Integrand size = 13, antiderivative size = 179 \[ \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx=\frac {a^9}{18 b^{10} \left (a+b x^2\right )^9}-\frac {9 a^8}{16 b^{10} \left (a+b x^2\right )^8}+\frac {18 a^7}{7 b^{10} \left (a+b x^2\right )^7}-\frac {7 a^6}{b^{10} \left (a+b x^2\right )^6}+\frac {63 a^5}{5 b^{10} \left (a+b x^2\right )^5}-\frac {63 a^4}{4 b^{10} \left (a+b x^2\right )^4}+\frac {14 a^3}{b^{10} \left (a+b x^2\right )^3}-\frac {9 a^2}{b^{10} \left (a+b x^2\right )^2}+\frac {9 a}{2 b^{10} \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^{10}} \] Output:
1/18*a^9/b^10/(b*x^2+a)^9-9/16*a^8/b^10/(b*x^2+a)^8+18/7*a^7/b^10/(b*x^2+a )^7-7*a^6/b^10/(b*x^2+a)^6+63/5*a^5/b^10/(b*x^2+a)^5-63/4*a^4/b^10/(b*x^2+ a)^4+14*a^3/b^10/(b*x^2+a)^3-9*a^2/b^10/(b*x^2+a)^2+9/2*a/b^10/(b*x^2+a)+1 /2*ln(b*x^2+a)/b^10
Time = 0.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.65 \[ \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {a \left (7129 a^8+61641 a^7 b x^2+235224 a^6 b^2 x^4+518616 a^5 b^3 x^6+725004 a^4 b^4 x^8+661500 a^3 b^5 x^{10}+388080 a^2 b^6 x^{12}+136080 a b^7 x^{14}+22680 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+2520 \log \left (a+b x^2\right )}{5040 b^{10}} \] Input:
Integrate[x^19/(a + b*x^2)^10,x]
Output:
((a*(7129*a^8 + 61641*a^7*b*x^2 + 235224*a^6*b^2*x^4 + 518616*a^5*b^3*x^6 + 725004*a^4*b^4*x^8 + 661500*a^3*b^5*x^10 + 388080*a^2*b^6*x^12 + 136080* a*b^7*x^14 + 22680*b^8*x^16))/(a + b*x^2)^9 + 2520*Log[a + b*x^2])/(5040*b ^10)
Time = 0.33 (sec) , antiderivative size = 178, 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, 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.
| \(\displaystyle \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int \frac {x^{18}}{\left (b x^2+a\right )^{10}}dx^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {a^9}{b^9 \left (b x^2+a\right )^{10}}+\frac {9 a^8}{b^9 \left (b x^2+a\right )^9}-\frac {36 a^7}{b^9 \left (b x^2+a\right )^8}+\frac {84 a^6}{b^9 \left (b x^2+a\right )^7}-\frac {126 a^5}{b^9 \left (b x^2+a\right )^6}+\frac {126 a^4}{b^9 \left (b x^2+a\right )^5}-\frac {84 a^3}{b^9 \left (b x^2+a\right )^4}+\frac {36 a^2}{b^9 \left (b x^2+a\right )^3}-\frac {9 a}{b^9 \left (b x^2+a\right )^2}+\frac {1}{b^9 \left (b x^2+a\right )}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {a^9}{9 b^{10} \left (a+b x^2\right )^9}-\frac {9 a^8}{8 b^{10} \left (a+b x^2\right )^8}+\frac {36 a^7}{7 b^{10} \left (a+b x^2\right )^7}-\frac {14 a^6}{b^{10} \left (a+b x^2\right )^6}+\frac {126 a^5}{5 b^{10} \left (a+b x^2\right )^5}-\frac {63 a^4}{2 b^{10} \left (a+b x^2\right )^4}+\frac {28 a^3}{b^{10} \left (a+b x^2\right )^3}-\frac {18 a^2}{b^{10} \left (a+b x^2\right )^2}+\frac {9 a}{b^{10} \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{b^{10}}\right )\) |
Input:
Int[x^19/(a + b*x^2)^10,x]
Output:
(a^9/(9*b^10*(a + b*x^2)^9) - (9*a^8)/(8*b^10*(a + b*x^2)^8) + (36*a^7)/(7 *b^10*(a + b*x^2)^7) - (14*a^6)/(b^10*(a + b*x^2)^6) + (126*a^5)/(5*b^10*( a + b*x^2)^5) - (63*a^4)/(2*b^10*(a + b*x^2)^4) + (28*a^3)/(b^10*(a + b*x^ 2)^3) - (18*a^2)/(b^10*(a + b*x^2)^2) + (9*a)/(b^10*(a + b*x^2)) + Log[a + b*x^2]/b^10)/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 = 0.30 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.67
| method | result | size |
| norman | \(\frac {\frac {7129 a^{9}}{5040 b^{10}}+\frac {9 a \,x^{16}}{2 b^{2}}+\frac {27 a^{2} x^{14}}{b^{3}}+\frac {77 a^{3} x^{12}}{b^{4}}+\frac {525 a^{4} x^{10}}{4 b^{5}}+\frac {2877 a^{5} x^{8}}{20 b^{6}}+\frac {1029 a^{6} x^{6}}{10 b^{7}}+\frac {3267 a^{7} x^{4}}{70 b^{8}}+\frac {6849 a^{8} x^{2}}{560 b^{9}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{10}}\) | \(120\) |
| risch | \(\frac {\frac {7129 a^{9}}{5040 b^{10}}+\frac {9 a \,x^{16}}{2 b^{2}}+\frac {27 a^{2} x^{14}}{b^{3}}+\frac {77 a^{3} x^{12}}{b^{4}}+\frac {525 a^{4} x^{10}}{4 b^{5}}+\frac {2877 a^{5} x^{8}}{20 b^{6}}+\frac {1029 a^{6} x^{6}}{10 b^{7}}+\frac {3267 a^{7} x^{4}}{70 b^{8}}+\frac {6849 a^{8} x^{2}}{560 b^{9}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{10}}\) | \(120\) |
| default | \(\frac {a^{9}}{18 b^{10} \left (b \,x^{2}+a \right )^{9}}-\frac {9 a^{8}}{16 b^{10} \left (b \,x^{2}+a \right )^{8}}+\frac {18 a^{7}}{7 b^{10} \left (b \,x^{2}+a \right )^{7}}-\frac {7 a^{6}}{b^{10} \left (b \,x^{2}+a \right )^{6}}+\frac {63 a^{5}}{5 b^{10} \left (b \,x^{2}+a \right )^{5}}-\frac {63 a^{4}}{4 b^{10} \left (b \,x^{2}+a \right )^{4}}+\frac {14 a^{3}}{b^{10} \left (b \,x^{2}+a \right )^{3}}-\frac {9 a^{2}}{b^{10} \left (b \,x^{2}+a \right )^{2}}+\frac {9 a}{2 b^{10} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{10}}\) | \(166\) |
| parallelrisch | \(\frac {22680 \ln \left (b \,x^{2}+a \right ) x^{16} a \,b^{8}+90720 \ln \left (b \,x^{2}+a \right ) x^{14} a^{2} b^{7}+211680 \ln \left (b \,x^{2}+a \right ) x^{12} a^{3} b^{6}+317520 \ln \left (b \,x^{2}+a \right ) x^{10} a^{4} b^{5}+317520 \ln \left (b \,x^{2}+a \right ) x^{8} a^{5} b^{4}+211680 \ln \left (b \,x^{2}+a \right ) x^{6} a^{6} b^{3}+90720 \ln \left (b \,x^{2}+a \right ) x^{4} a^{7} b^{2}+22680 \ln \left (b \,x^{2}+a \right ) x^{2} a^{8} b +7129 a^{9}+61641 a^{8} b \,x^{2}+2520 \ln \left (b \,x^{2}+a \right ) a^{9}+235224 a^{7} x^{4} b^{2}+136080 a^{2} x^{14} b^{7}+22680 a \,x^{16} b^{8}+2520 \ln \left (b \,x^{2}+a \right ) x^{18} b^{9}+518616 a^{6} x^{6} b^{3}+725004 a^{5} x^{8} b^{4}+661500 a^{4} x^{10} b^{5}+388080 a^{3} x^{12} b^{6}}{5040 b^{10} \left (b \,x^{2}+a \right )^{9}}\) | \(282\) |
Input:
int(x^19/(b*x^2+a)^10,x,method=_RETURNVERBOSE)
Output:
(7129/5040*a^9/b^10+9/2*a/b^2*x^16+27*a^2/b^3*x^14+77*a^3/b^4*x^12+525/4*a ^4/b^5*x^10+2877/20*a^5/b^6*x^8+1029/10*a^6/b^7*x^6+3267/70*a^7/b^8*x^4+68 49/560*a^8/b^9*x^2)/(b*x^2+a)^9+1/2*ln(b*x^2+a)/b^10
Time = 0.07 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.68 \[ \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx=\frac {22680 \, a b^{8} x^{16} + 136080 \, a^{2} b^{7} x^{14} + 388080 \, a^{3} b^{6} x^{12} + 661500 \, a^{4} b^{5} x^{10} + 725004 \, a^{5} b^{4} x^{8} + 518616 \, a^{6} b^{3} x^{6} + 235224 \, a^{7} b^{2} x^{4} + 61641 \, a^{8} b x^{2} + 7129 \, a^{9} + 2520 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (b x^{2} + a\right )}{5040 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}} \] Input:
integrate(x^19/(b*x^2+a)^10,x, algorithm="fricas")
Output:
1/5040*(22680*a*b^8*x^16 + 136080*a^2*b^7*x^14 + 388080*a^3*b^6*x^12 + 661 500*a^4*b^5*x^10 + 725004*a^5*b^4*x^8 + 518616*a^6*b^3*x^6 + 235224*a^7*b^ 2*x^4 + 61641*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))/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2*b^17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6 + 36*a^7*b^12*x^4 + 9*a^8*b^11*x^2 + a^9*b^10)
Time = 0.85 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.22 \[ \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx=\frac {7129 a^{9} + 61641 a^{8} b x^{2} + 235224 a^{7} b^{2} x^{4} + 518616 a^{6} b^{3} x^{6} + 725004 a^{5} b^{4} x^{8} + 661500 a^{4} b^{5} x^{10} + 388080 a^{3} b^{6} x^{12} + 136080 a^{2} b^{7} x^{14} + 22680 a b^{8} x^{16}}{5040 a^{9} b^{10} + 45360 a^{8} b^{11} x^{2} + 181440 a^{7} b^{12} x^{4} + 423360 a^{6} b^{13} x^{6} + 635040 a^{5} b^{14} x^{8} + 635040 a^{4} b^{15} x^{10} + 423360 a^{3} b^{16} x^{12} + 181440 a^{2} b^{17} x^{14} + 45360 a b^{18} x^{16} + 5040 b^{19} x^{18}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{10}} \] Input:
integrate(x**19/(b*x**2+a)**10,x)
Output:
(7129*a**9 + 61641*a**8*b*x**2 + 235224*a**7*b**2*x**4 + 518616*a**6*b**3* x**6 + 725004*a**5*b**4*x**8 + 661500*a**4*b**5*x**10 + 388080*a**3*b**6*x **12 + 136080*a**2*b**7*x**14 + 22680*a*b**8*x**16)/(5040*a**9*b**10 + 453 60*a**8*b**11*x**2 + 181440*a**7*b**12*x**4 + 423360*a**6*b**13*x**6 + 635 040*a**5*b**14*x**8 + 635040*a**4*b**15*x**10 + 423360*a**3*b**16*x**12 + 181440*a**2*b**17*x**14 + 45360*a*b**18*x**16 + 5040*b**19*x**18) + log(a + b*x**2)/(2*b**10)
Time = 0.04 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.17 \[ \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx=\frac {22680 \, a b^{8} x^{16} + 136080 \, a^{2} b^{7} x^{14} + 388080 \, a^{3} b^{6} x^{12} + 661500 \, a^{4} b^{5} x^{10} + 725004 \, a^{5} b^{4} x^{8} + 518616 \, a^{6} b^{3} x^{6} + 235224 \, a^{7} b^{2} x^{4} + 61641 \, a^{8} b x^{2} + 7129 \, a^{9}}{5040 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{10}} \] Input:
integrate(x^19/(b*x^2+a)^10,x, algorithm="maxima")
Output:
1/5040*(22680*a*b^8*x^16 + 136080*a^2*b^7*x^14 + 388080*a^3*b^6*x^12 + 661 500*a^4*b^5*x^10 + 725004*a^5*b^4*x^8 + 518616*a^6*b^3*x^6 + 235224*a^7*b^ 2*x^4 + 61641*a^8*b*x^2 + 7129*a^9)/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2*b^ 17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a^6 *b^13*x^6 + 36*a^7*b^12*x^4 + 9*a^8*b^11*x^2 + a^9*b^10) + 1/2*log(b*x^2 + a)/b^10
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{10}} - \frac {7129 \, b^{8} x^{18} + 41481 \, a b^{7} x^{16} + 120564 \, a^{2} b^{6} x^{14} + 210756 \, a^{3} b^{5} x^{12} + 236754 \, a^{4} b^{4} x^{10} + 173250 \, a^{5} b^{3} x^{8} + 80220 \, a^{6} b^{2} x^{6} + 21420 \, a^{7} b x^{4} + 2520 \, a^{8} x^{2}}{5040 \, {\left (b x^{2} + a\right )}^{9} b^{9}} \] Input:
integrate(x^19/(b*x^2+a)^10,x, algorithm="giac")
Output:
1/2*log(abs(b*x^2 + a))/b^10 - 1/5040*(7129*b^8*x^18 + 41481*a*b^7*x^16 + 120564*a^2*b^6*x^14 + 210756*a^3*b^5*x^12 + 236754*a^4*b^4*x^10 + 173250*a ^5*b^3*x^8 + 80220*a^6*b^2*x^6 + 21420*a^7*b*x^4 + 2520*a^8*x^2)/((b*x^2 + a)^9*b^9)
Time = 0.44 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.16 \[ \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx=\frac {\frac {7129\,a^9}{5040\,b^{10}}+\frac {9\,a\,x^{16}}{2\,b^2}+\frac {27\,a^2\,x^{14}}{b^3}+\frac {77\,a^3\,x^{12}}{b^4}+\frac {525\,a^4\,x^{10}}{4\,b^5}+\frac {2877\,a^5\,x^8}{20\,b^6}+\frac {1029\,a^6\,x^6}{10\,b^7}+\frac {3267\,a^7\,x^4}{70\,b^8}+\frac {6849\,a^8\,x^2}{560\,b^9}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {\ln \left (b\,x^2+a\right )}{2\,b^{10}} \] Input:
int(x^19/(a + b*x^2)^10,x)
Output:
((7129*a^9)/(5040*b^10) + (9*a*x^16)/(2*b^2) + (27*a^2*x^14)/b^3 + (77*a^3 *x^12)/b^4 + (525*a^4*x^10)/(4*b^5) + (2877*a^5*x^8)/(20*b^6) + (1029*a^6* x^6)/(10*b^7) + (3267*a^7*x^4)/(70*b^8) + (6849*a^8*x^2)/(560*b^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*b^10)
Time = 0.23 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.06 \[ \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx=\frac {2520 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{9}+22680 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{8} b \,x^{2}+90720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{7} b^{2} x^{4}+211680 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{6} b^{3} x^{6}+317520 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{5} b^{4} x^{8}+317520 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} b^{5} x^{10}+211680 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b^{6} x^{12}+90720 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{7} x^{14}+22680 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{8} x^{16}+2520 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{9} x^{18}+4609 a^{9}+38961 a^{8} b \,x^{2}+144504 a^{7} b^{2} x^{4}+306936 a^{6} b^{3} x^{6}+407484 a^{5} b^{4} x^{8}+343980 a^{4} b^{5} x^{10}+176400 a^{3} b^{6} x^{12}+45360 a^{2} b^{7} x^{14}-2520 b^{9} x^{18}}{5040 b^{10} \left (b^{9} x^{18}+9 a \,b^{8} x^{16}+36 a^{2} b^{7} x^{14}+84 a^{3} b^{6} x^{12}+126 a^{4} b^{5} x^{10}+126 a^{5} b^{4} x^{8}+84 a^{6} b^{3} x^{6}+36 a^{7} b^{2} x^{4}+9 a^{8} b \,x^{2}+a^{9}\right )} \] Input:
int(x^19/(b*x^2+a)^10,x)
Output:
(2520*log(a + b*x**2)*a**9 + 22680*log(a + b*x**2)*a**8*b*x**2 + 90720*log (a + b*x**2)*a**7*b**2*x**4 + 211680*log(a + b*x**2)*a**6*b**3*x**6 + 3175 20*log(a + b*x**2)*a**5*b**4*x**8 + 317520*log(a + b*x**2)*a**4*b**5*x**10 + 211680*log(a + b*x**2)*a**3*b**6*x**12 + 90720*log(a + b*x**2)*a**2*b** 7*x**14 + 22680*log(a + b*x**2)*a*b**8*x**16 + 2520*log(a + b*x**2)*b**9*x **18 + 4609*a**9 + 38961*a**8*b*x**2 + 144504*a**7*b**2*x**4 + 306936*a**6 *b**3*x**6 + 407484*a**5*b**4*x**8 + 343980*a**4*b**5*x**10 + 176400*a**3* b**6*x**12 + 45360*a**2*b**7*x**14 - 2520*b**9*x**18)/(5040*b**10*(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))