Integrand size = 13, antiderivative size = 74 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^{11}} \, dx=\frac {a^4}{4 b^5 \left (a+\frac {b}{x^2}\right )^2}-\frac {2 a^3}{b^5 \left (a+\frac {b}{x^2}\right )}-\frac {1}{4 b^3 x^4}+\frac {3 a}{2 b^4 x^2}-\frac {3 a^2 \log \left (a+\frac {b}{x^2}\right )}{b^5} \] Output:
1/4*a^4/b^5/(a+b/x^2)^2-2*a^3/b^5/(a+b/x^2)-1/4/b^3/x^4+3/2*a/b^4/x^2-3*a^ 2*ln(a+b/x^2)/b^5
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^{11}} \, dx=\frac {\frac {b \left (-b^3+4 a b^2 x^2+18 a^2 b x^4+12 a^3 x^6\right )}{x^4 \left (b+a x^2\right )^2}+24 a^2 \log (x)-12 a^2 \log \left (b+a x^2\right )}{4 b^5} \] Input:
Integrate[1/((a + b/x^2)^3*x^11),x]
Output:
((b*(-b^3 + 4*a*b^2*x^2 + 18*a^2*b*x^4 + 12*a^3*x^6))/(x^4*(b + a*x^2)^2) + 24*a^2*Log[x] - 12*a^2*Log[b + a*x^2])/(4*b^5)
Time = 0.39 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {795, 243, 54, 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.
\(\displaystyle \int \frac {1}{x^{11} \left (a+\frac {b}{x^2}\right )^3} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^5 \left (a x^2+b\right )^3}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (a x^2+b\right )^3}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {6 a^3}{b^5 \left (a x^2+b\right )}-\frac {3 a^3}{b^4 \left (a x^2+b\right )^2}-\frac {a^3}{b^3 \left (a x^2+b\right )^3}+\frac {6 a^2}{b^5 x^2}-\frac {3 a}{b^4 x^4}+\frac {1}{b^3 x^6}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {6 a^2 \log \left (x^2\right )}{b^5}-\frac {6 a^2 \log \left (a x^2+b\right )}{b^5}+\frac {3 a^2}{b^4 \left (a x^2+b\right )}+\frac {a^2}{2 b^3 \left (a x^2+b\right )^2}+\frac {3 a}{b^4 x^2}-\frac {1}{2 b^3 x^4}\right )\) |
Input:
Int[1/((a + b/x^2)^3*x^11),x]
Output:
(-1/2*1/(b^3*x^4) + (3*a)/(b^4*x^2) + a^2/(2*b^3*(b + a*x^2)^2) + (3*a^2)/ (b^4*(b + a*x^2)) + (6*a^2*Log[x^2])/b^5 - (6*a^2*Log[b + a*x^2])/b^5)/2
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {\frac {3 a^{3} x^{6}}{b^{4}}+\frac {9 a^{2} x^{4}}{2 b^{3}}+\frac {a \,x^{2}}{b^{2}}-\frac {1}{4 b}}{\left (a \,x^{2}+b \right )^{2} x^{4}}+\frac {6 a^{2} \ln \left (x \right )}{b^{5}}-\frac {3 a^{2} \ln \left (a \,x^{2}+b \right )}{b^{5}}\) | \(77\) |
norman | \(\frac {\frac {a \,x^{8}}{b^{2}}-\frac {x^{6}}{4 b}-\frac {6 a^{3} x^{12}}{b^{4}}-\frac {9 a^{4} x^{14}}{2 b^{5}}}{\left (a \,x^{2}+b \right )^{2} x^{10}}+\frac {6 a^{2} \ln \left (x \right )}{b^{5}}-\frac {3 a^{2} \ln \left (a \,x^{2}+b \right )}{b^{5}}\) | \(80\) |
default | \(-\frac {a^{3} \left (-\frac {3 b}{a \left (a \,x^{2}+b \right )}+\frac {6 \ln \left (a \,x^{2}+b \right )}{a}-\frac {b^{2}}{2 a \left (a \,x^{2}+b \right )^{2}}\right )}{2 b^{5}}-\frac {1}{4 b^{3} x^{4}}+\frac {6 a^{2} \ln \left (x \right )}{b^{5}}+\frac {3 a}{2 b^{4} x^{2}}\) | \(83\) |
parallelrisch | \(\frac {24 \ln \left (x \right ) x^{8} a^{4}-12 \ln \left (a \,x^{2}+b \right ) x^{8} a^{4}-18 a^{4} x^{8}+48 \ln \left (x \right ) x^{6} a^{3} b -24 \ln \left (a \,x^{2}+b \right ) x^{6} a^{3} b -24 a^{3} b \,x^{6}+24 \ln \left (x \right ) x^{4} a^{2} b^{2}-12 \ln \left (a \,x^{2}+b \right ) x^{4} a^{2} b^{2}+4 a \,b^{3} x^{2}-b^{4}}{4 b^{5} x^{4} \left (a \,x^{2}+b \right )^{2}}\) | \(136\) |
Input:
int(1/(a+b/x^2)^3/x^11,x,method=_RETURNVERBOSE)
Output:
(3*a^3/b^4*x^6+9/2*a^2/b^3*x^4+a*x^2/b^2-1/4/b)/(a*x^2+b)^2/x^4+6/b^5*a^2* ln(x)-3/b^5*a^2*ln(a*x^2+b)
Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.81 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^{11}} \, dx=\frac {12 \, a^{3} b x^{6} + 18 \, a^{2} b^{2} x^{4} + 4 \, a b^{3} x^{2} - b^{4} - 12 \, {\left (a^{4} x^{8} + 2 \, a^{3} b x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (a x^{2} + b\right ) + 24 \, {\left (a^{4} x^{8} + 2 \, a^{3} b x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{2} b^{5} x^{8} + 2 \, a b^{6} x^{6} + b^{7} x^{4}\right )}} \] Input:
integrate(1/(a+b/x^2)^3/x^11,x, algorithm="fricas")
Output:
1/4*(12*a^3*b*x^6 + 18*a^2*b^2*x^4 + 4*a*b^3*x^2 - b^4 - 12*(a^4*x^8 + 2*a ^3*b*x^6 + a^2*b^2*x^4)*log(a*x^2 + b) + 24*(a^4*x^8 + 2*a^3*b*x^6 + a^2*b ^2*x^4)*log(x))/(a^2*b^5*x^8 + 2*a*b^6*x^6 + b^7*x^4)
Time = 0.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^{11}} \, dx=\frac {6 a^{2} \log {\left (x \right )}}{b^{5}} - \frac {3 a^{2} \log {\left (x^{2} + \frac {b}{a} \right )}}{b^{5}} + \frac {12 a^{3} x^{6} + 18 a^{2} b x^{4} + 4 a b^{2} x^{2} - b^{3}}{4 a^{2} b^{4} x^{8} + 8 a b^{5} x^{6} + 4 b^{6} x^{4}} \] Input:
integrate(1/(a+b/x**2)**3/x**11,x)
Output:
6*a**2*log(x)/b**5 - 3*a**2*log(x**2 + b/a)/b**5 + (12*a**3*x**6 + 18*a**2 *b*x**4 + 4*a*b**2*x**2 - b**3)/(4*a**2*b**4*x**8 + 8*a*b**5*x**6 + 4*b**6 *x**4)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^{11}} \, dx=\frac {12 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} - b^{3}}{4 \, {\left (a^{2} b^{4} x^{8} + 2 \, a b^{5} x^{6} + b^{6} x^{4}\right )}} - \frac {3 \, a^{2} \log \left (a x^{2} + b\right )}{b^{5}} + \frac {3 \, a^{2} \log \left (x^{2}\right )}{b^{5}} \] Input:
integrate(1/(a+b/x^2)^3/x^11,x, algorithm="maxima")
Output:
1/4*(12*a^3*x^6 + 18*a^2*b*x^4 + 4*a*b^2*x^2 - b^3)/(a^2*b^4*x^8 + 2*a*b^5 *x^6 + b^6*x^4) - 3*a^2*log(a*x^2 + b)/b^5 + 3*a^2*log(x^2)/b^5
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^{11}} \, dx=\frac {3 \, a^{2} \log \left (x^{2}\right )}{b^{5}} - \frac {3 \, a^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{b^{5}} + \frac {12 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} - b^{3}}{4 \, {\left (a x^{4} + b x^{2}\right )}^{2} b^{4}} \] Input:
integrate(1/(a+b/x^2)^3/x^11,x, algorithm="giac")
Output:
3*a^2*log(x^2)/b^5 - 3*a^2*log(abs(a*x^2 + b))/b^5 + 1/4*(12*a^3*x^6 + 18* a^2*b*x^4 + 4*a*b^2*x^2 - b^3)/((a*x^4 + b*x^2)^2*b^4)
Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^{11}} \, dx=\frac {\frac {a\,x^2}{b^2}-\frac {1}{4\,b}+\frac {9\,a^2\,x^4}{2\,b^3}+\frac {3\,a^3\,x^6}{b^4}}{a^2\,x^8+2\,a\,b\,x^6+b^2\,x^4}-\frac {3\,a^2\,\ln \left (a\,x^2+b\right )}{b^5}+\frac {6\,a^2\,\ln \left (x\right )}{b^5} \] Input:
int(1/(x^11*(a + b/x^2)^3),x)
Output:
((a*x^2)/b^2 - 1/(4*b) + (9*a^2*x^4)/(2*b^3) + (3*a^3*x^6)/b^4)/(a^2*x^8 + b^2*x^4 + 2*a*b*x^6) - (3*a^2*log(b + a*x^2))/b^5 + (6*a^2*log(x))/b^5
Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^{11}} \, dx=\frac {-12 \,\mathrm {log}\left (a \,x^{2}+b \right ) a^{4} x^{8}-24 \,\mathrm {log}\left (a \,x^{2}+b \right ) a^{3} b \,x^{6}-12 \,\mathrm {log}\left (a \,x^{2}+b \right ) a^{2} b^{2} x^{4}+24 \,\mathrm {log}\left (x \right ) a^{4} x^{8}+48 \,\mathrm {log}\left (x \right ) a^{3} b \,x^{6}+24 \,\mathrm {log}\left (x \right ) a^{2} b^{2} x^{4}-6 a^{4} x^{8}+12 a^{2} b^{2} x^{4}+4 a \,b^{3} x^{2}-b^{4}}{4 b^{5} x^{4} \left (a^{2} x^{4}+2 a b \,x^{2}+b^{2}\right )} \] Input:
int(1/(a+b/x^2)^3/x^11,x)
Output:
( - 12*log(a*x**2 + b)*a**4*x**8 - 24*log(a*x**2 + b)*a**3*b*x**6 - 12*log (a*x**2 + b)*a**2*b**2*x**4 + 24*log(x)*a**4*x**8 + 48*log(x)*a**3*b*x**6 + 24*log(x)*a**2*b**2*x**4 - 6*a**4*x**8 + 12*a**2*b**2*x**4 + 4*a*b**3*x* *2 - b**4)/(4*b**5*x**4*(a**2*x**4 + 2*a*b*x**2 + b**2))