + 1995840*log(x)*a**7*b**4*x**8 + 4656960*log(x)*a** 6*b**5*x**10 + 6985440*log(x)*a**5*b**6*x**12 + 6985440*log(x)*a**4*b**7*x **14 + 4656960*log(x)*a**3*b**8*x**16 + 1995840*log(x)*a**2*b**9*x**18 + 4 98960*log(x)*a*b**10*x**20 + 55440*log(x)*b**11*x**22 - 252*a**11 + 2772*a **10*b*x**2 + 75339*a**9*b**2*x**4 + 428571*a**8*b**3*x**6 + 1215324*a**7* b**4*x**8 + 2059596*a**6*b**5*x**10 + 2216214*a**5*b**6*x**12 + 1517670*a* *4*b**7*x**14 + 623700*a**3*b**8*x**16 + 124740*a**2*b**9*x**18 - 3080*b** 11*x**22)/(1008*a**12*x**4*(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 + 36*a**2*b**7*x**14 + 9*a*b**8*x**16 + b**9*x**18))