Integrand size = 13, antiderivative size = 184 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx=-\frac {1}{2 a^{10} x^2}-\frac {b}{18 a^2 \left (a+b x^2\right )^9}-\frac {b}{8 a^3 \left (a+b x^2\right )^8}-\frac {3 b}{14 a^4 \left (a+b x^2\right )^7}-\frac {b}{3 a^5 \left (a+b x^2\right )^6}-\frac {b}{2 a^6 \left (a+b x^2\right )^5}-\frac {3 b}{4 a^7 \left (a+b x^2\right )^4}-\frac {7 b}{6 a^8 \left (a+b x^2\right )^3}-\frac {2 b}{a^9 \left (a+b x^2\right )^2}-\frac {9 b}{2 a^{10} \left (a+b x^2\right )}-\frac {10 b \log (x)}{a^{11}}+\frac {5 b \log \left (a+b x^2\right )}{a^{11}} \]
-1/2/a^10/x^2-1/18*b/a^2/(b*x^2+a)^9-1/8*b/a^3/(b*x^2+a)^8-3/14*b/a^4/(b*x ^2+a)^7-1/3*b/a^5/(b*x^2+a)^6-1/2*b/a^6/(b*x^2+a)^5-3/4*b/a^7/(b*x^2+a)^4- 7/6*b/a^8/(b*x^2+a)^3-2*b/a^9/(b*x^2+a)^2-9/2*b/a^10/(b*x^2+a)-10*b*ln(x)/ a^11+5*b*ln(b*x^2+a)/a^11
Time = 0.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx=-\frac {\frac {a \left (252 a^9+7129 a^8 b x^2+41481 a^7 b^2 x^4+120564 a^6 b^3 x^6+210756 a^5 b^4 x^8+236754 a^4 b^5 x^{10}+173250 a^3 b^6 x^{12}+80220 a^2 b^7 x^{14}+21420 a b^8 x^{16}+2520 b^9 x^{18}\right )}{x^2 \left (a+b x^2\right )^9}+5040 b \log (x)-2520 b \log \left (a+b x^2\right )}{504 a^{11}} \]
-1/504*((a*(252*a^9 + 7129*a^8*b*x^2 + 41481*a^7*b^2*x^4 + 120564*a^6*b^3* x^6 + 210756*a^5*b^4*x^8 + 236754*a^4*b^5*x^10 + 173250*a^3*b^6*x^12 + 802 20*a^2*b^7*x^14 + 21420*a*b^8*x^16 + 2520*b^9*x^18))/(x^2*(a + b*x^2)^9) + 5040*b*Log[x] - 2520*b*Log[a + b*x^2])/a^11
Time = 0.34 (sec) , antiderivative size = 184, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^2+a\right )^{10}}dx^2\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {10 b^2}{a^{11} \left (b x^2+a\right )}+\frac {9 b^2}{a^{10} \left (b x^2+a\right )^2}+\frac {8 b^2}{a^9 \left (b x^2+a\right )^3}+\frac {7 b^2}{a^8 \left (b x^2+a\right )^4}+\frac {6 b^2}{a^7 \left (b x^2+a\right )^5}+\frac {5 b^2}{a^6 \left (b x^2+a\right )^6}+\frac {4 b^2}{a^5 \left (b x^2+a\right )^7}+\frac {3 b^2}{a^4 \left (b x^2+a\right )^8}+\frac {2 b^2}{a^3 \left (b x^2+a\right )^9}+\frac {b^2}{a^2 \left (b x^2+a\right )^{10}}-\frac {10 b}{a^{11} x^2}+\frac {1}{a^{10} x^4}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {10 b \log \left (x^2\right )}{a^{11}}+\frac {10 b \log \left (a+b x^2\right )}{a^{11}}-\frac {9 b}{a^{10} \left (a+b x^2\right )}-\frac {1}{a^{10} x^2}-\frac {4 b}{a^9 \left (a+b x^2\right )^2}-\frac {7 b}{3 a^8 \left (a+b x^2\right )^3}-\frac {3 b}{2 a^7 \left (a+b x^2\right )^4}-\frac {b}{a^6 \left (a+b x^2\right )^5}-\frac {2 b}{3 a^5 \left (a+b x^2\right )^6}-\frac {3 b}{7 a^4 \left (a+b x^2\right )^7}-\frac {b}{4 a^3 \left (a+b x^2\right )^8}-\frac {b}{9 a^2 \left (a+b x^2\right )^9}\right )\) |
(-(1/(a^10*x^2)) - b/(9*a^2*(a + b*x^2)^9) - b/(4*a^3*(a + b*x^2)^8) - (3* b)/(7*a^4*(a + b*x^2)^7) - (2*b)/(3*a^5*(a + b*x^2)^6) - b/(a^6*(a + b*x^2 )^5) - (3*b)/(2*a^7*(a + b*x^2)^4) - (7*b)/(3*a^8*(a + b*x^2)^3) - (4*b)/( a^9*(a + b*x^2)^2) - (9*b)/(a^10*(a + b*x^2)) - (10*b*Log[x^2])/a^11 + (10 *b*Log[a + b*x^2])/a^11)/2
3.3.6.3.1 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 = 1.79 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.77
| method | result | size |
| norman | \(\frac {-\frac {1}{2 a}+\frac {45 b^{2} x^{4}}{a^{3}}+\frac {270 b^{3} x^{6}}{a^{4}}+\frac {770 b^{4} x^{8}}{a^{5}}+\frac {2625 b^{5} x^{10}}{2 a^{6}}+\frac {2877 b^{6} x^{12}}{2 a^{7}}+\frac {1029 b^{7} x^{14}}{a^{8}}+\frac {3267 b^{8} x^{16}}{7 a^{9}}+\frac {6849 b^{9} x^{18}}{56 a^{10}}+\frac {7129 b^{10} x^{20}}{504 a^{11}}}{x^{2} \left (b \,x^{2}+a \right )^{9}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {5 b \ln \left (b \,x^{2}+a \right )}{a^{11}}\) | \(142\) |
| risch | \(\frac {-\frac {1}{2 a}-\frac {7129 b \,x^{2}}{504 a^{2}}-\frac {4609 b^{2} x^{4}}{56 a^{3}}-\frac {3349 b^{3} x^{6}}{14 a^{4}}-\frac {2509 b^{4} x^{8}}{6 a^{5}}-\frac {1879 b^{5} x^{10}}{4 a^{6}}-\frac {1375 b^{6} x^{12}}{4 a^{7}}-\frac {955 b^{7} x^{14}}{6 a^{8}}-\frac {85 b^{8} x^{16}}{2 a^{9}}-\frac {5 b^{9} x^{18}}{a^{10}}}{x^{2} \left (b \,x^{2}+a \right )^{9}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {5 b \ln \left (-b \,x^{2}-a \right )}{a^{11}}\) | \(143\) |
| default | \(-\frac {1}{2 a^{10} x^{2}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {b^{2} \left (-\frac {7 a^{3}}{3 b \left (b \,x^{2}+a \right )^{3}}-\frac {a^{5}}{b \left (b \,x^{2}+a \right )^{5}}-\frac {a^{9}}{9 b \left (b \,x^{2}+a \right )^{9}}+\frac {10 \ln \left (b \,x^{2}+a \right )}{b}-\frac {2 a^{6}}{3 b \left (b \,x^{2}+a \right )^{6}}-\frac {3 a^{4}}{2 b \left (b \,x^{2}+a \right )^{4}}-\frac {3 a^{7}}{7 b \left (b \,x^{2}+a \right )^{7}}-\frac {a^{8}}{4 b \left (b \,x^{2}+a \right )^{8}}-\frac {4 a^{2}}{b \left (b \,x^{2}+a \right )^{2}}-\frac {9 a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{11}}\) | \(191\) |
| parallelrisch | \(-\frac {252 a^{10}-7129 b^{10} x^{20}-22680 a^{8} b^{2} x^{4}-136080 a^{7} b^{3} x^{6}-661500 a^{5} b^{5} x^{10}-61641 a \,b^{9} x^{18}-388080 a^{6} b^{4} x^{8}-725004 a^{4} b^{6} x^{12}-235224 a^{2} b^{8} x^{16}-518616 a^{3} b^{7} x^{14}+5040 \ln \left (x \right ) x^{2} a^{9} b +45360 \ln \left (x \right ) x^{18} a \,b^{9}+181440 \ln \left (x \right ) x^{16} a^{2} b^{8}+423360 \ln \left (x \right ) x^{14} a^{3} b^{7}+635040 \ln \left (x \right ) x^{12} a^{4} b^{6}+635040 \ln \left (x \right ) x^{10} a^{5} b^{5}+423360 \ln \left (x \right ) x^{8} a^{6} b^{4}+181440 \ln \left (x \right ) x^{6} a^{7} b^{3}+45360 \ln \left (x \right ) x^{4} a^{8} b^{2}-22680 \ln \left (b \,x^{2}+a \right ) x^{18} a \,b^{9}-90720 \ln \left (b \,x^{2}+a \right ) x^{16} a^{2} b^{8}-211680 \ln \left (b \,x^{2}+a \right ) x^{14} a^{3} b^{7}-317520 \ln \left (b \,x^{2}+a \right ) x^{12} a^{4} b^{6}-317520 \ln \left (b \,x^{2}+a \right ) x^{10} a^{5} b^{5}-211680 \ln \left (b \,x^{2}+a \right ) x^{8} a^{6} b^{4}-90720 \ln \left (b \,x^{2}+a \right ) x^{6} a^{7} b^{3}-22680 \ln \left (b \,x^{2}+a \right ) x^{4} a^{8} b^{2}-2520 \ln \left (b \,x^{2}+a \right ) x^{2} a^{9} b +5040 \ln \left (x \right ) x^{20} b^{10}-2520 \ln \left (b \,x^{2}+a \right ) x^{20} b^{10}}{504 a^{11} x^{2} \left (b \,x^{2}+a \right )^{9}}\) | \(424\) |
(-1/2/a+45*b^2/a^3*x^4+270*b^3/a^4*x^6+770*b^4/a^5*x^8+2625/2*b^5/a^6*x^10 +2877/2*b^6/a^7*x^12+1029*b^7/a^8*x^14+3267/7*b^8/a^9*x^16+6849/56*b^9/a^1 0*x^18+7129/504*b^10/a^11*x^20)/x^2/(b*x^2+a)^9-10*b*ln(x)/a^11+5*b*ln(b*x ^2+a)/a^11
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (166) = 332\).
Time = 0.25 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx=-\frac {2520 \, a b^{9} x^{18} + 21420 \, a^{2} b^{8} x^{16} + 80220 \, a^{3} b^{7} x^{14} + 173250 \, a^{4} b^{6} x^{12} + 236754 \, a^{5} b^{5} x^{10} + 210756 \, a^{6} b^{4} x^{8} + 120564 \, a^{7} b^{3} x^{6} + 41481 \, a^{8} b^{2} x^{4} + 7129 \, a^{9} b x^{2} + 252 \, a^{10} - 2520 \, {\left (b^{10} x^{20} + 9 \, a b^{9} x^{18} + 36 \, a^{2} b^{8} x^{16} + 84 \, a^{3} b^{7} x^{14} + 126 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 84 \, a^{6} b^{4} x^{8} + 36 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{4} + a^{9} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 5040 \, {\left (b^{10} x^{20} + 9 \, a b^{9} x^{18} + 36 \, a^{2} b^{8} x^{16} + 84 \, a^{3} b^{7} x^{14} + 126 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 84 \, a^{6} b^{4} x^{8} + 36 \, a^{7} b^{3} x^{6} + 9 \, a^{8} b^{2} x^{4} + a^{9} b x^{2}\right )} \log \left (x\right )}{504 \, {\left (a^{11} b^{9} x^{20} + 9 \, a^{12} b^{8} x^{18} + 36 \, a^{13} b^{7} x^{16} + 84 \, a^{14} b^{6} x^{14} + 126 \, a^{15} b^{5} x^{12} + 126 \, a^{16} b^{4} x^{10} + 84 \, a^{17} b^{3} x^{8} + 36 \, a^{18} b^{2} x^{6} + 9 \, a^{19} b x^{4} + a^{20} x^{2}\right )}} \]
-1/504*(2520*a*b^9*x^18 + 21420*a^2*b^8*x^16 + 80220*a^3*b^7*x^14 + 173250 *a^4*b^6*x^12 + 236754*a^5*b^5*x^10 + 210756*a^6*b^4*x^8 + 120564*a^7*b^3* x^6 + 41481*a^8*b^2*x^4 + 7129*a^9*b*x^2 + 252*a^10 - 2520*(b^10*x^20 + 9* a*b^9*x^18 + 36*a^2*b^8*x^16 + 84*a^3*b^7*x^14 + 126*a^4*b^6*x^12 + 126*a^ 5*b^5*x^10 + 84*a^6*b^4*x^8 + 36*a^7*b^3*x^6 + 9*a^8*b^2*x^4 + a^9*b*x^2)* log(b*x^2 + a) + 5040*(b^10*x^20 + 9*a*b^9*x^18 + 36*a^2*b^8*x^16 + 84*a^3 *b^7*x^14 + 126*a^4*b^6*x^12 + 126*a^5*b^5*x^10 + 84*a^6*b^4*x^8 + 36*a^7* b^3*x^6 + 9*a^8*b^2*x^4 + a^9*b*x^2)*log(x))/(a^11*b^9*x^20 + 9*a^12*b^8*x ^18 + 36*a^13*b^7*x^16 + 84*a^14*b^6*x^14 + 126*a^15*b^5*x^12 + 126*a^16*b ^4*x^10 + 84*a^17*b^3*x^8 + 36*a^18*b^2*x^6 + 9*a^19*b*x^4 + a^20*x^2)
Time = 0.72 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx=\frac {- 252 a^{9} - 7129 a^{8} b x^{2} - 41481 a^{7} b^{2} x^{4} - 120564 a^{6} b^{3} x^{6} - 210756 a^{5} b^{4} x^{8} - 236754 a^{4} b^{5} x^{10} - 173250 a^{3} b^{6} x^{12} - 80220 a^{2} b^{7} x^{14} - 21420 a b^{8} x^{16} - 2520 b^{9} x^{18}}{504 a^{19} x^{2} + 4536 a^{18} b x^{4} + 18144 a^{17} b^{2} x^{6} + 42336 a^{16} b^{3} x^{8} + 63504 a^{15} b^{4} x^{10} + 63504 a^{14} b^{5} x^{12} + 42336 a^{13} b^{6} x^{14} + 18144 a^{12} b^{7} x^{16} + 4536 a^{11} b^{8} x^{18} + 504 a^{10} b^{9} x^{20}} - \frac {10 b \log {\left (x \right )}}{a^{11}} + \frac {5 b \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{11}} \]
(-252*a**9 - 7129*a**8*b*x**2 - 41481*a**7*b**2*x**4 - 120564*a**6*b**3*x* *6 - 210756*a**5*b**4*x**8 - 236754*a**4*b**5*x**10 - 173250*a**3*b**6*x** 12 - 80220*a**2*b**7*x**14 - 21420*a*b**8*x**16 - 2520*b**9*x**18)/(504*a* *19*x**2 + 4536*a**18*b*x**4 + 18144*a**17*b**2*x**6 + 42336*a**16*b**3*x* *8 + 63504*a**15*b**4*x**10 + 63504*a**14*b**5*x**12 + 42336*a**13*b**6*x* *14 + 18144*a**12*b**7*x**16 + 4536*a**11*b**8*x**18 + 504*a**10*b**9*x**2 0) - 10*b*log(x)/a**11 + 5*b*log(a/b + x**2)/a**11
Time = 0.21 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx=-\frac {2520 \, b^{9} x^{18} + 21420 \, a b^{8} x^{16} + 80220 \, a^{2} b^{7} x^{14} + 173250 \, a^{3} b^{6} x^{12} + 236754 \, a^{4} b^{5} x^{10} + 210756 \, a^{5} b^{4} x^{8} + 120564 \, a^{6} b^{3} x^{6} + 41481 \, a^{7} b^{2} x^{4} + 7129 \, a^{8} b x^{2} + 252 \, a^{9}}{504 \, {\left (a^{10} b^{9} x^{20} + 9 \, a^{11} b^{8} x^{18} + 36 \, a^{12} b^{7} x^{16} + 84 \, a^{13} b^{6} x^{14} + 126 \, a^{14} b^{5} x^{12} + 126 \, a^{15} b^{4} x^{10} + 84 \, a^{16} b^{3} x^{8} + 36 \, a^{17} b^{2} x^{6} + 9 \, a^{18} b x^{4} + a^{19} x^{2}\right )}} + \frac {5 \, b \log \left (b x^{2} + a\right )}{a^{11}} - \frac {5 \, b \log \left (x^{2}\right )}{a^{11}} \]
-1/504*(2520*b^9*x^18 + 21420*a*b^8*x^16 + 80220*a^2*b^7*x^14 + 173250*a^3 *b^6*x^12 + 236754*a^4*b^5*x^10 + 210756*a^5*b^4*x^8 + 120564*a^6*b^3*x^6 + 41481*a^7*b^2*x^4 + 7129*a^8*b*x^2 + 252*a^9)/(a^10*b^9*x^20 + 9*a^11*b^ 8*x^18 + 36*a^12*b^7*x^16 + 84*a^13*b^6*x^14 + 126*a^14*b^5*x^12 + 126*a^1 5*b^4*x^10 + 84*a^16*b^3*x^8 + 36*a^17*b^2*x^6 + 9*a^18*b*x^4 + a^19*x^2) + 5*b*log(b*x^2 + a)/a^11 - 5*b*log(x^2)/a^11
Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx=-\frac {5 \, b \log \left (x^{2}\right )}{a^{11}} + \frac {5 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{11}} + \frac {10 \, b x^{2} - a}{2 \, a^{11} x^{2}} - \frac {7129 \, b^{10} x^{18} + 66429 \, a b^{9} x^{16} + 275796 \, a^{2} b^{8} x^{14} + 669984 \, a^{3} b^{7} x^{12} + 1050336 \, a^{4} b^{6} x^{10} + 1103256 \, a^{5} b^{5} x^{8} + 777840 \, a^{6} b^{4} x^{6} + 356040 \, a^{7} b^{3} x^{4} + 96570 \, a^{8} b^{2} x^{2} + 11990 \, a^{9} b}{504 \, {\left (b x^{2} + a\right )}^{9} a^{11}} \]
-5*b*log(x^2)/a^11 + 5*b*log(abs(b*x^2 + a))/a^11 + 1/2*(10*b*x^2 - a)/(a^ 11*x^2) - 1/504*(7129*b^10*x^18 + 66429*a*b^9*x^16 + 275796*a^2*b^8*x^14 + 669984*a^3*b^7*x^12 + 1050336*a^4*b^6*x^10 + 1103256*a^5*b^5*x^8 + 777840 *a^6*b^4*x^6 + 356040*a^7*b^3*x^4 + 96570*a^8*b^2*x^2 + 11990*a^9*b)/((b*x ^2 + a)^9*a^11)
Time = 0.52 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^3 \left (a+b x^2\right )^{10}} \, dx=\frac {5\,b\,\ln \left (b\,x^2+a\right )}{a^{11}}-\frac {\frac {1}{2\,a}+\frac {7129\,b\,x^2}{504\,a^2}+\frac {4609\,b^2\,x^4}{56\,a^3}+\frac {3349\,b^3\,x^6}{14\,a^4}+\frac {2509\,b^4\,x^8}{6\,a^5}+\frac {1879\,b^5\,x^{10}}{4\,a^6}+\frac {1375\,b^6\,x^{12}}{4\,a^7}+\frac {955\,b^7\,x^{14}}{6\,a^8}+\frac {85\,b^8\,x^{16}}{2\,a^9}+\frac {5\,b^9\,x^{18}}{a^{10}}}{a^9\,x^2+9\,a^8\,b\,x^4+36\,a^7\,b^2\,x^6+84\,a^6\,b^3\,x^8+126\,a^5\,b^4\,x^{10}+126\,a^4\,b^5\,x^{12}+84\,a^3\,b^6\,x^{14}+36\,a^2\,b^7\,x^{16}+9\,a\,b^8\,x^{18}+b^9\,x^{20}}-\frac {10\,b\,\ln \left (x\right )}{a^{11}} \]
(5*b*log(a + b*x^2))/a^11 - (1/(2*a) + (7129*b*x^2)/(504*a^2) + (4609*b^2* x^4)/(56*a^3) + (3349*b^3*x^6)/(14*a^4) + (2509*b^4*x^8)/(6*a^5) + (1879*b ^5*x^10)/(4*a^6) + (1375*b^6*x^12)/(4*a^7) + (955*b^7*x^14)/(6*a^8) + (85* b^8*x^16)/(2*a^9) + (5*b^9*x^18)/a^10)/(a^9*x^2 + b^9*x^20 + 9*a^8*b*x^4 + 9*a*b^8*x^18 + 36*a^7*b^2*x^6 + 84*a^6*b^3*x^8 + 126*a^5*b^4*x^10 + 126*a ^4*b^5*x^12 + 84*a^3*b^6*x^14 + 36*a^2*b^7*x^16) - (10*b*log(x))/a^11