Integrand size = 24, antiderivative size = 62 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx=-\frac {\left (a+b x^2\right )^7}{18 a x^{18}}+\frac {b \left (a+b x^2\right )^7}{72 a^2 x^{16}}-\frac {b^2 \left (a+b x^2\right )^7}{504 a^3 x^{14}} \] Output:
-1/18*(b*x^2+a)^7/a/x^18+1/72*b*(b*x^2+a)^7/a^2/x^16-1/504*b^2*(b*x^2+a)^7 /a^3/x^14
Time = 0.00 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx=-\frac {a^6}{18 x^{18}}-\frac {3 a^5 b}{8 x^{16}}-\frac {15 a^4 b^2}{14 x^{14}}-\frac {5 a^3 b^3}{3 x^{12}}-\frac {3 a^2 b^4}{2 x^{10}}-\frac {3 a b^5}{4 x^8}-\frac {b^6}{6 x^6} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^19,x]
Output:
-1/18*a^6/x^18 - (3*a^5*b)/(8*x^16) - (15*a^4*b^2)/(14*x^14) - (5*a^3*b^3) /(3*x^12) - (3*a^2*b^4)/(2*x^10) - (3*a*b^5)/(4*x^8) - b^6/(6*x^6)
Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1380, 27, 243, 55, 55, 48}
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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \frac {\int \frac {b^6 \left (b x^2+a\right )^6}{x^{19}}dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^6}{x^{19}}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^6}{x^{20}}dx^2\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^6}{x^{18}}dx^2}{9 a}-\frac {\left (a+b x^2\right )^7}{9 a x^{18}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 b \left (-\frac {b \int \frac {\left (b x^2+a\right )^6}{x^{16}}dx^2}{8 a}-\frac {\left (a+b x^2\right )^7}{8 a x^{16}}\right )}{9 a}-\frac {\left (a+b x^2\right )^7}{9 a x^{18}}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 b \left (\frac {b \left (a+b x^2\right )^7}{56 a^2 x^{14}}-\frac {\left (a+b x^2\right )^7}{8 a x^{16}}\right )}{9 a}-\frac {\left (a+b x^2\right )^7}{9 a x^{18}}\right )\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^19,x]
Output:
(-1/9*(a + b*x^2)^7/(a*x^18) - (2*b*(-1/8*(a + b*x^2)^7/(a*x^16) + (b*(a + b*x^2)^7)/(56*a^2*x^14)))/(9*a))/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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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 = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11
method | result | size |
default | \(-\frac {5 a^{3} b^{3}}{3 x^{12}}-\frac {b^{6}}{6 x^{6}}-\frac {3 a^{5} b}{8 x^{16}}-\frac {a^{6}}{18 x^{18}}-\frac {3 b^{5} a}{4 x^{8}}-\frac {3 a^{2} b^{4}}{2 x^{10}}-\frac {15 a^{4} b^{2}}{14 x^{14}}\) | \(69\) |
norman | \(\frac {-\frac {1}{18} a^{6}-\frac {3}{8} a^{5} b \,x^{2}-\frac {15}{14} a^{4} b^{2} x^{4}-\frac {5}{3} a^{3} b^{3} x^{6}-\frac {3}{2} a^{2} b^{4} x^{8}-\frac {3}{4} a \,b^{5} x^{10}-\frac {1}{6} b^{6} x^{12}}{x^{18}}\) | \(70\) |
risch | \(\frac {-\frac {1}{18} a^{6}-\frac {3}{8} a^{5} b \,x^{2}-\frac {15}{14} a^{4} b^{2} x^{4}-\frac {5}{3} a^{3} b^{3} x^{6}-\frac {3}{2} a^{2} b^{4} x^{8}-\frac {3}{4} a \,b^{5} x^{10}-\frac {1}{6} b^{6} x^{12}}{x^{18}}\) | \(70\) |
gosper | \(-\frac {84 b^{6} x^{12}+378 a \,b^{5} x^{10}+756 a^{2} b^{4} x^{8}+840 a^{3} b^{3} x^{6}+540 a^{4} b^{2} x^{4}+189 a^{5} b \,x^{2}+28 a^{6}}{504 x^{18}}\) | \(71\) |
parallelrisch | \(\frac {-84 b^{6} x^{12}-378 a \,b^{5} x^{10}-756 a^{2} b^{4} x^{8}-840 a^{3} b^{3} x^{6}-540 a^{4} b^{2} x^{4}-189 a^{5} b \,x^{2}-28 a^{6}}{504 x^{18}}\) | \(71\) |
orering | \(-\frac {\left (84 b^{6} x^{12}+378 a \,b^{5} x^{10}+756 a^{2} b^{4} x^{8}+840 a^{3} b^{3} x^{6}+540 a^{4} b^{2} x^{4}+189 a^{5} b \,x^{2}+28 a^{6}\right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}{504 x^{18} \left (b \,x^{2}+a \right )^{6}}\) | \(100\) |
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^3/x^19,x,method=_RETURNVERBOSE)
Output:
-5/3*a^3*b^3/x^12-1/6*b^6/x^6-3/8*a^5*b/x^16-1/18*a^6/x^18-3/4*b^5*a/x^8-3 /2*a^2*b^4/x^10-15/14*a^4*b^2/x^14
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx=-\frac {84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^19,x, algorithm="fricas")
Output:
-1/504*(84*b^6*x^12 + 378*a*b^5*x^10 + 756*a^2*b^4*x^8 + 840*a^3*b^3*x^6 + 540*a^4*b^2*x^4 + 189*a^5*b*x^2 + 28*a^6)/x^18
Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx=\frac {- 28 a^{6} - 189 a^{5} b x^{2} - 540 a^{4} b^{2} x^{4} - 840 a^{3} b^{3} x^{6} - 756 a^{2} b^{4} x^{8} - 378 a b^{5} x^{10} - 84 b^{6} x^{12}}{504 x^{18}} \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**19,x)
Output:
(-28*a**6 - 189*a**5*b*x**2 - 540*a**4*b**2*x**4 - 840*a**3*b**3*x**6 - 75 6*a**2*b**4*x**8 - 378*a*b**5*x**10 - 84*b**6*x**12)/(504*x**18)
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx=-\frac {84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^19,x, algorithm="maxima")
Output:
-1/504*(84*b^6*x^12 + 378*a*b^5*x^10 + 756*a^2*b^4*x^8 + 840*a^3*b^3*x^6 + 540*a^4*b^2*x^4 + 189*a^5*b*x^2 + 28*a^6)/x^18
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx=-\frac {84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^19,x, algorithm="giac")
Output:
-1/504*(84*b^6*x^12 + 378*a*b^5*x^10 + 756*a^2*b^4*x^8 + 840*a^3*b^3*x^6 + 540*a^4*b^2*x^4 + 189*a^5*b*x^2 + 28*a^6)/x^18
Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx=-\frac {\frac {a^6}{18}+\frac {3\,a^5\,b\,x^2}{8}+\frac {15\,a^4\,b^2\,x^4}{14}+\frac {5\,a^3\,b^3\,x^6}{3}+\frac {3\,a^2\,b^4\,x^8}{2}+\frac {3\,a\,b^5\,x^{10}}{4}+\frac {b^6\,x^{12}}{6}}{x^{18}} \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^19,x)
Output:
-(a^6/18 + (b^6*x^12)/6 + (3*a^5*b*x^2)/8 + (3*a*b^5*x^10)/4 + (15*a^4*b^2 *x^4)/14 + (5*a^3*b^3*x^6)/3 + (3*a^2*b^4*x^8)/2)/x^18
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx=\frac {-84 b^{6} x^{12}-378 a \,b^{5} x^{10}-756 a^{2} b^{4} x^{8}-840 a^{3} b^{3} x^{6}-540 a^{4} b^{2} x^{4}-189 a^{5} b \,x^{2}-28 a^{6}}{504 x^{18}} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^3/x^19,x)
Output:
( - 28*a**6 - 189*a**5*b*x**2 - 540*a**4*b**2*x**4 - 840*a**3*b**3*x**6 - 756*a**2*b**4*x**8 - 378*a*b**5*x**10 - 84*b**6*x**12)/(504*x**18)