Integrand size = 13, antiderivative size = 95 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx=-\frac {1}{6 a^3 x^6}+\frac {3 b}{4 a^4 x^4}-\frac {3 b^2}{a^5 x^2}-\frac {b^3}{4 a^4 \left (a+b x^2\right )^2}-\frac {2 b^3}{a^5 \left (a+b x^2\right )}-\frac {10 b^3 \log (x)}{a^6}+\frac {5 b^3 \log \left (a+b x^2\right )}{a^6} \] Output:
-1/6/a^3/x^6+3/4*b/a^4/x^4-3*b^2/a^5/x^2-1/4*b^3/a^4/(b*x^2+a)^2-2*b^3/a^5 /(b*x^2+a)-10*b^3*ln(x)/a^6+5*b^3*ln(b*x^2+a)/a^6
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx=-\frac {\frac {a \left (2 a^4-5 a^3 b x^2+20 a^2 b^2 x^4+90 a b^3 x^6+60 b^4 x^8\right )}{x^6 \left (a+b x^2\right )^2}+120 b^3 \log (x)-60 b^3 \log \left (a+b x^2\right )}{12 a^6} \] Input:
Integrate[1/(x^7*(a + b*x^2)^3),x]
Output:
-1/12*((a*(2*a^4 - 5*a^3*b*x^2 + 20*a^2*b^2*x^4 + 90*a*b^3*x^6 + 60*b^4*x^ 8))/(x^6*(a + b*x^2)^2) + 120*b^3*Log[x] - 60*b^3*Log[a + b*x^2])/a^6
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06, 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.
\(\displaystyle \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^8 \left (b x^2+a\right )^3}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (\frac {10 b^4}{a^6 \left (b x^2+a\right )}+\frac {4 b^4}{a^5 \left (b x^2+a\right )^2}+\frac {b^4}{a^4 \left (b x^2+a\right )^3}-\frac {10 b^3}{a^6 x^2}+\frac {6 b^2}{a^5 x^4}-\frac {3 b}{a^4 x^6}+\frac {1}{a^3 x^8}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {10 b^3 \log \left (x^2\right )}{a^6}+\frac {10 b^3 \log \left (a+b x^2\right )}{a^6}-\frac {4 b^3}{a^5 \left (a+b x^2\right )}-\frac {6 b^2}{a^5 x^2}-\frac {b^3}{2 a^4 \left (a+b x^2\right )^2}+\frac {3 b}{2 a^4 x^4}-\frac {1}{3 a^3 x^6}\right )\) |
Input:
Int[1/(x^7*(a + b*x^2)^3),x]
Output:
(-1/3*1/(a^3*x^6) + (3*b)/(2*a^4*x^4) - (6*b^2)/(a^5*x^2) - b^3/(2*a^4*(a + b*x^2)^2) - (4*b^3)/(a^5*(a + b*x^2)) - (10*b^3*Log[x^2])/a^6 + (10*b^3* 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 = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94
method | result | size |
norman | \(\frac {-\frac {1}{6 a}+\frac {5 b \,x^{2}}{12 a^{2}}-\frac {5 b^{2} x^{4}}{3 a^{3}}+\frac {10 b^{4} x^{8}}{a^{5}}+\frac {15 b^{5} x^{10}}{2 a^{6}}}{x^{6} \left (b \,x^{2}+a \right )^{2}}-\frac {10 b^{3} \ln \left (x \right )}{a^{6}}+\frac {5 b^{3} \ln \left (b \,x^{2}+a \right )}{a^{6}}\) | \(89\) |
risch | \(\frac {-\frac {5 b^{4} x^{8}}{a^{5}}-\frac {15 b^{3} x^{6}}{2 a^{4}}-\frac {5 b^{2} x^{4}}{3 a^{3}}+\frac {5 b \,x^{2}}{12 a^{2}}-\frac {1}{6 a}}{x^{6} \left (b \,x^{2}+a \right )^{2}}-\frac {10 b^{3} \ln \left (x \right )}{a^{6}}+\frac {5 b^{3} \ln \left (-b \,x^{2}-a \right )}{a^{6}}\) | \(92\) |
default | \(\frac {b^{4} \left (-\frac {a^{2}}{2 b \left (b \,x^{2}+a \right )^{2}}-\frac {4 a}{b \left (b \,x^{2}+a \right )}+\frac {10 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{6}}-\frac {1}{6 a^{3} x^{6}}+\frac {3 b}{4 a^{4} x^{4}}-\frac {3 b^{2}}{a^{5} x^{2}}-\frac {10 b^{3} \ln \left (x \right )}{a^{6}}\) | \(94\) |
parallelrisch | \(-\frac {120 \ln \left (x \right ) x^{10} b^{5}-60 \ln \left (b \,x^{2}+a \right ) x^{10} b^{5}-90 b^{5} x^{10}+240 a \,b^{4} \ln \left (x \right ) x^{8}-120 \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{4}-120 a \,b^{4} x^{8}+120 \ln \left (x \right ) x^{6} a^{2} b^{3}-60 \ln \left (b \,x^{2}+a \right ) x^{6} a^{2} b^{3}+20 a^{3} b^{2} x^{4}-5 a^{4} b \,x^{2}+2 a^{5}}{12 a^{6} x^{6} \left (b \,x^{2}+a \right )^{2}}\) | \(147\) |
Input:
int(1/x^7/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
(-1/6/a+5/12*b/a^2*x^2-5/3*b^2/a^3*x^4+10*b^4/a^5*x^8+15/2*b^5/a^6*x^10)/x ^6/(b*x^2+a)^2-10*b^3*ln(x)/a^6+5*b^3*ln(b*x^2+a)/a^6
Time = 0.06 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.53 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx=-\frac {60 \, a b^{4} x^{8} + 90 \, a^{2} b^{3} x^{6} + 20 \, a^{3} b^{2} x^{4} - 5 \, a^{4} b x^{2} + 2 \, a^{5} - 60 \, {\left (b^{5} x^{10} + 2 \, a b^{4} x^{8} + a^{2} b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{5} x^{10} + 2 \, a b^{4} x^{8} + a^{2} b^{3} x^{6}\right )} \log \left (x\right )}{12 \, {\left (a^{6} b^{2} x^{10} + 2 \, a^{7} b x^{8} + a^{8} x^{6}\right )}} \] Input:
integrate(1/x^7/(b*x^2+a)^3,x, algorithm="fricas")
Output:
-1/12*(60*a*b^4*x^8 + 90*a^2*b^3*x^6 + 20*a^3*b^2*x^4 - 5*a^4*b*x^2 + 2*a^ 5 - 60*(b^5*x^10 + 2*a*b^4*x^8 + a^2*b^3*x^6)*log(b*x^2 + a) + 120*(b^5*x^ 10 + 2*a*b^4*x^8 + a^2*b^3*x^6)*log(x))/(a^6*b^2*x^10 + 2*a^7*b*x^8 + a^8* x^6)
Time = 0.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {- 2 a^{4} + 5 a^{3} b x^{2} - 20 a^{2} b^{2} x^{4} - 90 a b^{3} x^{6} - 60 b^{4} x^{8}}{12 a^{7} x^{6} + 24 a^{6} b x^{8} + 12 a^{5} b^{2} x^{10}} - \frac {10 b^{3} \log {\left (x \right )}}{a^{6}} + \frac {5 b^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{6}} \] Input:
integrate(1/x**7/(b*x**2+a)**3,x)
Output:
(-2*a**4 + 5*a**3*b*x**2 - 20*a**2*b**2*x**4 - 90*a*b**3*x**6 - 60*b**4*x* *8)/(12*a**7*x**6 + 24*a**6*b*x**8 + 12*a**5*b**2*x**10) - 10*b**3*log(x)/ a**6 + 5*b**3*log(a/b + x**2)/a**6
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx=-\frac {60 \, b^{4} x^{8} + 90 \, a b^{3} x^{6} + 20 \, a^{2} b^{2} x^{4} - 5 \, a^{3} b x^{2} + 2 \, a^{4}}{12 \, {\left (a^{5} b^{2} x^{10} + 2 \, a^{6} b x^{8} + a^{7} x^{6}\right )}} + \frac {5 \, b^{3} \log \left (b x^{2} + a\right )}{a^{6}} - \frac {5 \, b^{3} \log \left (x^{2}\right )}{a^{6}} \] Input:
integrate(1/x^7/(b*x^2+a)^3,x, algorithm="maxima")
Output:
-1/12*(60*b^4*x^8 + 90*a*b^3*x^6 + 20*a^2*b^2*x^4 - 5*a^3*b*x^2 + 2*a^4)/( a^5*b^2*x^10 + 2*a^6*b*x^8 + a^7*x^6) + 5*b^3*log(b*x^2 + a)/a^6 - 5*b^3*l og(x^2)/a^6
Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx=-\frac {5 \, b^{3} \log \left (x^{2}\right )}{a^{6}} + \frac {5 \, b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{6}} - \frac {30 \, b^{5} x^{4} + 68 \, a b^{4} x^{2} + 39 \, a^{2} b^{3}}{4 \, {\left (b x^{2} + a\right )}^{2} a^{6}} + \frac {110 \, b^{3} x^{6} - 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{6} x^{6}} \] Input:
integrate(1/x^7/(b*x^2+a)^3,x, algorithm="giac")
Output:
-5*b^3*log(x^2)/a^6 + 5*b^3*log(abs(b*x^2 + a))/a^6 - 1/4*(30*b^5*x^4 + 68 *a*b^4*x^2 + 39*a^2*b^3)/((b*x^2 + a)^2*a^6) + 1/12*(110*b^3*x^6 - 36*a*b^ 2*x^4 + 9*a^2*b*x^2 - 2*a^3)/(a^6*x^6)
Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {5\,b^3\,\ln \left (b\,x^2+a\right )}{a^6}-\frac {\frac {1}{6\,a}-\frac {5\,b\,x^2}{12\,a^2}+\frac {5\,b^2\,x^4}{3\,a^3}+\frac {15\,b^3\,x^6}{2\,a^4}+\frac {5\,b^4\,x^8}{a^5}}{a^2\,x^6+2\,a\,b\,x^8+b^2\,x^{10}}-\frac {10\,b^3\,\ln \left (x\right )}{a^6} \] Input:
int(1/(x^7*(a + b*x^2)^3),x)
Output:
(5*b^3*log(a + b*x^2))/a^6 - (1/(6*a) - (5*b*x^2)/(12*a^2) + (5*b^2*x^4)/( 3*a^3) + (15*b^3*x^6)/(2*a^4) + (5*b^4*x^8)/a^5)/(a^2*x^6 + b^2*x^10 + 2*a *b*x^8) - (10*b^3*log(x))/a^6
Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.67 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^3} \, dx=\frac {60 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{3} x^{6}+120 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{4} x^{8}+60 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{5} x^{10}-120 \,\mathrm {log}\left (x \right ) a^{2} b^{3} x^{6}-240 \,\mathrm {log}\left (x \right ) a \,b^{4} x^{8}-120 \,\mathrm {log}\left (x \right ) b^{5} x^{10}-2 a^{5}+5 a^{4} b \,x^{2}-20 a^{3} b^{2} x^{4}-60 a^{2} b^{3} x^{6}+30 b^{5} x^{10}}{12 a^{6} x^{6} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int(1/x^7/(b*x^2+a)^3,x)
Output:
(60*log(a + b*x**2)*a**2*b**3*x**6 + 120*log(a + b*x**2)*a*b**4*x**8 + 60* log(a + b*x**2)*b**5*x**10 - 120*log(x)*a**2*b**3*x**6 - 240*log(x)*a*b**4 *x**8 - 120*log(x)*b**5*x**10 - 2*a**5 + 5*a**4*b*x**2 - 20*a**3*b**2*x**4 - 60*a**2*b**3*x**6 + 30*b**5*x**10)/(12*a**6*x**6*(a**2 + 2*a*b*x**2 + b **2*x**4))