Integrand size = 24, antiderivative size = 82 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx=-\frac {a^6}{22 x^{22}}-\frac {3 a^5 b}{10 x^{20}}-\frac {5 a^4 b^2}{6 x^{18}}-\frac {5 a^3 b^3}{4 x^{16}}-\frac {15 a^2 b^4}{14 x^{14}}-\frac {a b^5}{2 x^{12}}-\frac {b^6}{10 x^{10}} \] Output:
-1/22*a^6/x^22-3/10*a^5*b/x^20-5/6*a^4*b^2/x^18-5/4*a^3*b^3/x^16-15/14*a^2 *b^4/x^14-1/2*a*b^5/x^12-1/10*b^6/x^10
Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx=-\frac {a^6}{22 x^{22}}-\frac {3 a^5 b}{10 x^{20}}-\frac {5 a^4 b^2}{6 x^{18}}-\frac {5 a^3 b^3}{4 x^{16}}-\frac {15 a^2 b^4}{14 x^{14}}-\frac {a b^5}{2 x^{12}}-\frac {b^6}{10 x^{10}} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^23,x]
Output:
-1/22*a^6/x^22 - (3*a^5*b)/(10*x^20) - (5*a^4*b^2)/(6*x^18) - (5*a^3*b^3)/ (4*x^16) - (15*a^2*b^4)/(14*x^14) - (a*b^5)/(2*x^12) - b^6/(10*x^10)
Time = 0.38 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int \frac {b^6 \left (b x^2+a\right )^6}{x^{23}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^6}{x^{23}}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^6}{x^{24}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^6}{x^{24}}+\frac {6 b a^5}{x^{22}}+\frac {15 b^2 a^4}{x^{20}}+\frac {20 b^3 a^3}{x^{18}}+\frac {15 b^4 a^2}{x^{16}}+\frac {6 b^5 a}{x^{14}}+\frac {b^6}{x^{12}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^6}{11 x^{22}}-\frac {3 a^5 b}{5 x^{20}}-\frac {5 a^4 b^2}{3 x^{18}}-\frac {5 a^3 b^3}{2 x^{16}}-\frac {15 a^2 b^4}{7 x^{14}}-\frac {a b^5}{x^{12}}-\frac {b^6}{5 x^{10}}\right )\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^23,x]
Output:
(-1/11*a^6/x^22 - (3*a^5*b)/(5*x^20) - (5*a^4*b^2)/(3*x^18) - (5*a^3*b^3)/ (2*x^16) - (15*a^2*b^4)/(7*x^14) - (a*b^5)/x^12 - b^6/(5*x^10))/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 | \(-\frac {a^{6}}{22 x^{22}}-\frac {3 a^{5} b}{10 x^{20}}-\frac {5 a^{4} b^{2}}{6 x^{18}}-\frac {5 a^{3} b^{3}}{4 x^{16}}-\frac {15 a^{2} b^{4}}{14 x^{14}}-\frac {a \,b^{5}}{2 x^{12}}-\frac {b^{6}}{10 x^{10}}\) | \(69\) |
norman | \(\frac {-\frac {1}{22} a^{6}-\frac {3}{10} a^{5} b \,x^{2}-\frac {5}{6} a^{4} b^{2} x^{4}-\frac {5}{4} a^{3} b^{3} x^{6}-\frac {15}{14} a^{2} b^{4} x^{8}-\frac {1}{2} a \,b^{5} x^{10}-\frac {1}{10} b^{6} x^{12}}{x^{22}}\) | \(70\) |
risch | \(\frac {-\frac {1}{22} a^{6}-\frac {3}{10} a^{5} b \,x^{2}-\frac {5}{6} a^{4} b^{2} x^{4}-\frac {5}{4} a^{3} b^{3} x^{6}-\frac {15}{14} a^{2} b^{4} x^{8}-\frac {1}{2} a \,b^{5} x^{10}-\frac {1}{10} b^{6} x^{12}}{x^{22}}\) | \(70\) |
gosper | \(-\frac {462 b^{6} x^{12}+2310 a \,b^{5} x^{10}+4950 a^{2} b^{4} x^{8}+5775 a^{3} b^{3} x^{6}+3850 a^{4} b^{2} x^{4}+1386 a^{5} b \,x^{2}+210 a^{6}}{4620 x^{22}}\) | \(71\) |
parallelrisch | \(\frac {-462 b^{6} x^{12}-2310 a \,b^{5} x^{10}-4950 a^{2} b^{4} x^{8}-5775 a^{3} b^{3} x^{6}-3850 a^{4} b^{2} x^{4}-1386 a^{5} b \,x^{2}-210 a^{6}}{4620 x^{22}}\) | \(71\) |
orering | \(-\frac {\left (462 b^{6} x^{12}+2310 a \,b^{5} x^{10}+4950 a^{2} b^{4} x^{8}+5775 a^{3} b^{3} x^{6}+3850 a^{4} b^{2} x^{4}+1386 a^{5} b \,x^{2}+210 a^{6}\right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}{4620 x^{22} \left (b \,x^{2}+a \right )^{6}}\) | \(100\) |
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^3/x^23,x,method=_RETURNVERBOSE)
Output:
-1/22*a^6/x^22-3/10*a^5*b/x^20-5/6*a^4*b^2/x^18-5/4*a^3*b^3/x^16-15/14*a^2 *b^4/x^14-1/2*a*b^5/x^12-1/10*b^6/x^10
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx=-\frac {462 \, b^{6} x^{12} + 2310 \, a b^{5} x^{10} + 4950 \, a^{2} b^{4} x^{8} + 5775 \, a^{3} b^{3} x^{6} + 3850 \, a^{4} b^{2} x^{4} + 1386 \, a^{5} b x^{2} + 210 \, a^{6}}{4620 \, x^{22}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^23,x, algorithm="fricas")
Output:
-1/4620*(462*b^6*x^12 + 2310*a*b^5*x^10 + 4950*a^2*b^4*x^8 + 5775*a^3*b^3* x^6 + 3850*a^4*b^2*x^4 + 1386*a^5*b*x^2 + 210*a^6)/x^22
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx=\frac {- 210 a^{6} - 1386 a^{5} b x^{2} - 3850 a^{4} b^{2} x^{4} - 5775 a^{3} b^{3} x^{6} - 4950 a^{2} b^{4} x^{8} - 2310 a b^{5} x^{10} - 462 b^{6} x^{12}}{4620 x^{22}} \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**23,x)
Output:
(-210*a**6 - 1386*a**5*b*x**2 - 3850*a**4*b**2*x**4 - 5775*a**3*b**3*x**6 - 4950*a**2*b**4*x**8 - 2310*a*b**5*x**10 - 462*b**6*x**12)/(4620*x**22)
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx=-\frac {462 \, b^{6} x^{12} + 2310 \, a b^{5} x^{10} + 4950 \, a^{2} b^{4} x^{8} + 5775 \, a^{3} b^{3} x^{6} + 3850 \, a^{4} b^{2} x^{4} + 1386 \, a^{5} b x^{2} + 210 \, a^{6}}{4620 \, x^{22}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^23,x, algorithm="maxima")
Output:
-1/4620*(462*b^6*x^12 + 2310*a*b^5*x^10 + 4950*a^2*b^4*x^8 + 5775*a^3*b^3* x^6 + 3850*a^4*b^2*x^4 + 1386*a^5*b*x^2 + 210*a^6)/x^22
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx=-\frac {462 \, b^{6} x^{12} + 2310 \, a b^{5} x^{10} + 4950 \, a^{2} b^{4} x^{8} + 5775 \, a^{3} b^{3} x^{6} + 3850 \, a^{4} b^{2} x^{4} + 1386 \, a^{5} b x^{2} + 210 \, a^{6}}{4620 \, x^{22}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^23,x, algorithm="giac")
Output:
-1/4620*(462*b^6*x^12 + 2310*a*b^5*x^10 + 4950*a^2*b^4*x^8 + 5775*a^3*b^3* x^6 + 3850*a^4*b^2*x^4 + 1386*a^5*b*x^2 + 210*a^6)/x^22
Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx=-\frac {\frac {a^6}{22}+\frac {3\,a^5\,b\,x^2}{10}+\frac {5\,a^4\,b^2\,x^4}{6}+\frac {5\,a^3\,b^3\,x^6}{4}+\frac {15\,a^2\,b^4\,x^8}{14}+\frac {a\,b^5\,x^{10}}{2}+\frac {b^6\,x^{12}}{10}}{x^{22}} \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^23,x)
Output:
-(a^6/22 + (b^6*x^12)/10 + (3*a^5*b*x^2)/10 + (a*b^5*x^10)/2 + (5*a^4*b^2* x^4)/6 + (5*a^3*b^3*x^6)/4 + (15*a^2*b^4*x^8)/14)/x^22
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{23}} \, dx=\frac {-462 b^{6} x^{12}-2310 a \,b^{5} x^{10}-4950 a^{2} b^{4} x^{8}-5775 a^{3} b^{3} x^{6}-3850 a^{4} b^{2} x^{4}-1386 a^{5} b \,x^{2}-210 a^{6}}{4620 x^{22}} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^3/x^23,x)
Output:
( - 210*a**6 - 1386*a**5*b*x**2 - 3850*a**4*b**2*x**4 - 5775*a**3*b**3*x** 6 - 4950*a**2*b**4*x**8 - 2310*a*b**5*x**10 - 462*b**6*x**12)/(4620*x**22)