Integrand size = 13, antiderivative size = 86 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx=-\frac {1}{4 a^3 x^4}+\frac {3 b}{2 a^4 x^2}+\frac {b^2}{4 a^3 \left (a+b x^2\right )^2}+\frac {3 b^2}{2 a^4 \left (a+b x^2\right )}+\frac {6 b^2 \log (x)}{a^5}-\frac {3 b^2 \log \left (a+b x^2\right )}{a^5} \] Output:
-1/4/a^3/x^4+3/2*b/a^4/x^2+1/4*b^2/a^3/(b*x^2+a)^2+3/2*b^2/a^4/(b*x^2+a)+6 *b^2*ln(x)/a^5-3*b^2*ln(b*x^2+a)/a^5
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {\frac {a \left (-a^3+4 a^2 b x^2+18 a b^2 x^4+12 b^3 x^6\right )}{x^4 \left (a+b x^2\right )^2}+24 b^2 \log (x)-12 b^2 \log \left (a+b x^2\right )}{4 a^5} \] Input:
Integrate[1/(x^5*(a + b*x^2)^3),x]
Output:
((a*(-a^3 + 4*a^2*b*x^2 + 18*a*b^2*x^4 + 12*b^3*x^6))/(x^4*(a + b*x^2)^2) + 24*b^2*Log[x] - 12*b^2*Log[a + b*x^2])/(4*a^5)
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {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.
\(\displaystyle \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (b x^2+a\right )^3}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {6 b^3}{a^5 \left (b x^2+a\right )}-\frac {3 b^3}{a^4 \left (b x^2+a\right )^2}-\frac {b^3}{a^3 \left (b x^2+a\right )^3}+\frac {6 b^2}{a^5 x^2}-\frac {3 b}{a^4 x^4}+\frac {1}{a^3 x^6}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {6 b^2 \log \left (x^2\right )}{a^5}-\frac {6 b^2 \log \left (a+b x^2\right )}{a^5}+\frac {3 b^2}{a^4 \left (a+b x^2\right )}+\frac {3 b}{a^4 x^2}+\frac {b^2}{2 a^3 \left (a+b x^2\right )^2}-\frac {1}{2 a^3 x^4}\right )\) |
Input:
Int[1/(x^5*(a + b*x^2)^3),x]
Output:
(-1/2*1/(a^3*x^4) + (3*b)/(a^4*x^2) + b^2/(2*a^3*(a + b*x^2)^2) + (3*b^2)/ (a^4*(a + b*x^2)) + (6*b^2*Log[x^2])/a^5 - (6*b^2*Log[a + b*x^2])/a^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]
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90
method | result | size |
norman | \(\frac {\frac {b \,x^{2}}{a^{2}}-\frac {1}{4 a}-\frac {6 b^{3} x^{6}}{a^{4}}-\frac {9 b^{4} x^{8}}{2 a^{5}}}{x^{4} \left (b \,x^{2}+a \right )^{2}}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right )}{a^{5}}\) | \(77\) |
risch | \(\frac {\frac {3 b^{3} x^{6}}{a^{4}}+\frac {9 b^{2} x^{4}}{2 a^{3}}+\frac {b \,x^{2}}{a^{2}}-\frac {1}{4 a}}{x^{4} \left (b \,x^{2}+a \right )^{2}}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right )}{a^{5}}\) | \(77\) |
default | \(-\frac {b^{3} \left (-\frac {a^{2}}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {3 a}{b \left (b \,x^{2}+a \right )}+\frac {6 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{5}}-\frac {1}{4 a^{3} x^{4}}+\frac {3 b}{2 a^{4} x^{2}}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}\) | \(83\) |
parallelrisch | \(\frac {24 b^{4} \ln \left (x \right ) x^{8}-12 b^{4} \ln \left (b \,x^{2}+a \right ) x^{8}-18 b^{4} x^{8}+48 \ln \left (x \right ) x^{6} a \,b^{3}-24 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{3}-24 a \,b^{3} x^{6}+24 \ln \left (x \right ) x^{4} a^{2} b^{2}-12 \ln \left (b \,x^{2}+a \right ) x^{4} a^{2} b^{2}+4 a^{3} b \,x^{2}-a^{4}}{4 a^{5} x^{4} \left (b \,x^{2}+a \right )^{2}}\) | \(136\) |
Input:
int(1/x^5/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
(b/a^2*x^2-1/4/a-6*b^3/a^4*x^6-9/2*b^4/a^5*x^8)/x^4/(b*x^2+a)^2+6*b^2*ln(x )/a^5-3*b^2*ln(b*x^2+a)/a^5
Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.56 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {12 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} - a^{4} - 12 \, {\left (b^{4} x^{8} + 2 \, a b^{3} x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 24 \, {\left (b^{4} x^{8} + 2 \, a b^{3} x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}} \] Input:
integrate(1/x^5/(b*x^2+a)^3,x, algorithm="fricas")
Output:
1/4*(12*a*b^3*x^6 + 18*a^2*b^2*x^4 + 4*a^3*b*x^2 - a^4 - 12*(b^4*x^8 + 2*a *b^3*x^6 + a^2*b^2*x^4)*log(b*x^2 + a) + 24*(b^4*x^8 + 2*a*b^3*x^6 + a^2*b ^2*x^4)*log(x))/(a^5*b^2*x^8 + 2*a^6*b*x^6 + a^7*x^4)
Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {- a^{3} + 4 a^{2} b x^{2} + 18 a b^{2} x^{4} + 12 b^{3} x^{6}}{4 a^{6} x^{4} + 8 a^{5} b x^{6} + 4 a^{4} b^{2} x^{8}} + \frac {6 b^{2} \log {\left (x \right )}}{a^{5}} - \frac {3 b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{5}} \] Input:
integrate(1/x**5/(b*x**2+a)**3,x)
Output:
(-a**3 + 4*a**2*b*x**2 + 18*a*b**2*x**4 + 12*b**3*x**6)/(4*a**6*x**4 + 8*a **5*b*x**6 + 4*a**4*b**2*x**8) + 6*b**2*log(x)/a**5 - 3*b**2*log(a/b + x** 2)/a**5
Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {12 \, b^{3} x^{6} + 18 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - a^{3}}{4 \, {\left (a^{4} b^{2} x^{8} + 2 \, a^{5} b x^{6} + a^{6} x^{4}\right )}} - \frac {3 \, b^{2} \log \left (b x^{2} + a\right )}{a^{5}} + \frac {3 \, b^{2} \log \left (x^{2}\right )}{a^{5}} \] Input:
integrate(1/x^5/(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/4*(12*b^3*x^6 + 18*a*b^2*x^4 + 4*a^2*b*x^2 - a^3)/(a^4*b^2*x^8 + 2*a^5*b *x^6 + a^6*x^4) - 3*b^2*log(b*x^2 + a)/a^5 + 3*b^2*log(x^2)/a^5
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {3 \, b^{2} \log \left (x^{2}\right )}{a^{5}} - \frac {3 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{5}} + \frac {12 \, b^{3} x^{6} + 18 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - a^{3}}{4 \, {\left (b x^{4} + a x^{2}\right )}^{2} a^{4}} \] Input:
integrate(1/x^5/(b*x^2+a)^3,x, algorithm="giac")
Output:
3*b^2*log(x^2)/a^5 - 3*b^2*log(abs(b*x^2 + a))/a^5 + 1/4*(12*b^3*x^6 + 18* a*b^2*x^4 + 4*a^2*b*x^2 - a^3)/((b*x^4 + a*x^2)^2*a^4)
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {\frac {b\,x^2}{a^2}-\frac {1}{4\,a}+\frac {9\,b^2\,x^4}{2\,a^3}+\frac {3\,b^3\,x^6}{a^4}}{a^2\,x^4+2\,a\,b\,x^6+b^2\,x^8}-\frac {3\,b^2\,\ln \left (b\,x^2+a\right )}{a^5}+\frac {6\,b^2\,\ln \left (x\right )}{a^5} \] Input:
int(1/(x^5*(a + b*x^2)^3),x)
Output:
((b*x^2)/a^2 - 1/(4*a) + (9*b^2*x^4)/(2*a^3) + (3*b^3*x^6)/a^4)/(a^2*x^4 + b^2*x^8 + 2*a*b*x^6) - (3*b^2*log(a + b*x^2))/a^5 + (6*b^2*log(x))/a^5
Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {1}{x^5 \left (a+b x^2\right )^3} \, dx=\frac {-12 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} x^{4}-24 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{3} x^{6}-12 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{4} x^{8}+24 \,\mathrm {log}\left (x \right ) a^{2} b^{2} x^{4}+48 \,\mathrm {log}\left (x \right ) a \,b^{3} x^{6}+24 \,\mathrm {log}\left (x \right ) b^{4} x^{8}-a^{4}+4 a^{3} b \,x^{2}+12 a^{2} b^{2} x^{4}-6 b^{4} x^{8}}{4 a^{5} x^{4} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int(1/x^5/(b*x^2+a)^3,x)
Output:
( - 12*log(a + b*x**2)*a**2*b**2*x**4 - 24*log(a + b*x**2)*a*b**3*x**6 - 1 2*log(a + b*x**2)*b**4*x**8 + 24*log(x)*a**2*b**2*x**4 + 48*log(x)*a*b**3* x**6 + 24*log(x)*b**4*x**8 - a**4 + 4*a**3*b*x**2 + 12*a**2*b**2*x**4 - 6* b**4*x**8)/(4*a**5*x**4*(a**2 + 2*a*b*x**2 + b**2*x**4))