Integrand size = 24, antiderivative size = 82 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx=-\frac {a^6}{24 x^{24}}-\frac {3 a^5 b}{11 x^{22}}-\frac {3 a^4 b^2}{4 x^{20}}-\frac {10 a^3 b^3}{9 x^{18}}-\frac {15 a^2 b^4}{16 x^{16}}-\frac {3 a b^5}{7 x^{14}}-\frac {b^6}{12 x^{12}} \] Output:
-1/24*a^6/x^24-3/11*a^5*b/x^22-3/4*a^4*b^2/x^20-10/9*a^3*b^3/x^18-15/16*a^ 2*b^4/x^16-3/7*a*b^5/x^14-1/12*b^6/x^12
Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx=-\frac {a^6}{24 x^{24}}-\frac {3 a^5 b}{11 x^{22}}-\frac {3 a^4 b^2}{4 x^{20}}-\frac {10 a^3 b^3}{9 x^{18}}-\frac {15 a^2 b^4}{16 x^{16}}-\frac {3 a b^5}{7 x^{14}}-\frac {b^6}{12 x^{12}} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^25,x]
Output:
-1/24*a^6/x^24 - (3*a^5*b)/(11*x^22) - (3*a^4*b^2)/(4*x^20) - (10*a^3*b^3) /(9*x^18) - (15*a^2*b^4)/(16*x^16) - (3*a*b^5)/(7*x^14) - b^6/(12*x^12)
Time = 0.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05, 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, 53, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int \frac {b^6 \left (b x^2+a\right )^6}{x^{25}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^6}{x^{25}}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^6}{x^{26}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^6}{x^{26}}+\frac {6 b a^5}{x^{24}}+\frac {15 b^2 a^4}{x^{22}}+\frac {20 b^3 a^3}{x^{20}}+\frac {15 b^4 a^2}{x^{18}}+\frac {6 b^5 a}{x^{16}}+\frac {b^6}{x^{14}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^6}{12 x^{24}}-\frac {6 a^5 b}{11 x^{22}}-\frac {3 a^4 b^2}{2 x^{20}}-\frac {20 a^3 b^3}{9 x^{18}}-\frac {15 a^2 b^4}{8 x^{16}}-\frac {6 a b^5}{7 x^{14}}-\frac {b^6}{6 x^{12}}\right )\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^25,x]
Output:
(-1/12*a^6/x^24 - (6*a^5*b)/(11*x^22) - (3*a^4*b^2)/(2*x^20) - (20*a^3*b^3 )/(9*x^18) - (15*a^2*b^4)/(8*x^16) - (6*a*b^5)/(7*x^14) - b^6/(6*x^12))/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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {a^{6}}{24 x^{24}}-\frac {3 a^{5} b}{11 x^{22}}-\frac {3 a^{4} b^{2}}{4 x^{20}}-\frac {10 a^{3} b^{3}}{9 x^{18}}-\frac {15 a^{2} b^{4}}{16 x^{16}}-\frac {3 a \,b^{5}}{7 x^{14}}-\frac {b^{6}}{12 x^{12}}\) | \(69\) |
norman | \(\frac {-\frac {1}{24} a^{6}-\frac {15}{16} a^{2} b^{4} x^{8}-\frac {3}{7} a \,b^{5} x^{10}-\frac {1}{12} b^{6} x^{12}-\frac {3}{4} a^{4} b^{2} x^{4}-\frac {10}{9} a^{3} b^{3} x^{6}-\frac {3}{11} a^{5} b \,x^{2}}{x^{24}}\) | \(70\) |
risch | \(\frac {-\frac {1}{24} a^{6}-\frac {15}{16} a^{2} b^{4} x^{8}-\frac {3}{7} a \,b^{5} x^{10}-\frac {1}{12} b^{6} x^{12}-\frac {3}{4} a^{4} b^{2} x^{4}-\frac {10}{9} a^{3} b^{3} x^{6}-\frac {3}{11} a^{5} b \,x^{2}}{x^{24}}\) | \(70\) |
gosper | \(-\frac {924 b^{6} x^{12}+4752 a \,b^{5} x^{10}+10395 a^{2} b^{4} x^{8}+12320 a^{3} b^{3} x^{6}+8316 a^{4} b^{2} x^{4}+3024 a^{5} b \,x^{2}+462 a^{6}}{11088 x^{24}}\) | \(71\) |
parallelrisch | \(\frac {-924 b^{6} x^{12}-4752 a \,b^{5} x^{10}-10395 a^{2} b^{4} x^{8}-12320 a^{3} b^{3} x^{6}-8316 a^{4} b^{2} x^{4}-3024 a^{5} b \,x^{2}-462 a^{6}}{11088 x^{24}}\) | \(71\) |
orering | \(-\frac {\left (924 b^{6} x^{12}+4752 a \,b^{5} x^{10}+10395 a^{2} b^{4} x^{8}+12320 a^{3} b^{3} x^{6}+8316 a^{4} b^{2} x^{4}+3024 a^{5} b \,x^{2}+462 a^{6}\right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}{11088 x^{24} \left (b \,x^{2}+a \right )^{6}}\) | \(100\) |
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^3/x^25,x,method=_RETURNVERBOSE)
Output:
-1/24*a^6/x^24-3/11*a^5*b/x^22-3/4*a^4*b^2/x^20-10/9*a^3*b^3/x^18-15/16*a^ 2*b^4/x^16-3/7*a*b^5/x^14-1/12*b^6/x^12
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx=-\frac {924 \, b^{6} x^{12} + 4752 \, a b^{5} x^{10} + 10395 \, a^{2} b^{4} x^{8} + 12320 \, a^{3} b^{3} x^{6} + 8316 \, a^{4} b^{2} x^{4} + 3024 \, a^{5} b x^{2} + 462 \, a^{6}}{11088 \, x^{24}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^25,x, algorithm="fricas")
Output:
-1/11088*(924*b^6*x^12 + 4752*a*b^5*x^10 + 10395*a^2*b^4*x^8 + 12320*a^3*b ^3*x^6 + 8316*a^4*b^2*x^4 + 3024*a^5*b*x^2 + 462*a^6)/x^24
Time = 0.38 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx=\frac {- 462 a^{6} - 3024 a^{5} b x^{2} - 8316 a^{4} b^{2} x^{4} - 12320 a^{3} b^{3} x^{6} - 10395 a^{2} b^{4} x^{8} - 4752 a b^{5} x^{10} - 924 b^{6} x^{12}}{11088 x^{24}} \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**25,x)
Output:
(-462*a**6 - 3024*a**5*b*x**2 - 8316*a**4*b**2*x**4 - 12320*a**3*b**3*x**6 - 10395*a**2*b**4*x**8 - 4752*a*b**5*x**10 - 924*b**6*x**12)/(11088*x**24 )
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx=-\frac {924 \, b^{6} x^{12} + 4752 \, a b^{5} x^{10} + 10395 \, a^{2} b^{4} x^{8} + 12320 \, a^{3} b^{3} x^{6} + 8316 \, a^{4} b^{2} x^{4} + 3024 \, a^{5} b x^{2} + 462 \, a^{6}}{11088 \, x^{24}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^25,x, algorithm="maxima")
Output:
-1/11088*(924*b^6*x^12 + 4752*a*b^5*x^10 + 10395*a^2*b^4*x^8 + 12320*a^3*b ^3*x^6 + 8316*a^4*b^2*x^4 + 3024*a^5*b*x^2 + 462*a^6)/x^24
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx=-\frac {924 \, b^{6} x^{12} + 4752 \, a b^{5} x^{10} + 10395 \, a^{2} b^{4} x^{8} + 12320 \, a^{3} b^{3} x^{6} + 8316 \, a^{4} b^{2} x^{4} + 3024 \, a^{5} b x^{2} + 462 \, a^{6}}{11088 \, x^{24}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^25,x, algorithm="giac")
Output:
-1/11088*(924*b^6*x^12 + 4752*a*b^5*x^10 + 10395*a^2*b^4*x^8 + 12320*a^3*b ^3*x^6 + 8316*a^4*b^2*x^4 + 3024*a^5*b*x^2 + 462*a^6)/x^24
Time = 19.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx=-\frac {\frac {a^6}{24}+\frac {3\,a^5\,b\,x^2}{11}+\frac {3\,a^4\,b^2\,x^4}{4}+\frac {10\,a^3\,b^3\,x^6}{9}+\frac {15\,a^2\,b^4\,x^8}{16}+\frac {3\,a\,b^5\,x^{10}}{7}+\frac {b^6\,x^{12}}{12}}{x^{24}} \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^25,x)
Output:
-(a^6/24 + (b^6*x^12)/12 + (3*a^5*b*x^2)/11 + (3*a*b^5*x^10)/7 + (3*a^4*b^ 2*x^4)/4 + (10*a^3*b^3*x^6)/9 + (15*a^2*b^4*x^8)/16)/x^24
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{25}} \, dx=\frac {-924 b^{6} x^{12}-4752 a \,b^{5} x^{10}-10395 a^{2} b^{4} x^{8}-12320 a^{3} b^{3} x^{6}-8316 a^{4} b^{2} x^{4}-3024 a^{5} b \,x^{2}-462 a^{6}}{11088 x^{24}} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^3/x^25,x)
Output:
( - 462*a**6 - 3024*a**5*b*x**2 - 8316*a**4*b**2*x**4 - 12320*a**3*b**3*x* *6 - 10395*a**2*b**4*x**8 - 4752*a*b**5*x**10 - 924*b**6*x**12)/(11088*x** 24)