Integrand size = 14, antiderivative size = 69 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx=-\frac {1}{2 a^3 x^2}+\frac {b}{4 a^2 \left (a-b x^2\right )^2}+\frac {b}{a^3 \left (a-b x^2\right )}+\frac {3 b \log (x)}{a^4}-\frac {3 b \log \left (a-b x^2\right )}{2 a^4} \] Output:
-1/2/a^3/x^2+1/4*b/a^2/(-b*x^2+a)^2+b/a^3/(-b*x^2+a)+3*b*ln(x)/a^4-3/2*b*l n(-b*x^2+a)/a^4
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx=\frac {\frac {a \left (-2 a^2+9 a b x^2-6 b^2 x^4\right )}{\left (a x-b x^3\right )^2}+12 b \log (x)-6 b \log \left (a-b x^2\right )}{4 a^4} \] Input:
Integrate[1/(x^3*(a - b*x^2)^3),x]
Output:
((a*(-2*a^2 + 9*a*b*x^2 - 6*b^2*x^4))/(a*x - b*x^3)^2 + 12*b*Log[x] - 6*b* Log[a - b*x^2])/(4*a^4)
Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (a-b x^2\right )^3}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (\frac {3 b^2}{a^4 \left (a-b x^2\right )}+\frac {2 b^2}{a^3 \left (a-b x^2\right )^2}+\frac {b^2}{a^2 \left (a-b x^2\right )^3}+\frac {3 b}{a^4 x^2}+\frac {1}{a^3 x^4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {3 b \log \left (x^2\right )}{a^4}-\frac {3 b \log \left (a-b x^2\right )}{a^4}+\frac {2 b}{a^3 \left (a-b x^2\right )}-\frac {1}{a^3 x^2}+\frac {b}{2 a^2 \left (a-b x^2\right )^2}\right )\) |
Input:
Int[1/(x^3*(a - b*x^2)^3),x]
Output:
(-(1/(a^3*x^2)) + b/(2*a^2*(a - b*x^2)^2) + (2*b)/(a^3*(a - b*x^2)) + (3*b *Log[x^2])/a^4 - (3*b*Log[a - b*x^2])/a^4)/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 = 65, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {-\frac {3 b^{2} x^{4}}{2 a^{3}}+\frac {9 b \,x^{2}}{4 a^{2}}-\frac {1}{2 a}}{x^{2} \left (-b \,x^{2}+a \right )^{2}}+\frac {3 b \ln \left (x \right )}{a^{4}}-\frac {3 b \ln \left (-b \,x^{2}+a \right )}{2 a^{4}}\) | \(65\) |
norman | \(\frac {-\frac {1}{2 a}+\frac {3 b^{2} x^{4}}{a^{3}}-\frac {9 b^{3} x^{6}}{4 a^{4}}}{x^{2} \left (-b \,x^{2}+a \right )^{2}}+\frac {3 b \ln \left (x \right )}{a^{4}}-\frac {3 b \ln \left (-b \,x^{2}+a \right )}{2 a^{4}}\) | \(67\) |
default | \(\frac {b^{2} \left (\frac {a^{2}}{2 b \left (-b \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (-b \,x^{2}+a \right )}{b}+\frac {2 a}{b \left (-b \,x^{2}+a \right )}\right )}{2 a^{4}}-\frac {1}{2 a^{3} x^{2}}+\frac {3 b \ln \left (x \right )}{a^{4}}\) | \(75\) |
parallelrisch | \(\frac {12 b^{3} \ln \left (x \right ) x^{6}-6 \ln \left (b \,x^{2}-a \right ) x^{6} b^{3}-9 b^{3} x^{6}-24 a \,b^{2} \ln \left (x \right ) x^{4}+12 \ln \left (b \,x^{2}-a \right ) x^{4} a \,b^{2}+12 a \,b^{2} x^{4}+12 a^{2} b \ln \left (x \right ) x^{2}-6 \ln \left (b \,x^{2}-a \right ) x^{2} a^{2} b -2 a^{3}}{4 a^{4} x^{2} \left (b \,x^{2}-a \right )^{2}}\) | \(131\) |
Input:
int(1/x^3/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
(-3/2*b^2/a^3*x^4+9/4*b/a^2*x^2-1/2/a)/x^2/(-b*x^2+a)^2+3*b*ln(x)/a^4-3/2* b*ln(-b*x^2+a)/a^4
Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.75 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx=-\frac {6 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} + 2 \, a^{3} + 6 \, {\left (b^{3} x^{6} - 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (b x^{2} - a\right ) - 12 \, {\left (b^{3} x^{6} - 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b^{2} x^{6} - 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \] Input:
integrate(1/x^3/(-b*x^2+a)^3,x, algorithm="fricas")
Output:
-1/4*(6*a*b^2*x^4 - 9*a^2*b*x^2 + 2*a^3 + 6*(b^3*x^6 - 2*a*b^2*x^4 + a^2*b *x^2)*log(b*x^2 - a) - 12*(b^3*x^6 - 2*a*b^2*x^4 + a^2*b*x^2)*log(x))/(a^4 *b^2*x^6 - 2*a^5*b*x^4 + a^6*x^2)
Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx=- \frac {2 a^{2} - 9 a b x^{2} + 6 b^{2} x^{4}}{4 a^{5} x^{2} - 8 a^{4} b x^{4} + 4 a^{3} b^{2} x^{6}} + \frac {3 b \log {\left (x \right )}}{a^{4}} - \frac {3 b \log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{4}} \] Input:
integrate(1/x**3/(-b*x**2+a)**3,x)
Output:
-(2*a**2 - 9*a*b*x**2 + 6*b**2*x**4)/(4*a**5*x**2 - 8*a**4*b*x**4 + 4*a**3 *b**2*x**6) + 3*b*log(x)/a**4 - 3*b*log(-a/b + x**2)/(2*a**4)
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx=-\frac {6 \, b^{2} x^{4} - 9 \, a b x^{2} + 2 \, a^{2}}{4 \, {\left (a^{3} b^{2} x^{6} - 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} - \frac {3 \, b \log \left (b x^{2} - a\right )}{2 \, a^{4}} + \frac {3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} \] Input:
integrate(1/x^3/(-b*x^2+a)^3,x, algorithm="maxima")
Output:
-1/4*(6*b^2*x^4 - 9*a*b*x^2 + 2*a^2)/(a^3*b^2*x^6 - 2*a^4*b*x^4 + a^5*x^2) - 3/2*b*log(b*x^2 - a)/a^4 + 3/2*b*log(x^2)/a^4
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx=\frac {3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {3 \, b \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{4}} + \frac {9 \, b^{3} x^{4} - 22 \, a b^{2} x^{2} + 14 \, a^{2} b}{4 \, {\left (b x^{2} - a\right )}^{2} a^{4}} - \frac {3 \, b x^{2} + a}{2 \, a^{4} x^{2}} \] Input:
integrate(1/x^3/(-b*x^2+a)^3,x, algorithm="giac")
Output:
3/2*b*log(x^2)/a^4 - 3/2*b*log(abs(b*x^2 - a))/a^4 + 1/4*(9*b^3*x^4 - 22*a *b^2*x^2 + 14*a^2*b)/((b*x^2 - a)^2*a^4) - 1/2*(3*b*x^2 + a)/(a^4*x^2)
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx=\frac {3\,b\,\ln \left (x\right )}{a^4}-\frac {3\,b\,\ln \left (a-b\,x^2\right )}{2\,a^4}-\frac {\frac {1}{2\,a}-\frac {9\,b\,x^2}{4\,a^2}+\frac {3\,b^2\,x^4}{2\,a^3}}{a^2\,x^2-2\,a\,b\,x^4+b^2\,x^6} \] Input:
int(1/(x^3*(a - b*x^2)^3),x)
Output:
(3*b*log(x))/a^4 - (3*b*log(a - b*x^2))/(2*a^4) - (1/(2*a) - (9*b*x^2)/(4* a^2) + (3*b^2*x^4)/(2*a^3))/(a^2*x^2 + b^2*x^6 - 2*a*b*x^4)
Time = 0.23 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.96 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^3} \, dx=\frac {-6 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b \,x^{2}+12 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{2} x^{4}-6 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) b^{3} x^{6}-6 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b \,x^{2}+12 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{2} x^{4}-6 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) b^{3} x^{6}+12 \,\mathrm {log}\left (x \right ) a^{2} b \,x^{2}-24 \,\mathrm {log}\left (x \right ) a \,b^{2} x^{4}+12 \,\mathrm {log}\left (x \right ) b^{3} x^{6}-2 a^{3}+6 a^{2} b \,x^{2}-3 b^{3} x^{6}}{4 a^{4} x^{2} \left (b^{2} x^{4}-2 a b \,x^{2}+a^{2}\right )} \] Input:
int(1/x^3/(-b*x^2+a)^3,x)
Output:
( - 6*log( - sqrt(b)*sqrt(a) - b*x)*a**2*b*x**2 + 12*log( - sqrt(b)*sqrt(a ) - b*x)*a*b**2*x**4 - 6*log( - sqrt(b)*sqrt(a) - b*x)*b**3*x**6 - 6*log(s qrt(b)*sqrt(a) - b*x)*a**2*b*x**2 + 12*log(sqrt(b)*sqrt(a) - b*x)*a*b**2*x **4 - 6*log(sqrt(b)*sqrt(a) - b*x)*b**3*x**6 + 12*log(x)*a**2*b*x**2 - 24* log(x)*a*b**2*x**4 + 12*log(x)*b**3*x**6 - 2*a**3 + 6*a**2*b*x**2 - 3*b**3 *x**6)/(4*a**4*x**2*(a**2 - 2*a*b*x**2 + b**2*x**4))