Integrand size = 22, antiderivative size = 78 \[ \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {1}{2 a^4 c^2 x^2}+\frac {b^2}{2 a^4 c^2 \left (a^2-b^2 x^2\right )}+\frac {2 b^2 \log (x)}{a^6 c^2}-\frac {b^2 \log \left (a^2-b^2 x^2\right )}{a^6 c^2} \] Output:
-1/2/a^4/c^2/x^2+1/2*b^2/a^4/c^2/(-b^2*x^2+a^2)+2*b^2*ln(x)/a^6/c^2-b^2*ln (-b^2*x^2+a^2)/a^6/c^2
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx=\frac {-\frac {2 a^2}{x^2}+\frac {a b^2}{a-b x}+\frac {a b^2}{a+b x}+8 b^2 \log (x)-4 b^2 \log (a-b x)-4 b^2 \log (a+b x)}{4 a^6 c^2} \] Input:
Integrate[1/(x^3*(a + b*x)^2*(a*c - b*c*x)^2),x]
Output:
((-2*a^2)/x^2 + (a*b^2)/(a - b*x) + (a*b^2)/(a + b*x) + 8*b^2*Log[x] - 4*b ^2*Log[a - b*x] - 4*b^2*Log[a + b*x])/(4*a^6*c^2)
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {82, 243, 27, 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.
\(\displaystyle \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {1}{x^3 \left (a^2 c-b^2 c x^2\right )^2}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{c^2 x^4 \left (a^2-b^2 x^2\right )^2}dx^2\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1}{x^4 \left (a^2-b^2 x^2\right )^2}dx^2}{2 c^2}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\int \left (\frac {2 b^4}{a^6 \left (a^2-b^2 x^2\right )}+\frac {b^4}{a^4 \left (a^2-b^2 x^2\right )^2}+\frac {2 b^2}{a^6 x^2}+\frac {1}{a^4 x^4}\right )dx^2}{2 c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {2 b^2 \log \left (x^2\right )}{a^6}-\frac {1}{a^4 x^2}-\frac {2 b^2 \log \left (a^2-b^2 x^2\right )}{a^6}+\frac {b^2}{a^4 \left (a^2-b^2 x^2\right )}}{2 c^2}\) |
Input:
Int[1/(x^3*(a + b*x)^2*(a*c - b*c*x)^2),x]
Output:
(-(1/(a^4*x^2)) + b^2/(a^4*(a^2 - b^2*x^2)) + (2*b^2*Log[x^2])/a^6 - (2*b^ 2*Log[a^2 - b^2*x^2])/a^6)/(2*c^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
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 = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {\frac {b^{2} x^{2}}{a^{4}}-\frac {1}{2 a^{2}}}{x^{2} \left (b x +a \right ) c^{2} \left (-b x +a \right )}+\frac {2 b^{2} \ln \left (x \right )}{a^{6} c^{2}}-\frac {b^{2} \ln \left (-b^{2} x^{2}+a^{2}\right )}{a^{6} c^{2}}\) | \(77\) |
default | \(\frac {-\frac {1}{2 a^{4} x^{2}}+\frac {2 b^{2} \ln \left (x \right )}{a^{6}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{6}}+\frac {b^{2}}{4 a^{5} \left (b x +a \right )}-\frac {b^{2} \ln \left (-b x +a \right )}{a^{6}}+\frac {b^{2}}{4 a^{5} \left (-b x +a \right )}}{c^{2}}\) | \(84\) |
norman | \(\frac {-\frac {1}{2 a^{2} c}+\frac {b^{4} x^{4}}{a^{6} c}}{x^{2} \left (b x +a \right ) c \left (-b x +a \right )}+\frac {2 b^{2} \ln \left (x \right )}{a^{6} c^{2}}-\frac {b^{2} \ln \left (-b x +a \right )}{a^{6} c^{2}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{6} c^{2}}\) | \(94\) |
parallelrisch | \(\frac {4 b^{4} \ln \left (x \right ) x^{4}-2 \ln \left (b x -a \right ) x^{4} b^{4}-2 \ln \left (b x +a \right ) x^{4} b^{4}-2 b^{4} x^{4}-4 a^{2} b^{2} \ln \left (x \right ) x^{2}+2 \ln \left (b x -a \right ) x^{2} a^{2} b^{2}+2 \ln \left (b x +a \right ) x^{2} a^{2} b^{2}+a^{4}}{2 a^{6} c^{2} x^{2} \left (b x +a \right ) \left (b x -a \right )}\) | \(129\) |
Input:
int(1/x^3/(b*x+a)^2/(-b*c*x+a*c)^2,x,method=_RETURNVERBOSE)
Output:
(1/a^4*b^2*x^2-1/2/a^2)/x^2/(b*x+a)/c^2/(-b*x+a)+2*b^2*ln(x)/a^6/c^2-b^2*l n(-b^2*x^2+a^2)/a^6/c^2
Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {2 \, a^{2} b^{2} x^{2} - a^{4} + 2 \, {\left (b^{4} x^{4} - a^{2} b^{2} x^{2}\right )} \log \left (b^{2} x^{2} - a^{2}\right ) - 4 \, {\left (b^{4} x^{4} - a^{2} b^{2} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{6} b^{2} c^{2} x^{4} - a^{8} c^{2} x^{2}\right )}} \] Input:
integrate(1/x^3/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="fricas")
Output:
-1/2*(2*a^2*b^2*x^2 - a^4 + 2*(b^4*x^4 - a^2*b^2*x^2)*log(b^2*x^2 - a^2) - 4*(b^4*x^4 - a^2*b^2*x^2)*log(x))/(a^6*b^2*c^2*x^4 - a^8*c^2*x^2)
Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx=\frac {a^{2} - 2 b^{2} x^{2}}{- 2 a^{6} c^{2} x^{2} + 2 a^{4} b^{2} c^{2} x^{4}} + \frac {2 b^{2} \log {\left (x \right )}}{a^{6} c^{2}} - \frac {b^{2} \log {\left (- \frac {a^{2}}{b^{2}} + x^{2} \right )}}{a^{6} c^{2}} \] Input:
integrate(1/x**3/(b*x+a)**2/(-b*c*x+a*c)**2,x)
Output:
(a**2 - 2*b**2*x**2)/(-2*a**6*c**2*x**2 + 2*a**4*b**2*c**2*x**4) + 2*b**2* log(x)/(a**6*c**2) - b**2*log(-a**2/b**2 + x**2)/(a**6*c**2)
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {2 \, b^{2} x^{2} - a^{2}}{2 \, {\left (a^{4} b^{2} c^{2} x^{4} - a^{6} c^{2} x^{2}\right )}} - \frac {b^{2} \log \left (b x + a\right )}{a^{6} c^{2}} - \frac {b^{2} \log \left (b x - a\right )}{a^{6} c^{2}} + \frac {2 \, b^{2} \log \left (x\right )}{a^{6} c^{2}} \] Input:
integrate(1/x^3/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="maxima")
Output:
-1/2*(2*b^2*x^2 - a^2)/(a^4*b^2*c^2*x^4 - a^6*c^2*x^2) - b^2*log(b*x + a)/ (a^6*c^2) - b^2*log(b*x - a)/(a^6*c^2) + 2*b^2*log(x)/(a^6*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (75) = 150\).
Time = 0.13 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.23 \[ \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {b^{2}}{4 \, {\left (b c x - a c\right )} a^{5} c} + \frac {2 \, b^{2} \log \left ({\left | -\frac {a c}{b c x - a c} - 1 \right |}\right )}{a^{6} c^{2}} - \frac {b^{2} \log \left ({\left | -\frac {2 \, a c}{b c x - a c} - 1 \right |}\right )}{a^{6} c^{2}} + \frac {\frac {15 \, a^{2} b^{2} c^{2}}{{\left (b c x - a c\right )}^{2}} + \frac {14 \, a b^{2} c}{b c x - a c} + 3 \, b^{2}}{8 \, a^{6} {\left (\frac {2 \, a c}{b c x - a c} + 1\right )} {\left (\frac {a c}{b c x - a c} + 1\right )}^{2} c^{2}} \] Input:
integrate(1/x^3/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="giac")
Output:
-1/4*b^2/((b*c*x - a*c)*a^5*c) + 2*b^2*log(abs(-a*c/(b*c*x - a*c) - 1))/(a ^6*c^2) - b^2*log(abs(-2*a*c/(b*c*x - a*c) - 1))/(a^6*c^2) + 1/8*(15*a^2*b ^2*c^2/(b*c*x - a*c)^2 + 14*a*b^2*c/(b*c*x - a*c) + 3*b^2)/(a^6*(2*a*c/(b* c*x - a*c) + 1)*(a*c/(b*c*x - a*c) + 1)^2*c^2)
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx=\frac {2\,b^2\,\ln \left (x\right )}{a^6\,c^2}-\frac {b^2\,\ln \left (a^2-b^2\,x^2\right )}{a^6\,c^2}-\frac {\frac {1}{2\,a^2}-\frac {b^2\,x^2}{a^4}}{a^2\,c^2\,x^2-b^2\,c^2\,x^4} \] Input:
int(1/(x^3*(a*c - b*c*x)^2*(a + b*x)^2),x)
Output:
(2*b^2*log(x))/(a^6*c^2) - (b^2*log(a^2 - b^2*x^2))/(a^6*c^2) - (1/(2*a^2) - (b^2*x^2)/a^4)/(a^2*c^2*x^2 - b^2*c^2*x^4)
Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^3 (a+b x)^2 (a c-b c x)^2} \, dx=\frac {-2 \,\mathrm {log}\left (-b x -a \right ) a^{2} b^{2} x^{2}+2 \,\mathrm {log}\left (-b x -a \right ) b^{4} x^{4}-2 \,\mathrm {log}\left (-b x +a \right ) a^{2} b^{2} x^{2}+2 \,\mathrm {log}\left (-b x +a \right ) b^{4} x^{4}+4 \,\mathrm {log}\left (x \right ) a^{2} b^{2} x^{2}-4 \,\mathrm {log}\left (x \right ) b^{4} x^{4}-a^{4}+2 b^{4} x^{4}}{2 a^{6} c^{2} x^{2} \left (-b^{2} x^{2}+a^{2}\right )} \] Input:
int(1/x^3/(b*x+a)^2/(-b*c*x+a*c)^2,x)
Output:
( - 2*log( - a - b*x)*a**2*b**2*x**2 + 2*log( - a - b*x)*b**4*x**4 - 2*log (a - b*x)*a**2*b**2*x**2 + 2*log(a - b*x)*b**4*x**4 + 4*log(x)*a**2*b**2*x **2 - 4*log(x)*b**4*x**4 - a**4 + 2*b**4*x**4)/(2*a**6*c**2*x**2*(a**2 - b **2*x**2))