Integrand size = 14, antiderivative size = 91 \[ \int \frac {1}{x \left (a-b x^2\right )^5} \, dx=\frac {1}{8 a \left (a-b x^2\right )^4}+\frac {1}{6 a^2 \left (a-b x^2\right )^3}+\frac {1}{4 a^3 \left (a-b x^2\right )^2}+\frac {1}{2 a^4 \left (a-b x^2\right )}+\frac {\log (x)}{a^5}-\frac {\log \left (a-b x^2\right )}{2 a^5} \] Output:
1/8/a/(-b*x^2+a)^4+1/6/a^2/(-b*x^2+a)^3+1/4/a^3/(-b*x^2+a)^2+1/2/a^4/(-b*x ^2+a)+ln(x)/a^5-1/2*ln(-b*x^2+a)/a^5
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x \left (a-b x^2\right )^5} \, dx=\frac {\frac {a \left (25 a^3-52 a^2 b x^2+42 a b^2 x^4-12 b^3 x^6\right )}{\left (a-b x^2\right )^4}+24 \log (x)-12 \log \left (a-b x^2\right )}{24 a^5} \] Input:
Integrate[1/(x*(a - b*x^2)^5),x]
Output:
((a*(25*a^3 - 52*a^2*b*x^2 + 42*a*b^2*x^4 - 12*b^3*x^6))/(a - b*x^2)^4 + 2 4*Log[x] - 12*Log[a - b*x^2])/(24*a^5)
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, 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.
\(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^5} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (a-b x^2\right )^5}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (\frac {b}{a^5 \left (a-b x^2\right )}+\frac {b}{a^4 \left (a-b x^2\right )^2}+\frac {b}{a^3 \left (a-b x^2\right )^3}+\frac {b}{a^2 \left (a-b x^2\right )^4}+\frac {b}{a \left (a-b x^2\right )^5}+\frac {1}{a^5 x^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {\log \left (a-b x^2\right )}{a^5}+\frac {\log \left (x^2\right )}{a^5}+\frac {1}{a^4 \left (a-b x^2\right )}+\frac {1}{2 a^3 \left (a-b x^2\right )^2}+\frac {1}{3 a^2 \left (a-b x^2\right )^3}+\frac {1}{4 a \left (a-b x^2\right )^4}\right )\) |
Input:
Int[1/(x*(a - b*x^2)^5),x]
Output:
(1/(4*a*(a - b*x^2)^4) + 1/(3*a^2*(a - b*x^2)^3) + 1/(2*a^3*(a - b*x^2)^2) + 1/(a^4*(a - b*x^2)) + Log[x^2]/a^5 - 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 = 70, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {-\frac {b^{3} x^{6}}{2 a^{4}}+\frac {7 b^{2} x^{4}}{4 a^{3}}-\frac {13 b \,x^{2}}{6 a^{2}}+\frac {25}{24 a}}{\left (-b \,x^{2}+a \right )^{4}}+\frac {\ln \left (x \right )}{a^{5}}-\frac {\ln \left (-b \,x^{2}+a \right )}{2 a^{5}}\) | \(70\) |
norman | \(\frac {\frac {2 b \,x^{2}}{a^{2}}-\frac {9 b^{2} x^{4}}{2 a^{3}}+\frac {11 b^{3} x^{6}}{3 a^{4}}-\frac {25 b^{4} x^{8}}{24 a^{5}}}{\left (-b \,x^{2}+a \right )^{4}}+\frac {\ln \left (x \right )}{a^{5}}-\frac {\ln \left (-b \,x^{2}+a \right )}{2 a^{5}}\) | \(76\) |
default | \(\frac {b \left (\frac {a^{2}}{2 b \left (-b \,x^{2}+a \right )^{2}}-\frac {\ln \left (-b \,x^{2}+a \right )}{b}+\frac {a^{3}}{3 b \left (-b \,x^{2}+a \right )^{3}}+\frac {a}{b \left (-b \,x^{2}+a \right )}+\frac {a^{4}}{4 b \left (-b \,x^{2}+a \right )^{4}}\right )}{2 a^{5}}+\frac {\ln \left (x \right )}{a^{5}}\) | \(98\) |
parallelrisch | \(\frac {24 b^{4} \ln \left (x \right ) x^{8}-12 \ln \left (b \,x^{2}-a \right ) x^{8} b^{4}-25 b^{4} x^{8}-96 \ln \left (x \right ) x^{6} a \,b^{3}+48 \ln \left (b \,x^{2}-a \right ) x^{6} a \,b^{3}+88 a \,b^{3} x^{6}+144 \ln \left (x \right ) x^{4} a^{2} b^{2}-72 \ln \left (b \,x^{2}-a \right ) x^{4} a^{2} b^{2}-108 a^{2} b^{2} x^{4}-96 \ln \left (x \right ) x^{2} a^{3} b +48 \ln \left (b \,x^{2}-a \right ) x^{2} a^{3} b +48 a^{3} b \,x^{2}+24 \ln \left (x \right ) a^{4}-12 \ln \left (b \,x^{2}-a \right ) a^{4}}{24 a^{5} \left (b \,x^{2}-a \right )^{4}}\) | \(199\) |
Input:
int(1/x/(-b*x^2+a)^5,x,method=_RETURNVERBOSE)
Output:
(-1/2*b^3/a^4*x^6+7/4*b^2/a^3*x^4-13/6*b/a^2*x^2+25/24/a)/(-b*x^2+a)^4+ln( x)/a^5-1/2*ln(-b*x^2+a)/a^5
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (85) = 170\).
Time = 0.07 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.98 \[ \int \frac {1}{x \left (a-b x^2\right )^5} \, dx=-\frac {12 \, a b^{3} x^{6} - 42 \, a^{2} b^{2} x^{4} + 52 \, a^{3} b x^{2} - 25 \, a^{4} + 12 \, {\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} - a\right ) - 24 \, {\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (x\right )}{24 \, {\left (a^{5} b^{4} x^{8} - 4 \, a^{6} b^{3} x^{6} + 6 \, a^{7} b^{2} x^{4} - 4 \, a^{8} b x^{2} + a^{9}\right )}} \] Input:
integrate(1/x/(-b*x^2+a)^5,x, algorithm="fricas")
Output:
-1/24*(12*a*b^3*x^6 - 42*a^2*b^2*x^4 + 52*a^3*b*x^2 - 25*a^4 + 12*(b^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a^3*b*x^2 + a^4)*log(b*x^2 - a) - 24*(b ^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a^3*b*x^2 + a^4)*log(x))/(a^5*b^4 *x^8 - 4*a^6*b^3*x^6 + 6*a^7*b^2*x^4 - 4*a^8*b*x^2 + a^9)
Time = 0.35 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (a-b x^2\right )^5} \, dx=- \frac {- 25 a^{3} + 52 a^{2} b x^{2} - 42 a b^{2} x^{4} + 12 b^{3} x^{6}}{24 a^{8} - 96 a^{7} b x^{2} + 144 a^{6} b^{2} x^{4} - 96 a^{5} b^{3} x^{6} + 24 a^{4} b^{4} x^{8}} + \frac {\log {\left (x \right )}}{a^{5}} - \frac {\log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{5}} \] Input:
integrate(1/x/(-b*x**2+a)**5,x)
Output:
-(-25*a**3 + 52*a**2*b*x**2 - 42*a*b**2*x**4 + 12*b**3*x**6)/(24*a**8 - 96 *a**7*b*x**2 + 144*a**6*b**2*x**4 - 96*a**5*b**3*x**6 + 24*a**4*b**4*x**8) + log(x)/a**5 - log(-a/b + x**2)/(2*a**5)
Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x \left (a-b x^2\right )^5} \, dx=-\frac {12 \, b^{3} x^{6} - 42 \, a b^{2} x^{4} + 52 \, a^{2} b x^{2} - 25 \, a^{3}}{24 \, {\left (a^{4} b^{4} x^{8} - 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} - 4 \, a^{7} b x^{2} + a^{8}\right )}} - \frac {\log \left (b x^{2} - a\right )}{2 \, a^{5}} + \frac {\log \left (x^{2}\right )}{2 \, a^{5}} \] Input:
integrate(1/x/(-b*x^2+a)^5,x, algorithm="maxima")
Output:
-1/24*(12*b^3*x^6 - 42*a*b^2*x^4 + 52*a^2*b*x^2 - 25*a^3)/(a^4*b^4*x^8 - 4 *a^5*b^3*x^6 + 6*a^6*b^2*x^4 - 4*a^7*b*x^2 + a^8) - 1/2*log(b*x^2 - a)/a^5 + 1/2*log(x^2)/a^5
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \left (a-b x^2\right )^5} \, dx=\frac {\log \left (x^{2}\right )}{2 \, a^{5}} - \frac {\log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{5}} + \frac {25 \, b^{4} x^{8} - 112 \, a b^{3} x^{6} + 192 \, a^{2} b^{2} x^{4} - 152 \, a^{3} b x^{2} + 50 \, a^{4}}{24 \, {\left (b x^{2} - a\right )}^{4} a^{5}} \] Input:
integrate(1/x/(-b*x^2+a)^5,x, algorithm="giac")
Output:
1/2*log(x^2)/a^5 - 1/2*log(abs(b*x^2 - a))/a^5 + 1/24*(25*b^4*x^8 - 112*a* b^3*x^6 + 192*a^2*b^2*x^4 - 152*a^3*b*x^2 + 50*a^4)/((b*x^2 - a)^4*a^5)
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a-b x^2\right )^5} \, dx=\frac {\ln \left (x\right )}{a^5}+\frac {\frac {25}{24\,a}-\frac {13\,b\,x^2}{6\,a^2}+\frac {7\,b^2\,x^4}{4\,a^3}-\frac {b^3\,x^6}{2\,a^4}}{a^4-4\,a^3\,b\,x^2+6\,a^2\,b^2\,x^4-4\,a\,b^3\,x^6+b^4\,x^8}-\frac {\ln \left (a-b\,x^2\right )}{2\,a^5} \] Input:
int(1/(x*(a - b*x^2)^5),x)
Output:
log(x)/a^5 + (25/(24*a) - (13*b*x^2)/(6*a^2) + (7*b^2*x^4)/(4*a^3) - (b^3* x^6)/(2*a^4))/(a^4 + b^4*x^8 - 4*a^3*b*x^2 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4) - log(a - b*x^2)/(2*a^5)
Time = 0.22 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.65 \[ \int \frac {1}{x \left (a-b x^2\right )^5} \, dx=\frac {-12 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{4}+48 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{3} b \,x^{2}-72 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b^{2} x^{4}+48 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{3} x^{6}-12 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) b^{4} x^{8}-12 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{4}+48 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{3} b \,x^{2}-72 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b^{2} x^{4}+48 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{3} x^{6}-12 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) b^{4} x^{8}+24 \,\mathrm {log}\left (x \right ) a^{4}-96 \,\mathrm {log}\left (x \right ) a^{3} b \,x^{2}+144 \,\mathrm {log}\left (x \right ) a^{2} b^{2} x^{4}-96 \,\mathrm {log}\left (x \right ) a \,b^{3} x^{6}+24 \,\mathrm {log}\left (x \right ) b^{4} x^{8}+22 a^{4}-40 a^{3} b \,x^{2}+24 a^{2} b^{2} x^{4}-3 b^{4} x^{8}}{24 a^{5} \left (b^{4} x^{8}-4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}-4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(1/x/(-b*x^2+a)^5,x)
Output:
( - 12*log( - sqrt(b)*sqrt(a) - b*x)*a**4 + 48*log( - sqrt(b)*sqrt(a) - b* x)*a**3*b*x**2 - 72*log( - sqrt(b)*sqrt(a) - b*x)*a**2*b**2*x**4 + 48*log( - sqrt(b)*sqrt(a) - b*x)*a*b**3*x**6 - 12*log( - sqrt(b)*sqrt(a) - b*x)*b **4*x**8 - 12*log(sqrt(b)*sqrt(a) - b*x)*a**4 + 48*log(sqrt(b)*sqrt(a) - b *x)*a**3*b*x**2 - 72*log(sqrt(b)*sqrt(a) - b*x)*a**2*b**2*x**4 + 48*log(sq rt(b)*sqrt(a) - b*x)*a*b**3*x**6 - 12*log(sqrt(b)*sqrt(a) - b*x)*b**4*x**8 + 24*log(x)*a**4 - 96*log(x)*a**3*b*x**2 + 144*log(x)*a**2*b**2*x**4 - 96 *log(x)*a*b**3*x**6 + 24*log(x)*b**4*x**8 + 22*a**4 - 40*a**3*b*x**2 + 24* a**2*b**2*x**4 - 3*b**4*x**8)/(24*a**5*(a**4 - 4*a**3*b*x**2 + 6*a**2*b**2 *x**4 - 4*a*b**3*x**6 + b**4*x**8))