Integrand size = 24, antiderivative size = 133 \[ \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {3 a x^2}{b^7}+\frac {x^4}{4 b^6}+\frac {a^7}{10 b^8 \left (a+b x^2\right )^5}-\frac {7 a^6}{8 b^8 \left (a+b x^2\right )^4}+\frac {7 a^5}{2 b^8 \left (a+b x^2\right )^3}-\frac {35 a^4}{4 b^8 \left (a+b x^2\right )^2}+\frac {35 a^3}{2 b^8 \left (a+b x^2\right )}+\frac {21 a^2 \log \left (a+b x^2\right )}{2 b^8} \] Output:
-3*a*x^2/b^7+1/4*x^4/b^6+1/10*a^7/b^8/(b*x^2+a)^5-7/8*a^6/b^8/(b*x^2+a)^4+ 7/2*a^5/b^8/(b*x^2+a)^3-35/4*a^4/b^8/(b*x^2+a)^2+35/2*a^3/b^8/(b*x^2+a)+21 /2*a^2*ln(b*x^2+a)/b^8
Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {459 a^7+1875 a^6 b x^2+2700 a^5 b^2 x^4+1300 a^4 b^3 x^6-400 a^3 b^4 x^8-500 a^2 b^5 x^{10}-70 a b^6 x^{12}+10 b^7 x^{14}+420 a^2 \left (a+b x^2\right )^5 \log \left (a+b x^2\right )}{40 b^8 \left (a+b x^2\right )^5} \] Input:
Integrate[x^15/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
(459*a^7 + 1875*a^6*b*x^2 + 2700*a^5*b^2*x^4 + 1300*a^4*b^3*x^6 - 400*a^3* b^4*x^8 - 500*a^2*b^5*x^10 - 70*a*b^6*x^12 + 10*b^7*x^14 + 420*a^2*(a + b* x^2)^5*Log[a + b*x^2])/(40*b^8*(a + b*x^2)^5)
Time = 0.51 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1380, 27, 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.
\(\displaystyle \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle b^6 \int \frac {x^{15}}{b^6 \left (b x^2+a\right )^6}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x^{15}}{\left (a+b x^2\right )^6}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {x^{14}}{\left (b x^2+a\right )^6}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {a^7}{b^7 \left (b x^2+a\right )^6}+\frac {7 a^6}{b^7 \left (b x^2+a\right )^5}-\frac {21 a^5}{b^7 \left (b x^2+a\right )^4}+\frac {35 a^4}{b^7 \left (b x^2+a\right )^3}-\frac {35 a^3}{b^7 \left (b x^2+a\right )^2}+\frac {21 a^2}{b^7 \left (b x^2+a\right )}-\frac {6 a}{b^7}+\frac {x^2}{b^6}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a^7}{5 b^8 \left (a+b x^2\right )^5}-\frac {7 a^6}{4 b^8 \left (a+b x^2\right )^4}+\frac {7 a^5}{b^8 \left (a+b x^2\right )^3}-\frac {35 a^4}{2 b^8 \left (a+b x^2\right )^2}+\frac {35 a^3}{b^8 \left (a+b x^2\right )}+\frac {21 a^2 \log \left (a+b x^2\right )}{b^8}-\frac {6 a x^2}{b^7}+\frac {x^4}{2 b^6}\right )\) |
Input:
Int[x^15/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
Output:
((-6*a*x^2)/b^7 + x^4/(2*b^6) + a^7/(5*b^8*(a + b*x^2)^5) - (7*a^6)/(4*b^8 *(a + b*x^2)^4) + (7*a^5)/(b^8*(a + b*x^2)^3) - (35*a^4)/(2*b^8*(a + b*x^2 )^2) + (35*a^3)/(b^8*(a + b*x^2)) + (21*a^2*Log[a + b*x^2])/b^8)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74
method | result | size |
norman | \(\frac {\frac {x^{14}}{4 b}-\frac {7 a \,x^{12}}{4 b^{2}}+\frac {959 a^{7}}{40 b^{8}}+\frac {105 a^{3} x^{8}}{2 b^{4}}+\frac {315 a^{4} x^{6}}{2 b^{5}}+\frac {385 a^{5} x^{4}}{2 b^{6}}+\frac {875 a^{6} x^{2}}{8 b^{7}}}{\left (b \,x^{2}+a \right )^{5}}+\frac {21 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{8}}\) | \(98\) |
default | \(\frac {\left (-b \,x^{2}+6 a \right )^{2}}{4 b^{8}}+\frac {a^{2} \left (\frac {21 \ln \left (b \,x^{2}+a \right )}{b}+\frac {7 a^{3}}{b \left (b \,x^{2}+a \right )^{3}}+\frac {a^{5}}{5 b \left (b \,x^{2}+a \right )^{5}}-\frac {35 a^{2}}{2 b \left (b \,x^{2}+a \right )^{2}}+\frac {35 a}{b \left (b \,x^{2}+a \right )}-\frac {7 a^{4}}{4 b \left (b \,x^{2}+a \right )^{4}}\right )}{2 b^{7}}\) | \(124\) |
risch | \(\frac {x^{4}}{4 b^{6}}-\frac {3 a \,x^{2}}{b^{7}}+\frac {9 a^{2}}{b^{8}}+\frac {\frac {35 a^{3} b^{3} x^{8}}{2}+\frac {245 a^{4} b^{2} x^{6}}{4}+\frac {329 a^{5} b \,x^{4}}{4}+\frac {399 a^{6} x^{2}}{8}+\frac {459 a^{7}}{40 b}}{b^{7} \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}+\frac {21 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{8}}\) | \(124\) |
parallelrisch | \(\frac {10 b^{7} x^{14}-70 a \,x^{12} b^{6}+420 \ln \left (b \,x^{2}+a \right ) x^{10} a^{2} b^{5}+2100 \ln \left (b \,x^{2}+a \right ) x^{8} a^{3} b^{4}+2100 x^{8} a^{3} b^{4}+4200 \ln \left (b \,x^{2}+a \right ) x^{6} a^{4} b^{3}+6300 x^{6} a^{4} b^{3}+4200 \ln \left (b \,x^{2}+a \right ) x^{4} a^{5} b^{2}+7700 x^{4} a^{5} b^{2}+2100 \ln \left (b \,x^{2}+a \right ) x^{2} a^{6} b +4375 x^{2} a^{6} b +420 \ln \left (b \,x^{2}+a \right ) a^{7}+959 a^{7}}{40 b^{8} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} \left (b \,x^{2}+a \right )}\) | \(206\) |
Input:
int(x^15/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)
Output:
(1/4/b*x^14-7/4*a/b^2*x^12+959/40*a^7/b^8+105/2*a^3/b^4*x^8+315/2*a^4/b^5* x^6+385/2*a^5/b^6*x^4+875/8*a^6/b^7*x^2)/(b*x^2+a)^5+21/2*a^2*ln(b*x^2+a)/ b^8
Time = 0.06 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.53 \[ \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {10 \, b^{7} x^{14} - 70 \, a b^{6} x^{12} - 500 \, a^{2} b^{5} x^{10} - 400 \, a^{3} b^{4} x^{8} + 1300 \, a^{4} b^{3} x^{6} + 2700 \, a^{5} b^{2} x^{4} + 1875 \, a^{6} b x^{2} + 459 \, a^{7} + 420 \, {\left (a^{2} b^{5} x^{10} + 5 \, a^{3} b^{4} x^{8} + 10 \, a^{4} b^{3} x^{6} + 10 \, a^{5} b^{2} x^{4} + 5 \, a^{6} b x^{2} + a^{7}\right )} \log \left (b x^{2} + a\right )}{40 \, {\left (b^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}} \] Input:
integrate(x^15/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
Output:
1/40*(10*b^7*x^14 - 70*a*b^6*x^12 - 500*a^2*b^5*x^10 - 400*a^3*b^4*x^8 + 1 300*a^4*b^3*x^6 + 2700*a^5*b^2*x^4 + 1875*a^6*b*x^2 + 459*a^7 + 420*(a^2*b ^5*x^10 + 5*a^3*b^4*x^8 + 10*a^4*b^3*x^6 + 10*a^5*b^2*x^4 + 5*a^6*b*x^2 + a^7)*log(b*x^2 + a))/(b^13*x^10 + 5*a*b^12*x^8 + 10*a^2*b^11*x^6 + 10*a^3* b^10*x^4 + 5*a^4*b^9*x^2 + a^5*b^8)
Time = 0.45 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13 \[ \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {21 a^{2} \log {\left (a + b x^{2} \right )}}{2 b^{8}} - \frac {3 a x^{2}}{b^{7}} + \frac {459 a^{7} + 1995 a^{6} b x^{2} + 3290 a^{5} b^{2} x^{4} + 2450 a^{4} b^{3} x^{6} + 700 a^{3} b^{4} x^{8}}{40 a^{5} b^{8} + 200 a^{4} b^{9} x^{2} + 400 a^{3} b^{10} x^{4} + 400 a^{2} b^{11} x^{6} + 200 a b^{12} x^{8} + 40 b^{13} x^{10}} + \frac {x^{4}}{4 b^{6}} \] Input:
integrate(x**15/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
Output:
21*a**2*log(a + b*x**2)/(2*b**8) - 3*a*x**2/b**7 + (459*a**7 + 1995*a**6*b *x**2 + 3290*a**5*b**2*x**4 + 2450*a**4*b**3*x**6 + 700*a**3*b**4*x**8)/(4 0*a**5*b**8 + 200*a**4*b**9*x**2 + 400*a**3*b**10*x**4 + 400*a**2*b**11*x* *6 + 200*a*b**12*x**8 + 40*b**13*x**10) + x**4/(4*b**6)
Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {700 \, a^{3} b^{4} x^{8} + 2450 \, a^{4} b^{3} x^{6} + 3290 \, a^{5} b^{2} x^{4} + 1995 \, a^{6} b x^{2} + 459 \, a^{7}}{40 \, {\left (b^{13} x^{10} + 5 \, a b^{12} x^{8} + 10 \, a^{2} b^{11} x^{6} + 10 \, a^{3} b^{10} x^{4} + 5 \, a^{4} b^{9} x^{2} + a^{5} b^{8}\right )}} + \frac {21 \, a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{8}} + \frac {b x^{4} - 12 \, a x^{2}}{4 \, b^{7}} \] Input:
integrate(x^15/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
Output:
1/40*(700*a^3*b^4*x^8 + 2450*a^4*b^3*x^6 + 3290*a^5*b^2*x^4 + 1995*a^6*b*x ^2 + 459*a^7)/(b^13*x^10 + 5*a*b^12*x^8 + 10*a^2*b^11*x^6 + 10*a^3*b^10*x^ 4 + 5*a^4*b^9*x^2 + a^5*b^8) + 21/2*a^2*log(b*x^2 + a)/b^8 + 1/4*(b*x^4 - 12*a*x^2)/b^7
Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.85 \[ \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {21 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{8}} + \frac {b^{6} x^{4} - 12 \, a b^{5} x^{2}}{4 \, b^{12}} - \frac {959 \, a^{2} b^{5} x^{10} + 4095 \, a^{3} b^{4} x^{8} + 7140 \, a^{4} b^{3} x^{6} + 6300 \, a^{5} b^{2} x^{4} + 2800 \, a^{6} b x^{2} + 500 \, a^{7}}{40 \, {\left (b x^{2} + a\right )}^{5} b^{8}} \] Input:
integrate(x^15/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
Output:
21/2*a^2*log(abs(b*x^2 + a))/b^8 + 1/4*(b^6*x^4 - 12*a*b^5*x^2)/b^12 - 1/4 0*(959*a^2*b^5*x^10 + 4095*a^3*b^4*x^8 + 7140*a^4*b^3*x^6 + 6300*a^5*b^2*x ^4 + 2800*a^6*b*x^2 + 500*a^7)/((b*x^2 + a)^5*b^8)
Time = 17.75 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.07 \[ \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {459\,a^7}{40\,b}+\frac {399\,a^6\,x^2}{8}+\frac {329\,a^5\,b\,x^4}{4}+\frac {245\,a^4\,b^2\,x^6}{4}+\frac {35\,a^3\,b^3\,x^8}{2}}{a^5\,b^7+5\,a^4\,b^8\,x^2+10\,a^3\,b^9\,x^4+10\,a^2\,b^{10}\,x^6+5\,a\,b^{11}\,x^8+b^{12}\,x^{10}}+\frac {x^4}{4\,b^6}-\frac {3\,a\,x^2}{b^7}+\frac {21\,a^2\,\ln \left (b\,x^2+a\right )}{2\,b^8} \] Input:
int(x^15/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
Output:
((459*a^7)/(40*b) + (399*a^6*x^2)/8 + (329*a^5*b*x^4)/4 + (245*a^4*b^2*x^6 )/4 + (35*a^3*b^3*x^8)/2)/(a^5*b^7 + b^12*x^10 + 5*a*b^11*x^8 + 5*a^4*b^8* x^2 + 10*a^3*b^9*x^4 + 10*a^2*b^10*x^6) + x^4/(4*b^6) - (3*a*x^2)/b^7 + (2 1*a^2*log(a + b*x^2))/(2*b^8)
Time = 0.15 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.72 \[ \int \frac {x^{15}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {420 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{7}+2100 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{6} b \,x^{2}+4200 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{5} b^{2} x^{4}+4200 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} b^{3} x^{6}+2100 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b^{4} x^{8}+420 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{5} x^{10}+539 a^{7}+2275 a^{6} b \,x^{2}+3500 a^{5} b^{2} x^{4}+2100 a^{4} b^{3} x^{6}-420 a^{2} b^{5} x^{10}-70 a \,b^{6} x^{12}+10 b^{7} x^{14}}{40 b^{8} \left (b^{5} x^{10}+5 a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}+5 a^{4} b \,x^{2}+a^{5}\right )} \] Input:
int(x^15/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
Output:
(420*log(a + b*x**2)*a**7 + 2100*log(a + b*x**2)*a**6*b*x**2 + 4200*log(a + b*x**2)*a**5*b**2*x**4 + 4200*log(a + b*x**2)*a**4*b**3*x**6 + 2100*log( a + b*x**2)*a**3*b**4*x**8 + 420*log(a + b*x**2)*a**2*b**5*x**10 + 539*a** 7 + 2275*a**6*b*x**2 + 3500*a**5*b**2*x**4 + 2100*a**4*b**3*x**6 - 420*a** 2*b**5*x**10 - 70*a*b**6*x**12 + 10*b**7*x**14)/(40*b**8*(a**5 + 5*a**4*b* x**2 + 10*a**3*b**2*x**4 + 10*a**2*b**3*x**6 + 5*a*b**4*x**8 + b**5*x**10) )