Integrand size = 14, antiderivative size = 106 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx=-\frac {1}{2 a^5 x^2}+\frac {b}{8 a^2 \left (a-b x^2\right )^4}+\frac {b}{3 a^3 \left (a-b x^2\right )^3}+\frac {3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac {2 b}{a^5 \left (a-b x^2\right )}+\frac {5 b \log (x)}{a^6}-\frac {5 b \log \left (a-b x^2\right )}{2 a^6} \] Output:
-1/2/a^5/x^2+1/8*b/a^2/(-b*x^2+a)^4+1/3*b/a^3/(-b*x^2+a)^3+3/4*b/a^4/(-b*x ^2+a)^2+2*b/a^5/(-b*x^2+a)+5*b*ln(x)/a^6-5/2*b*ln(-b*x^2+a)/a^6
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx=\frac {\frac {a \left (-12 a^4+125 a^3 b x^2-260 a^2 b^2 x^4+210 a b^3 x^6-60 b^4 x^8\right )}{x^2 \left (a-b x^2\right )^4}+120 b \log (x)-60 b \log \left (a-b x^2\right )}{24 a^6} \] Input:
Integrate[1/(x^3*(a - b*x^2)^5),x]
Output:
((a*(-12*a^4 + 125*a^3*b*x^2 - 260*a^2*b^2*x^4 + 210*a*b^3*x^6 - 60*b^4*x^ 8))/(x^2*(a - b*x^2)^4) + 120*b*Log[x] - 60*b*Log[a - b*x^2])/(24*a^6)
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (a-b x^2\right )^5}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (\frac {5 b^2}{a^6 \left (a-b x^2\right )}+\frac {4 b^2}{a^5 \left (a-b x^2\right )^2}+\frac {3 b^2}{a^4 \left (a-b x^2\right )^3}+\frac {2 b^2}{a^3 \left (a-b x^2\right )^4}+\frac {b^2}{a^2 \left (a-b x^2\right )^5}+\frac {5 b}{a^6 x^2}+\frac {1}{a^5 x^4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {5 b \log \left (x^2\right )}{a^6}-\frac {5 b \log \left (a-b x^2\right )}{a^6}+\frac {4 b}{a^5 \left (a-b x^2\right )}-\frac {1}{a^5 x^2}+\frac {3 b}{2 a^4 \left (a-b x^2\right )^2}+\frac {2 b}{3 a^3 \left (a-b x^2\right )^3}+\frac {b}{4 a^2 \left (a-b x^2\right )^4}\right )\) |
Input:
Int[1/(x^3*(a - b*x^2)^5),x]
Output:
(-(1/(a^5*x^2)) + b/(4*a^2*(a - b*x^2)^4) + (2*b)/(3*a^3*(a - b*x^2)^3) + (3*b)/(2*a^4*(a - b*x^2)^2) + (4*b)/(a^5*(a - b*x^2)) + (5*b*Log[x^2])/a^6 - (5*b*Log[a - b*x^2])/a^6)/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 = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {-\frac {5 b^{4} x^{8}}{2 a^{5}}+\frac {35 b^{3} x^{6}}{4 a^{4}}-\frac {65 b^{2} x^{4}}{6 a^{3}}+\frac {125 b \,x^{2}}{24 a^{2}}-\frac {1}{2 a}}{x^{2} \left (-b \,x^{2}+a \right )^{4}}+\frac {5 b \ln \left (x \right )}{a^{6}}-\frac {5 b \ln \left (-b \,x^{2}+a \right )}{2 a^{6}}\) | \(87\) |
norman | \(\frac {-\frac {1}{2 a}+\frac {10 b^{2} x^{4}}{a^{3}}-\frac {45 b^{3} x^{6}}{2 a^{4}}+\frac {55 b^{4} x^{8}}{3 a^{5}}-\frac {125 b^{5} x^{10}}{24 a^{6}}}{x^{2} \left (-b \,x^{2}+a \right )^{4}}+\frac {5 b \ln \left (x \right )}{a^{6}}-\frac {5 b \ln \left (-b \,x^{2}+a \right )}{2 a^{6}}\) | \(89\) |
default | \(\frac {b^{2} \left (\frac {3 a^{2}}{2 b \left (-b \,x^{2}+a \right )^{2}}-\frac {5 \ln \left (-b \,x^{2}+a \right )}{b}+\frac {2 a^{3}}{3 b \left (-b \,x^{2}+a \right )^{3}}+\frac {4 a}{b \left (-b \,x^{2}+a \right )}+\frac {a^{4}}{4 b \left (-b \,x^{2}+a \right )^{4}}\right )}{2 a^{6}}-\frac {1}{2 a^{5} x^{2}}+\frac {5 b \ln \left (x \right )}{a^{6}}\) | \(111\) |
parallelrisch | \(\frac {120 \ln \left (x \right ) x^{10} b^{5}-60 \ln \left (b \,x^{2}-a \right ) x^{10} b^{5}-125 b^{5} x^{10}-480 a \,b^{4} \ln \left (x \right ) x^{8}+240 \ln \left (b \,x^{2}-a \right ) x^{8} a \,b^{4}+440 a \,b^{4} x^{8}+720 \ln \left (x \right ) x^{6} a^{2} b^{3}-360 \ln \left (b \,x^{2}-a \right ) x^{6} a^{2} b^{3}-540 a^{2} b^{3} x^{6}-480 \ln \left (x \right ) x^{4} a^{3} b^{2}+240 \ln \left (b \,x^{2}-a \right ) x^{4} a^{3} b^{2}+240 a^{3} b^{2} x^{4}+120 a^{4} b \ln \left (x \right ) x^{2}-60 \ln \left (b \,x^{2}-a \right ) x^{2} a^{4} b -12 a^{5}}{24 a^{6} x^{2} \left (b \,x^{2}-a \right )^{4}}\) | \(221\) |
Input:
int(1/x^3/(-b*x^2+a)^5,x,method=_RETURNVERBOSE)
Output:
(-5/2*b^4/a^5*x^8+35/4*b^3/a^4*x^6-65/6*b^2/a^3*x^4+125/24*b/a^2*x^2-1/2/a )/x^2/(-b*x^2+a)^4+5*b*ln(x)/a^6-5/2*b*ln(-b*x^2+a)/a^6
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (100) = 200\).
Time = 0.06 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.97 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx=-\frac {60 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} - 125 \, a^{4} b x^{2} + 12 \, a^{5} + 60 \, {\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (b x^{2} - a\right ) - 120 \, {\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (x\right )}{24 \, {\left (a^{6} b^{4} x^{10} - 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} - 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}} \] Input:
integrate(1/x^3/(-b*x^2+a)^5,x, algorithm="fricas")
Output:
-1/24*(60*a*b^4*x^8 - 210*a^2*b^3*x^6 + 260*a^3*b^2*x^4 - 125*a^4*b*x^2 + 12*a^5 + 60*(b^5*x^10 - 4*a*b^4*x^8 + 6*a^2*b^3*x^6 - 4*a^3*b^2*x^4 + a^4* b*x^2)*log(b*x^2 - a) - 120*(b^5*x^10 - 4*a*b^4*x^8 + 6*a^2*b^3*x^6 - 4*a^ 3*b^2*x^4 + a^4*b*x^2)*log(x))/(a^6*b^4*x^10 - 4*a^7*b^3*x^8 + 6*a^8*b^2*x ^6 - 4*a^9*b*x^4 + a^10*x^2)
Time = 0.44 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx=- \frac {12 a^{4} - 125 a^{3} b x^{2} + 260 a^{2} b^{2} x^{4} - 210 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{9} x^{2} - 96 a^{8} b x^{4} + 144 a^{7} b^{2} x^{6} - 96 a^{6} b^{3} x^{8} + 24 a^{5} b^{4} x^{10}} + \frac {5 b \log {\left (x \right )}}{a^{6}} - \frac {5 b \log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{6}} \] Input:
integrate(1/x**3/(-b*x**2+a)**5,x)
Output:
-(12*a**4 - 125*a**3*b*x**2 + 260*a**2*b**2*x**4 - 210*a*b**3*x**6 + 60*b* *4*x**8)/(24*a**9*x**2 - 96*a**8*b*x**4 + 144*a**7*b**2*x**6 - 96*a**6*b** 3*x**8 + 24*a**5*b**4*x**10) + 5*b*log(x)/a**6 - 5*b*log(-a/b + x**2)/(2*a **6)
Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx=-\frac {60 \, b^{4} x^{8} - 210 \, a b^{3} x^{6} + 260 \, a^{2} b^{2} x^{4} - 125 \, a^{3} b x^{2} + 12 \, a^{4}}{24 \, {\left (a^{5} b^{4} x^{10} - 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} - 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} - \frac {5 \, b \log \left (b x^{2} - a\right )}{2 \, a^{6}} + \frac {5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} \] Input:
integrate(1/x^3/(-b*x^2+a)^5,x, algorithm="maxima")
Output:
-1/24*(60*b^4*x^8 - 210*a*b^3*x^6 + 260*a^2*b^2*x^4 - 125*a^3*b*x^2 + 12*a ^4)/(a^5*b^4*x^10 - 4*a^6*b^3*x^8 + 6*a^7*b^2*x^6 - 4*a^8*b*x^4 + a^9*x^2) - 5/2*b*log(b*x^2 - a)/a^6 + 5/2*b*log(x^2)/a^6
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx=\frac {5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} - \frac {5 \, b \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{6}} - \frac {5 \, b x^{2} + a}{2 \, a^{6} x^{2}} + \frac {125 \, b^{5} x^{8} - 548 \, a b^{4} x^{6} + 912 \, a^{2} b^{3} x^{4} - 688 \, a^{3} b^{2} x^{2} + 202 \, a^{4} b}{24 \, {\left (b x^{2} - a\right )}^{4} a^{6}} \] Input:
integrate(1/x^3/(-b*x^2+a)^5,x, algorithm="giac")
Output:
5/2*b*log(x^2)/a^6 - 5/2*b*log(abs(b*x^2 - a))/a^6 - 1/2*(5*b*x^2 + a)/(a^ 6*x^2) + 1/24*(125*b^5*x^8 - 548*a*b^4*x^6 + 912*a^2*b^3*x^4 - 688*a^3*b^2 *x^2 + 202*a^4*b)/((b*x^2 - a)^4*a^6)
Time = 0.30 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx=\frac {5\,b\,\ln \left (x\right )}{a^6}-\frac {5\,b\,\ln \left (a-b\,x^2\right )}{2\,a^6}-\frac {\frac {1}{2\,a}-\frac {125\,b\,x^2}{24\,a^2}+\frac {65\,b^2\,x^4}{6\,a^3}-\frac {35\,b^3\,x^6}{4\,a^4}+\frac {5\,b^4\,x^8}{2\,a^5}}{a^4\,x^2-4\,a^3\,b\,x^4+6\,a^2\,b^2\,x^6-4\,a\,b^3\,x^8+b^4\,x^{10}} \] Input:
int(1/(x^3*(a - b*x^2)^5),x)
Output:
(5*b*log(x))/a^6 - (5*b*log(a - b*x^2))/(2*a^6) - (1/(2*a) - (125*b*x^2)/( 24*a^2) + (65*b^2*x^4)/(6*a^3) - (35*b^3*x^6)/(4*a^4) + (5*b^4*x^8)/(2*a^5 ))/(a^4*x^2 + b^4*x^10 - 4*a^3*b*x^4 - 4*a*b^3*x^8 + 6*a^2*b^2*x^6)
Time = 0.22 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.43 \[ \int \frac {1}{x^3 \left (a-b x^2\right )^5} \, dx=\frac {-60 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{4} b \,x^{2}+240 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{3} b^{2} x^{4}-360 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b^{3} x^{6}+240 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{4} x^{8}-60 \,\mathrm {log}\left (-\sqrt {b}\, \sqrt {a}-b x \right ) b^{5} x^{10}-60 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{4} b \,x^{2}+240 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{3} b^{2} x^{4}-360 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a^{2} b^{3} x^{6}+240 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) a \,b^{4} x^{8}-60 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}-b x \right ) b^{5} x^{10}+120 \,\mathrm {log}\left (x \right ) a^{4} b \,x^{2}-480 \,\mathrm {log}\left (x \right ) a^{3} b^{2} x^{4}+720 \,\mathrm {log}\left (x \right ) a^{2} b^{3} x^{6}-480 \,\mathrm {log}\left (x \right ) a \,b^{4} x^{8}+120 \,\mathrm {log}\left (x \right ) b^{5} x^{10}-12 a^{5}+110 a^{4} b \,x^{2}-200 a^{3} b^{2} x^{4}+120 a^{2} b^{3} x^{6}-15 b^{5} x^{10}}{24 a^{6} x^{2} \left (b^{4} x^{8}-4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}-4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(1/x^3/(-b*x^2+a)^5,x)
Output:
( - 60*log( - sqrt(b)*sqrt(a) - b*x)*a**4*b*x**2 + 240*log( - sqrt(b)*sqrt (a) - b*x)*a**3*b**2*x**4 - 360*log( - sqrt(b)*sqrt(a) - b*x)*a**2*b**3*x* *6 + 240*log( - sqrt(b)*sqrt(a) - b*x)*a*b**4*x**8 - 60*log( - sqrt(b)*sqr t(a) - b*x)*b**5*x**10 - 60*log(sqrt(b)*sqrt(a) - b*x)*a**4*b*x**2 + 240*l og(sqrt(b)*sqrt(a) - b*x)*a**3*b**2*x**4 - 360*log(sqrt(b)*sqrt(a) - b*x)* a**2*b**3*x**6 + 240*log(sqrt(b)*sqrt(a) - b*x)*a*b**4*x**8 - 60*log(sqrt( b)*sqrt(a) - b*x)*b**5*x**10 + 120*log(x)*a**4*b*x**2 - 480*log(x)*a**3*b* *2*x**4 + 720*log(x)*a**2*b**3*x**6 - 480*log(x)*a*b**4*x**8 + 120*log(x)* b**5*x**10 - 12*a**5 + 110*a**4*b*x**2 - 200*a**3*b**2*x**4 + 120*a**2*b** 3*x**6 - 15*b**5*x**10)/(24*a**6*x**2*(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))