Integrand size = 23, antiderivative size = 119 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx=\frac {1}{6561 a^4 b (3 a-b x)}-\frac {2}{243 a^2 b (3 a+2 b x)^3}-\frac {2}{729 a^3 b (3 a+2 b x)^2}-\frac {2}{2187 a^4 b (3 a+2 b x)}-\frac {8 \log (3 a-b x)}{59049 a^5 b}+\frac {8 \log (3 a+2 b x)}{59049 a^5 b} \] Output:
1/6561/a^4/b/(-b*x+3*a)-2/243/a^2/b/(2*b*x+3*a)^3-2/729/a^3/b/(2*b*x+3*a)^ 2-2/2187/a^4/b/(2*b*x+3*a)-8/59049*ln(-b*x+3*a)/a^5/b+8/59049*ln(2*b*x+3*a )/a^5/b
Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx=\frac {\frac {9 a \left (-459 a^3-108 a^2 b x+72 a b^2 x^2+32 b^3 x^3\right )}{(3 a-b x) (3 a+2 b x)^3}-8 \log (3 a-b x)+8 \log (3 a+2 b x)}{59049 a^5 b} \] Input:
Integrate[(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(-2),x]
Output:
((9*a*(-459*a^3 - 108*a^2*b*x + 72*a*b^2*x^2 + 32*b^3*x^3))/((3*a - b*x)*( 3*a + 2*b*x)^3) - 8*Log[3*a - b*x] + 8*Log[3*a + 2*b*x])/(59049*a^5*b)
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2462, 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.
\(\displaystyle \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {16}{59049 a^5 (3 a+2 b x)}+\frac {8}{59049 a^5 (3 a-b x)}+\frac {4}{2187 a^4 (3 a+2 b x)^2}+\frac {1}{6561 a^4 (3 a-b x)^2}+\frac {8}{729 a^3 (3 a+2 b x)^3}+\frac {4}{81 a^2 (3 a+2 b x)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {8 \log (3 a-b x)}{59049 a^5 b}+\frac {8 \log (3 a+2 b x)}{59049 a^5 b}+\frac {1}{6561 a^4 b (3 a-b x)}-\frac {2}{2187 a^4 b (3 a+2 b x)}-\frac {2}{729 a^3 b (3 a+2 b x)^2}-\frac {2}{243 a^2 b (3 a+2 b x)^3}\) |
Input:
Int[(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(-2),x]
Output:
1/(6561*a^4*b*(3*a - b*x)) - 2/(243*a^2*b*(3*a + 2*b*x)^3) - 2/(729*a^3*b* (3*a + 2*b*x)^2) - 2/(2187*a^4*b*(3*a + 2*b*x)) - (8*Log[3*a - b*x])/(5904 9*a^5*b) + (8*Log[3*a + 2*b*x])/(59049*a^5*b)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77
method | result | size |
norman | \(\frac {-\frac {17}{243 b a}+\frac {8 b \,x^{2}}{729 a^{3}}+\frac {32 b^{2} x^{3}}{6561 a^{4}}-\frac {4 x}{243 a^{2}}}{\left (2 b x +3 a \right )^{3} \left (-b x +3 a \right )}-\frac {8 \ln \left (-b x +3 a \right )}{59049 a^{5} b}+\frac {8 \ln \left (2 b x +3 a \right )}{59049 a^{5} b}\) | \(92\) |
risch | \(\frac {-\frac {17}{243 b a}+\frac {8 b \,x^{2}}{729 a^{3}}+\frac {32 b^{2} x^{3}}{6561 a^{4}}-\frac {4 x}{243 a^{2}}}{\left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right ) \left (2 b x +3 a \right )}-\frac {8 \ln \left (-b x +3 a \right )}{59049 a^{5} b}+\frac {8 \ln \left (2 b x +3 a \right )}{59049 a^{5} b}\) | \(106\) |
default | \(\frac {1}{6561 a^{4} b \left (-b x +3 a \right )}-\frac {2}{243 a^{2} b \left (2 b x +3 a \right )^{3}}-\frac {2}{729 a^{3} b \left (2 b x +3 a \right )^{2}}-\frac {2}{2187 a^{4} b \left (2 b x +3 a \right )}-\frac {8 \ln \left (-b x +3 a \right )}{59049 a^{5} b}+\frac {8 \ln \left (2 b x +3 a \right )}{59049 a^{5} b}\) | \(108\) |
parallelrisch | \(\frac {512 \ln \left (b x +\frac {3 a}{2}\right ) x^{4} b^{7}-512 \ln \left (b x -3 a \right ) x^{4} b^{7}+768 \ln \left (b x +\frac {3 a}{2}\right ) x^{3} a \,b^{6}-768 \ln \left (b x -3 a \right ) x^{3} a \,b^{6}-3456 \ln \left (b x +\frac {3 a}{2}\right ) x^{2} a^{2} b^{5}+3456 \ln \left (b x -3 a \right ) x^{2} a^{2} b^{5}-2304 x^{3} a \,b^{6}-8640 \ln \left (b x +\frac {3 a}{2}\right ) x \,a^{3} b^{4}+8640 \ln \left (b x -3 a \right ) x \,a^{3} b^{4}-5184 x^{2} a^{2} b^{5}-5184 \ln \left (b x +\frac {3 a}{2}\right ) a^{4} b^{3}+5184 \ln \left (b x -3 a \right ) a^{4} b^{3}+7776 a^{3} b^{4} x +33048 a^{4} b^{3}}{472392 a^{5} b^{4} \left (4 b^{3} x^{3}-27 b \,a^{2} x -27 a^{3}\right ) \left (2 b x +3 a \right )}\) | \(250\) |
Input:
int(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^2,x,method=_RETURNVERBOSE)
Output:
(-17/243/b/a+8/729/a^3*b*x^2+32/6561*b^2/a^4*x^3-4/243/a^2*x)/(2*b*x+3*a)^ 3/(-b*x+3*a)-8/59049*ln(-b*x+3*a)/a^5/b+8/59049*ln(2*b*x+3*a)/a^5/b
Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.59 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx=-\frac {288 \, a b^{3} x^{3} + 648 \, a^{2} b^{2} x^{2} - 972 \, a^{3} b x - 4131 \, a^{4} - 8 \, {\left (8 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} - 54 \, a^{2} b^{2} x^{2} - 135 \, a^{3} b x - 81 \, a^{4}\right )} \log \left (2 \, b x + 3 \, a\right ) + 8 \, {\left (8 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} - 54 \, a^{2} b^{2} x^{2} - 135 \, a^{3} b x - 81 \, a^{4}\right )} \log \left (b x - 3 \, a\right )}{59049 \, {\left (8 \, a^{5} b^{5} x^{4} + 12 \, a^{6} b^{4} x^{3} - 54 \, a^{7} b^{3} x^{2} - 135 \, a^{8} b^{2} x - 81 \, a^{9} b\right )}} \] Input:
integrate(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^2,x, algorithm="fricas")
Output:
-1/59049*(288*a*b^3*x^3 + 648*a^2*b^2*x^2 - 972*a^3*b*x - 4131*a^4 - 8*(8* b^4*x^4 + 12*a*b^3*x^3 - 54*a^2*b^2*x^2 - 135*a^3*b*x - 81*a^4)*log(2*b*x + 3*a) + 8*(8*b^4*x^4 + 12*a*b^3*x^3 - 54*a^2*b^2*x^2 - 135*a^3*b*x - 81*a ^4)*log(b*x - 3*a))/(8*a^5*b^5*x^4 + 12*a^6*b^4*x^3 - 54*a^7*b^3*x^2 - 135 *a^8*b^2*x - 81*a^9*b)
Time = 0.33 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx=\frac {459 a^{3} + 108 a^{2} b x - 72 a b^{2} x^{2} - 32 b^{3} x^{3}}{- 531441 a^{8} b - 885735 a^{7} b^{2} x - 354294 a^{6} b^{3} x^{2} + 78732 a^{5} b^{4} x^{3} + 52488 a^{4} b^{5} x^{4}} + \frac {- \frac {8 \log {\left (- \frac {3 a}{b} + x \right )}}{59049} + \frac {8 \log {\left (\frac {3 a}{2 b} + x \right )}}{59049}}{a^{5} b} \] Input:
integrate(1/(-4*b**3*x**3+27*a**2*b*x+27*a**3)**2,x)
Output:
(459*a**3 + 108*a**2*b*x - 72*a*b**2*x**2 - 32*b**3*x**3)/(-531441*a**8*b - 885735*a**7*b**2*x - 354294*a**6*b**3*x**2 + 78732*a**5*b**4*x**3 + 5248 8*a**4*b**5*x**4) + (-8*log(-3*a/b + x)/59049 + 8*log(3*a/(2*b) + x)/59049 )/(a**5*b)
Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx=-\frac {32 \, b^{3} x^{3} + 72 \, a b^{2} x^{2} - 108 \, a^{2} b x - 459 \, a^{3}}{6561 \, {\left (8 \, a^{4} b^{5} x^{4} + 12 \, a^{5} b^{4} x^{3} - 54 \, a^{6} b^{3} x^{2} - 135 \, a^{7} b^{2} x - 81 \, a^{8} b\right )}} + \frac {8 \, \log \left (2 \, b x + 3 \, a\right )}{59049 \, a^{5} b} - \frac {8 \, \log \left (b x - 3 \, a\right )}{59049 \, a^{5} b} \] Input:
integrate(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^2,x, algorithm="maxima")
Output:
-1/6561*(32*b^3*x^3 + 72*a*b^2*x^2 - 108*a^2*b*x - 459*a^3)/(8*a^4*b^5*x^4 + 12*a^5*b^4*x^3 - 54*a^6*b^3*x^2 - 135*a^7*b^2*x - 81*a^8*b) + 8/59049*l og(2*b*x + 3*a)/(a^5*b) - 8/59049*log(b*x - 3*a)/(a^5*b)
Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx=\frac {8 \, \log \left ({\left | 2 \, b x + 3 \, a \right |}\right )}{59049 \, a^{5} b} - \frac {8 \, \log \left ({\left | b x - 3 \, a \right |}\right )}{59049 \, a^{5} b} - \frac {32 \, a b^{3} x^{3} + 72 \, a^{2} b^{2} x^{2} - 108 \, a^{3} b x - 459 \, a^{4}}{6561 \, {\left (2 \, b x + 3 \, a\right )}^{3} {\left (b x - 3 \, a\right )} a^{5} b} \] Input:
integrate(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^2,x, algorithm="giac")
Output:
8/59049*log(abs(2*b*x + 3*a))/(a^5*b) - 8/59049*log(abs(b*x - 3*a))/(a^5*b ) - 1/6561*(32*a*b^3*x^3 + 72*a^2*b^2*x^2 - 108*a^3*b*x - 459*a^4)/((2*b*x + 3*a)^3*(b*x - 3*a)*a^5*b)
Time = 12.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx=\frac {16\,\mathrm {atanh}\left (\frac {4\,b\,x}{9\,a}-\frac {1}{3}\right )}{59049\,a^5\,b}-\frac {\frac {4\,x}{243\,a^2}+\frac {17}{243\,a\,b}-\frac {8\,b\,x^2}{729\,a^3}-\frac {32\,b^2\,x^3}{6561\,a^4}}{81\,a^4+135\,a^3\,b\,x+54\,a^2\,b^2\,x^2-12\,a\,b^3\,x^3-8\,b^4\,x^4} \] Input:
int(1/(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^2,x)
Output:
(16*atanh((4*b*x)/(9*a) - 1/3))/(59049*a^5*b) - ((4*x)/(243*a^2) + 17/(243 *a*b) - (8*b*x^2)/(729*a^3) - (32*b^2*x^3)/(6561*a^4))/(81*a^4 - 8*b^4*x^4 - 12*a*b^3*x^3 + 54*a^2*b^2*x^2 + 135*a^3*b*x)
Time = 0.17 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.13 \[ \int \frac {1}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^2} \, dx=\frac {-648 \,\mathrm {log}\left (-b x +3 a \right ) a^{4}-1080 \,\mathrm {log}\left (-b x +3 a \right ) a^{3} b x -432 \,\mathrm {log}\left (-b x +3 a \right ) a^{2} b^{2} x^{2}+96 \,\mathrm {log}\left (-b x +3 a \right ) a \,b^{3} x^{3}+64 \,\mathrm {log}\left (-b x +3 a \right ) b^{4} x^{4}+648 \,\mathrm {log}\left (2 b x +3 a \right ) a^{4}+1080 \,\mathrm {log}\left (2 b x +3 a \right ) a^{3} b x +432 \,\mathrm {log}\left (2 b x +3 a \right ) a^{2} b^{2} x^{2}-96 \,\mathrm {log}\left (2 b x +3 a \right ) a \,b^{3} x^{3}-64 \,\mathrm {log}\left (2 b x +3 a \right ) b^{4} x^{4}-2187 a^{4}+2268 a^{3} b x +1944 a^{2} b^{2} x^{2}-192 b^{4} x^{4}}{59049 a^{5} b \left (-8 b^{4} x^{4}-12 a \,b^{3} x^{3}+54 a^{2} b^{2} x^{2}+135 a^{3} b x +81 a^{4}\right )} \] Input:
int(1/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^2,x)
Output:
( - 648*log(3*a - b*x)*a**4 - 1080*log(3*a - b*x)*a**3*b*x - 432*log(3*a - b*x)*a**2*b**2*x**2 + 96*log(3*a - b*x)*a*b**3*x**3 + 64*log(3*a - b*x)*b **4*x**4 + 648*log(3*a + 2*b*x)*a**4 + 1080*log(3*a + 2*b*x)*a**3*b*x + 43 2*log(3*a + 2*b*x)*a**2*b**2*x**2 - 96*log(3*a + 2*b*x)*a*b**3*x**3 - 64*l og(3*a + 2*b*x)*b**4*x**4 - 2187*a**4 + 2268*a**3*b*x + 1944*a**2*b**2*x** 2 - 192*b**4*x**4)/(59049*a**5*b*(81*a**4 + 135*a**3*b*x + 54*a**2*b**2*x* *2 - 12*a*b**3*x**3 - 8*b**4*x**4))