Integrand size = 13, antiderivative size = 166 \[ \int \frac {1}{x \left (a+b x^2\right )^{10}} \, dx=\frac {1}{18 a \left (a+b x^2\right )^9}+\frac {1}{16 a^2 \left (a+b x^2\right )^8}+\frac {1}{14 a^3 \left (a+b x^2\right )^7}+\frac {1}{12 a^4 \left (a+b x^2\right )^6}+\frac {1}{10 a^5 \left (a+b x^2\right )^5}+\frac {1}{8 a^6 \left (a+b x^2\right )^4}+\frac {1}{6 a^7 \left (a+b x^2\right )^3}+\frac {1}{4 a^8 \left (a+b x^2\right )^2}+\frac {1}{2 a^9 \left (a+b x^2\right )}+\frac {\log (x)}{a^{10}}-\frac {\log \left (a+b x^2\right )}{2 a^{10}} \]
1/18/a/(b*x^2+a)^9+1/16/a^2/(b*x^2+a)^8+1/14/a^3/(b*x^2+a)^7+1/12/a^4/(b*x ^2+a)^6+1/10/a^5/(b*x^2+a)^5+1/8/a^6/(b*x^2+a)^4+1/6/a^7/(b*x^2+a)^3+1/4/a ^8/(b*x^2+a)^2+1/2/a^9/(b*x^2+a)+ln(x)/a^10-1/2*ln(b*x^2+a)/a^10
Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x \left (a+b x^2\right )^{10}} \, dx=\frac {\frac {a \left (7129 a^8+41481 a^7 b x^2+120564 a^6 b^2 x^4+210756 a^5 b^3 x^6+236754 a^4 b^4 x^8+173250 a^3 b^5 x^{10}+80220 a^2 b^6 x^{12}+21420 a b^7 x^{14}+2520 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+5040 \log (x)-2520 \log \left (a+b x^2\right )}{5040 a^{10}} \]
((a*(7129*a^8 + 41481*a^7*b*x^2 + 120564*a^6*b^2*x^4 + 210756*a^5*b^3*x^6 + 236754*a^4*b^4*x^8 + 173250*a^3*b^5*x^10 + 80220*a^2*b^6*x^12 + 21420*a* b^7*x^14 + 2520*b^8*x^16))/(a + b*x^2)^9 + 5040*Log[x] - 2520*Log[a + b*x^ 2])/(5040*a^10)
Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int \frac {1}{x \left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (b x^2+a\right )^{10}}dx^2\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {b}{a^{10} \left (b x^2+a\right )}-\frac {b}{a^9 \left (b x^2+a\right )^2}-\frac {b}{a^8 \left (b x^2+a\right )^3}-\frac {b}{a^7 \left (b x^2+a\right )^4}-\frac {b}{a^6 \left (b x^2+a\right )^5}-\frac {b}{a^5 \left (b x^2+a\right )^6}-\frac {b}{a^4 \left (b x^2+a\right )^7}-\frac {b}{a^3 \left (b x^2+a\right )^8}-\frac {b}{a^2 \left (b x^2+a\right )^9}-\frac {b}{a \left (b x^2+a\right )^{10}}+\frac {1}{a^{10} x^2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\log \left (a+b x^2\right )}{a^{10}}+\frac {\log \left (x^2\right )}{a^{10}}+\frac {1}{a^9 \left (a+b x^2\right )}+\frac {1}{2 a^8 \left (a+b x^2\right )^2}+\frac {1}{3 a^7 \left (a+b x^2\right )^3}+\frac {1}{4 a^6 \left (a+b x^2\right )^4}+\frac {1}{5 a^5 \left (a+b x^2\right )^5}+\frac {1}{6 a^4 \left (a+b x^2\right )^6}+\frac {1}{7 a^3 \left (a+b x^2\right )^7}+\frac {1}{8 a^2 \left (a+b x^2\right )^8}+\frac {1}{9 a \left (a+b x^2\right )^9}\right )\) |
(1/(9*a*(a + b*x^2)^9) + 1/(8*a^2*(a + b*x^2)^8) + 1/(7*a^3*(a + b*x^2)^7) + 1/(6*a^4*(a + b*x^2)^6) + 1/(5*a^5*(a + b*x^2)^5) + 1/(4*a^6*(a + b*x^2 )^4) + 1/(3*a^7*(a + b*x^2)^3) + 1/(2*a^8*(a + b*x^2)^2) + 1/(a^9*(a + b*x ^2)) + Log[x^2]/a^10 - Log[a + b*x^2]/a^10)/2
3.3.5.3.1 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 = 6.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(\frac {\frac {7129}{5040 a}+\frac {4609 b \,x^{2}}{560 a^{2}}+\frac {3349 b^{2} x^{4}}{140 a^{3}}+\frac {2509 b^{3} x^{6}}{60 a^{4}}+\frac {1879 b^{4} x^{8}}{40 a^{5}}+\frac {275 b^{5} x^{10}}{8 a^{6}}+\frac {191 b^{6} x^{12}}{12 a^{7}}+\frac {17 b^{7} x^{14}}{4 a^{8}}+\frac {b^{8} x^{16}}{2 a^{9}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {\ln \left (x \right )}{a^{10}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{10}}\) | \(123\) |
| norman | \(\frac {-\frac {9 b \,x^{2}}{2 a^{2}}-\frac {27 b^{2} x^{4}}{a^{3}}-\frac {77 b^{3} x^{6}}{a^{4}}-\frac {525 b^{4} x^{8}}{4 a^{5}}-\frac {2877 b^{5} x^{10}}{20 a^{6}}-\frac {1029 b^{6} x^{12}}{10 a^{7}}-\frac {3267 b^{7} x^{14}}{70 a^{8}}-\frac {6849 b^{8} x^{16}}{560 a^{9}}-\frac {7129 b^{9} x^{18}}{5040 a^{10}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {\ln \left (x \right )}{a^{10}}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a^{10}}\) | \(129\) |
| default | \(\frac {\ln \left (x \right )}{a^{10}}-\frac {b \left (-\frac {a^{3}}{3 b \left (b \,x^{2}+a \right )^{3}}-\frac {a^{5}}{5 b \left (b \,x^{2}+a \right )^{5}}-\frac {a^{9}}{9 b \left (b \,x^{2}+a \right )^{9}}+\frac {\ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{6}}{6 b \left (b \,x^{2}+a \right )^{6}}-\frac {a^{4}}{4 b \left (b \,x^{2}+a \right )^{4}}-\frac {a^{8}}{8 b \left (b \,x^{2}+a \right )^{8}}-\frac {a^{7}}{7 b \left (b \,x^{2}+a \right )^{7}}-\frac {a^{2}}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{10}}\) | \(178\) |
| parallelrisch | \(\frac {423360 \ln \left (x \right ) x^{6} a^{6} b^{3}+181440 \ln \left (x \right ) x^{4} a^{7} b^{2}+45360 \ln \left (x \right ) x^{2} a^{8} b +45360 \ln \left (x \right ) x^{16} a \,b^{8}+181440 \ln \left (x \right ) x^{14} a^{2} b^{7}+423360 \ln \left (x \right ) x^{12} a^{3} b^{6}+635040 \ln \left (x \right ) x^{10} a^{4} b^{5}+635040 \ln \left (x \right ) x^{8} a^{5} b^{4}-7129 b^{9} x^{18}-22680 \ln \left (b \,x^{2}+a \right ) x^{16} a \,b^{8}-90720 \ln \left (b \,x^{2}+a \right ) x^{14} a^{2} b^{7}+5040 \ln \left (x \right ) a^{9}-2520 \ln \left (b \,x^{2}+a \right ) a^{9}-211680 \ln \left (b \,x^{2}+a \right ) x^{12} a^{3} b^{6}-317520 \ln \left (b \,x^{2}+a \right ) x^{10} a^{4} b^{5}-317520 \ln \left (b \,x^{2}+a \right ) x^{8} a^{5} b^{4}-211680 \ln \left (b \,x^{2}+a \right ) x^{6} a^{6} b^{3}-90720 \ln \left (b \,x^{2}+a \right ) x^{4} a^{7} b^{2}-22680 \ln \left (b \,x^{2}+a \right ) x^{2} a^{8} b -22680 a^{8} b \,x^{2}-661500 a^{5} x^{8} b^{4}-725004 a^{4} x^{10} b^{5}-518616 a^{3} x^{12} b^{6}-388080 a^{6} x^{6} b^{3}-136080 a^{7} x^{4} b^{2}-61641 a \,x^{16} b^{8}-2520 \ln \left (b \,x^{2}+a \right ) x^{18} b^{9}+5040 \ln \left (x \right ) x^{18} b^{9}-235224 a^{2} x^{14} b^{7}}{5040 a^{10} \left (b \,x^{2}+a \right )^{9}}\) | \(402\) |
(7129/5040/a+4609/560*b/a^2*x^2+3349/140*b^2/a^3*x^4+2509/60*b^3/a^4*x^6+1 879/40*b^4/a^5*x^8+275/8*b^5/a^6*x^10+191/12*b^6/a^7*x^12+17/4*b^7/a^8*x^1 4+1/2*b^8/a^9*x^16)/(b*x^2+a)^9+ln(x)/a^10-1/2*ln(b*x^2+a)/a^10
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (146) = 292\).
Time = 0.25 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.40 \[ \int \frac {1}{x \left (a+b x^2\right )^{10}} \, dx=\frac {2520 \, a b^{8} x^{16} + 21420 \, a^{2} b^{7} x^{14} + 80220 \, a^{3} b^{6} x^{12} + 173250 \, a^{4} b^{5} x^{10} + 236754 \, a^{5} b^{4} x^{8} + 210756 \, a^{6} b^{3} x^{6} + 120564 \, a^{7} b^{2} x^{4} + 41481 \, a^{8} b x^{2} + 7129 \, a^{9} - 2520 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (b x^{2} + a\right ) + 5040 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (x\right )}{5040 \, {\left (a^{10} b^{9} x^{18} + 9 \, a^{11} b^{8} x^{16} + 36 \, a^{12} b^{7} x^{14} + 84 \, a^{13} b^{6} x^{12} + 126 \, a^{14} b^{5} x^{10} + 126 \, a^{15} b^{4} x^{8} + 84 \, a^{16} b^{3} x^{6} + 36 \, a^{17} b^{2} x^{4} + 9 \, a^{18} b x^{2} + a^{19}\right )}} \]
1/5040*(2520*a*b^8*x^16 + 21420*a^2*b^7*x^14 + 80220*a^3*b^6*x^12 + 173250 *a^4*b^5*x^10 + 236754*a^5*b^4*x^8 + 210756*a^6*b^3*x^6 + 120564*a^7*b^2*x ^4 + 41481*a^8*b*x^2 + 7129*a^9 - 2520*(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)*log(b*x^2 + a) + 5040*(b^9*x^1 8 + 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)*log (x))/(a^10*b^9*x^18 + 9*a^11*b^8*x^16 + 36*a^12*b^7*x^14 + 84*a^13*b^6*x^1 2 + 126*a^14*b^5*x^10 + 126*a^15*b^4*x^8 + 84*a^16*b^3*x^6 + 36*a^17*b^2*x ^4 + 9*a^18*b*x^2 + a^19)
Time = 0.62 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x \left (a+b x^2\right )^{10}} \, dx=\frac {7129 a^{8} + 41481 a^{7} b x^{2} + 120564 a^{6} b^{2} x^{4} + 210756 a^{5} b^{3} x^{6} + 236754 a^{4} b^{4} x^{8} + 173250 a^{3} b^{5} x^{10} + 80220 a^{2} b^{6} x^{12} + 21420 a b^{7} x^{14} + 2520 b^{8} x^{16}}{5040 a^{18} + 45360 a^{17} b x^{2} + 181440 a^{16} b^{2} x^{4} + 423360 a^{15} b^{3} x^{6} + 635040 a^{14} b^{4} x^{8} + 635040 a^{13} b^{5} x^{10} + 423360 a^{12} b^{6} x^{12} + 181440 a^{11} b^{7} x^{14} + 45360 a^{10} b^{8} x^{16} + 5040 a^{9} b^{9} x^{18}} + \frac {\log {\left (x \right )}}{a^{10}} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{10}} \]
(7129*a**8 + 41481*a**7*b*x**2 + 120564*a**6*b**2*x**4 + 210756*a**5*b**3* x**6 + 236754*a**4*b**4*x**8 + 173250*a**3*b**5*x**10 + 80220*a**2*b**6*x* *12 + 21420*a*b**7*x**14 + 2520*b**8*x**16)/(5040*a**18 + 45360*a**17*b*x* *2 + 181440*a**16*b**2*x**4 + 423360*a**15*b**3*x**6 + 635040*a**14*b**4*x **8 + 635040*a**13*b**5*x**10 + 423360*a**12*b**6*x**12 + 181440*a**11*b** 7*x**14 + 45360*a**10*b**8*x**16 + 5040*a**9*b**9*x**18) + log(x)/a**10 - log(a/b + x**2)/(2*a**10)
Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \left (a+b x^2\right )^{10}} \, dx=\frac {2520 \, b^{8} x^{16} + 21420 \, a b^{7} x^{14} + 80220 \, a^{2} b^{6} x^{12} + 173250 \, a^{3} b^{5} x^{10} + 236754 \, a^{4} b^{4} x^{8} + 210756 \, a^{5} b^{3} x^{6} + 120564 \, a^{6} b^{2} x^{4} + 41481 \, a^{7} b x^{2} + 7129 \, a^{8}}{5040 \, {\left (a^{9} b^{9} x^{18} + 9 \, a^{10} b^{8} x^{16} + 36 \, a^{11} b^{7} x^{14} + 84 \, a^{12} b^{6} x^{12} + 126 \, a^{13} b^{5} x^{10} + 126 \, a^{14} b^{4} x^{8} + 84 \, a^{15} b^{3} x^{6} + 36 \, a^{16} b^{2} x^{4} + 9 \, a^{17} b x^{2} + a^{18}\right )}} - \frac {\log \left (b x^{2} + a\right )}{2 \, a^{10}} + \frac {\log \left (x^{2}\right )}{2 \, a^{10}} \]
1/5040*(2520*b^8*x^16 + 21420*a*b^7*x^14 + 80220*a^2*b^6*x^12 + 173250*a^3 *b^5*x^10 + 236754*a^4*b^4*x^8 + 210756*a^5*b^3*x^6 + 120564*a^6*b^2*x^4 + 41481*a^7*b*x^2 + 7129*a^8)/(a^9*b^9*x^18 + 9*a^10*b^8*x^16 + 36*a^11*b^7 *x^14 + 84*a^12*b^6*x^12 + 126*a^13*b^5*x^10 + 126*a^14*b^4*x^8 + 84*a^15* b^3*x^6 + 36*a^16*b^2*x^4 + 9*a^17*b*x^2 + a^18) - 1/2*log(b*x^2 + a)/a^10 + 1/2*log(x^2)/a^10
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \left (a+b x^2\right )^{10}} \, dx=\frac {\log \left (x^{2}\right )}{2 \, a^{10}} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{10}} + \frac {7129 \, b^{9} x^{18} + 66681 \, a b^{8} x^{16} + 278064 \, a^{2} b^{7} x^{14} + 679056 \, a^{3} b^{6} x^{12} + 1071504 \, a^{4} b^{5} x^{10} + 1135008 \, a^{5} b^{4} x^{8} + 809592 \, a^{6} b^{3} x^{6} + 377208 \, a^{7} b^{2} x^{4} + 105642 \, a^{8} b x^{2} + 14258 \, a^{9}}{5040 \, {\left (b x^{2} + a\right )}^{9} a^{10}} \]
1/2*log(x^2)/a^10 - 1/2*log(abs(b*x^2 + a))/a^10 + 1/5040*(7129*b^9*x^18 + 66681*a*b^8*x^16 + 278064*a^2*b^7*x^14 + 679056*a^3*b^6*x^12 + 1071504*a^ 4*b^5*x^10 + 1135008*a^5*b^4*x^8 + 809592*a^6*b^3*x^6 + 377208*a^7*b^2*x^4 + 105642*a^8*b*x^2 + 14258*a^9)/((b*x^2 + a)^9*a^10)
Time = 4.96 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \left (a+b x^2\right )^{10}} \, dx=\frac {\ln \left (x\right )}{a^{10}}+\frac {\frac {7129}{5040\,a}+\frac {4609\,b\,x^2}{560\,a^2}+\frac {3349\,b^2\,x^4}{140\,a^3}+\frac {2509\,b^3\,x^6}{60\,a^4}+\frac {1879\,b^4\,x^8}{40\,a^5}+\frac {275\,b^5\,x^{10}}{8\,a^6}+\frac {191\,b^6\,x^{12}}{12\,a^7}+\frac {17\,b^7\,x^{14}}{4\,a^8}+\frac {b^8\,x^{16}}{2\,a^9}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}-\frac {\ln \left (b\,x^2+a\right )}{2\,a^{10}} \]
log(x)/a^10 + (7129/(5040*a) + (4609*b*x^2)/(560*a^2) + (3349*b^2*x^4)/(14 0*a^3) + (2509*b^3*x^6)/(60*a^4) + (1879*b^4*x^8)/(40*a^5) + (275*b^5*x^10 )/(8*a^6) + (191*b^6*x^12)/(12*a^7) + (17*b^7*x^14)/(4*a^8) + (b^8*x^16)/( 2*a^9))/(a^9 + b^9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 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) - log(a + b*x^2)/(2*a^10)