Integrand size = 46, antiderivative size = 645 \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=-\frac {1}{27 a^3 x}+\frac {\left (2 (-1)^{2/3} b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {\left (2 b^2-12 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{243 \sqrt {3} a^{23/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {(-1)^{2/3} \left (2 b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 (-1)^{2/3} a^{2/3} c^{4/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac {\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}} \] Output:
-1/27/a^3/x+1/243*(2*(-1)^(2/3)*b^2+12*(-1)^(1/3)*a^(1/3)*b*c^(2/3)+9*a^(2 /3)*c^(4/3))*arctan(1/3*(3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)*3^(1/2)/a^(1/ 2)/(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))^(1/2))*3^(1/2)/(1+(-1)^(1/3))^2/a^(2 3/6)/(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))^(1/2)/c^(2/3)+1/729*(2*b^2-12*a^(1 /3)*b*c^(2/3)+9*a^(2/3)*c^(4/3))*arctan(1/3*(3*a^(2/3)*c^(1/3)+2*b*x)*3^(1 /2)/a^(1/2)/(4*b-3*a^(1/3)*c^(2/3))^(1/2))*3^(1/2)/a^(23/6)/(4*b-3*a^(1/3) *c^(2/3))^(1/2)/c^(2/3)+1/243*(-1)^(2/3)*(2*b^2+12*(-1)^(1/3)*a^(1/3)*b*c^ (2/3)+9*(-1)^(2/3)*a^(2/3)*c^(4/3))*arctan(1/3*(3*(-1)^(2/3)*a^(2/3)*c^(1/ 3)+2*b*x)*3^(1/2)/a^(1/2)/(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2))*3^(1/2 )/(1-(-1)^(1/3))/(1+(-1)^(1/3))^2/a^(23/6)/(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/ 3))^(1/2)/c^(2/3)-1/486*(2*b-3*a^(1/3)*c^(2/3))*ln(3*a+3*a^(2/3)*c^(1/3)*x +b*x^2)/a^(11/3)/c^(1/3)+1/162*(2*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))*ln(3*a-3 *(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(1+(-1)^(1/3))^2/a^(11/3)/c^(1/3)+1/4 86*(-1)^(1/3)*(2*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))*ln(3*a+3*(-1)^(2/3)*a^(2/ 3)*c^(1/3)*x+b*x^2)/a^(11/3)/c^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=-\frac {3+x \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {27 a^2 b \log (x-\text {$\#$1})+27 a^2 c \log (x-\text {$\#$1}) \text {$\#$1}+9 a b^2 \log (x-\text {$\#$1}) \text {$\#$1}^2+b^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{18 a^2 b \text {$\#$1}+27 a^2 c \text {$\#$1}^2+12 a b^2 \text {$\#$1}^3+2 b^3 \text {$\#$1}^5}\&\right ]}{81 a^3 x} \] Input:
Integrate[1/(x^2*(27*a^3 + 27*a^2*b*x^2 + 27*a^2*c*x^3 + 9*a*b^2*x^4 + b^3 *x^6)),x]
Output:
-1/81*(3 + x*RootSum[27*a^3 + 27*a^2*b*#1^2 + 27*a^2*c*#1^3 + 9*a*b^2*#1^4 + b^3*#1^6 & , (27*a^2*b*Log[x - #1] + 27*a^2*c*Log[x - #1]*#1 + 9*a*b^2* Log[x - #1]*#1^2 + b^3*Log[x - #1]*#1^4)/(18*a^2*b*#1 + 27*a^2*c*#1^2 + 12 *a*b^2*#1^3 + 2*b^3*#1^5) & ])/(a^3*x)
Time = 2.41 (sec) , antiderivative size = 625, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2466, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 19683 a^6 \int \left (\frac {\sqrt [3]{a} \left (b^2-9 \sqrt [3]{a} c^{2/3} b+9 a^{2/3} c^{4/3}\right )-b \left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{4782969 a^{29/3} c^{2/3} \left (b x^2+3 a^{2/3} \sqrt [3]{c} x+3 a\right )}-\frac {\sqrt [3]{a} \left ((-1)^{2/3} b^2+9 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3} b+9 a^{2/3} c^{4/3}\right )-b \left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{1594323 \left (1+\sqrt [3]{-1}\right )^2 a^{29/3} c^{2/3} \left (b x^2-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a\right )}+\frac {\sqrt [3]{a} \left ((-1)^{2/3} b^2-9 \sqrt [3]{a} c^{2/3} b-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right )+\sqrt [3]{-1} b \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{4782969 a^{29/3} c^{2/3} \left (b x^2+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a\right )}+\frac {1}{531441 a^9 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 19683 a^6 \left (\frac {\left (9 a^{2/3} c^{4/3}+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{1594323 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{59/6} c^{2/3} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac {\left (9 a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 b^2\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{4782969 \sqrt {3} a^{59/6} c^{2/3} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {\left (-9 \sqrt [3]{-1} a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{4782969 \sqrt {3} a^{59/6} c^{2/3} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{9565938 a^{29/3} \sqrt [3]{c}}+\frac {\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{3188646 \left (1+\sqrt [3]{-1}\right )^2 a^{29/3} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+2 b\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{9565938 a^{29/3} \sqrt [3]{c}}-\frac {1}{531441 a^9 x}\right )\) |
Input:
Int[1/(x^2*(27*a^3 + 27*a^2*b*x^2 + 27*a^2*c*x^3 + 9*a*b^2*x^4 + b^3*x^6)) ,x]
Output:
19683*a^6*(-1/531441*1/(a^9*x) + ((2*(-1)^(2/3)*b^2 + 12*(-1)^(1/3)*a^(1/3 )*b*c^(2/3) + 9*a^(2/3)*c^(4/3))*ArcTan[(3*(-1)^(1/3)*a^(2/3)*c^(1/3) - 2* b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)])])/(1594323 *Sqrt[3]*(1 + (-1)^(1/3))^2*a^(59/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/ 3)]*c^(2/3)) + ((2*b^2 - 12*a^(1/3)*b*c^(2/3) + 9*a^(2/3)*c^(4/3))*ArcTan[ (3*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*a^(1/3)*c^(2/3)] )])/(4782969*Sqrt[3]*a^(59/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(2/3)) + ((2 *(-1)^(2/3)*b^2 - 12*a^(1/3)*b*c^(2/3) - 9*(-1)^(1/3)*a^(2/3)*c^(4/3))*Arc Tan[(3*(-1)^(2/3)*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b + 3*( -1)^(1/3)*a^(1/3)*c^(2/3)])])/(4782969*Sqrt[3]*a^(59/6)*Sqrt[4*b + 3*(-1)^ (1/3)*a^(1/3)*c^(2/3)]*c^(2/3)) - ((2*b - 3*a^(1/3)*c^(2/3))*Log[3*a + 3*a ^(2/3)*c^(1/3)*x + b*x^2])/(9565938*a^(29/3)*c^(1/3)) + ((2*b - 3*(-1)^(2/ 3)*a^(1/3)*c^(2/3))*Log[3*a - 3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(31 88646*(1 + (-1)^(1/3))^2*a^(29/3)*c^(1/3)) + ((-1)^(1/3)*(2*b + 3*(-1)^(1/ 3)*a^(1/3)*c^(2/3))*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(95 65938*a^(29/3)*c^(1/3)))
Int[(u_.)*(Q6_)^(p_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Sim p[1/(3^(3*p)*a^(2*p)) Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c, 3]* x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3* (-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coef f[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.21
| method | result | size |
| default | \(-\frac {1}{27 a^{3} x}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\left (-\textit {\_R}^{4} b^{3}-9 \textit {\_R}^{2} a \,b^{2}-27 \textit {\_R} \,a^{2} c -27 b \,a^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}}{81 a^{3}}\) | \(133\) |
| risch | \(-\frac {1}{27 a^{3} x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (729 a^{24} c^{6}-1728 a^{23} b^{3} c^{4}\right ) \textit {\_Z}^{6}+\left (13122 a^{17} b \,c^{6}-31347 a^{16} b^{4} c^{4}\right ) \textit {\_Z}^{4}+\left (-19683 c^{7} a^{14}+52488 b^{3} c^{5} a^{13}-14472 b^{6} c^{3} a^{12}\right ) \textit {\_Z}^{3}+\left (-4374 a^{9} b^{5} c^{4}-1701 a^{8} b^{8} c^{2}\right ) \textit {\_Z}^{2}-72 a^{4} b^{10} c \textit {\_Z} -b^{12}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-8748 a^{24} c^{6}+20574 a^{23} b^{3} c^{4}\right ) \textit {\_R}^{6}+\left (-3645 a^{20} b^{2} c^{5}+8100 a^{19} b^{5} c^{3}\right ) \textit {\_R}^{5}+\left (-118098 a^{17} b \,c^{6}+278478 a^{16} b^{4} c^{4}+1728 a^{15} b^{7} c^{2}\right ) \textit {\_R}^{4}+\left (177147 c^{7} a^{14}-472392 b^{3} c^{5} a^{13}+130329 b^{6} c^{3} a^{12}+108 a^{11} b^{9} c \right ) \textit {\_R}^{3}+\left (39366 a^{9} b^{5} c^{4}+15309 a^{8} b^{8} c^{2}+2 a^{7} b^{11}\right ) \textit {\_R}^{2}+648 \textit {\_R} \,a^{4} b^{10} c +9 b^{12}\right ) x +\left (729 a^{24} b \,c^{5}-2160 a^{23} b^{4} c^{3}\right ) \textit {\_R}^{6}+\left (-6561 a^{21} c^{6}+15066 a^{20} b^{3} c^{4}-144 a^{19} b^{6} c^{2}\right ) \textit {\_R}^{5}+\left (-6561 a^{17} b^{2} c^{5}+5832 a^{16} b^{5} c^{3}\right ) \textit {\_R}^{4}+54 a^{12} b^{7} c^{2} \textit {\_R}^{3}-9 a^{8} b^{9} c \,\textit {\_R}^{2}\right )\right )}{243}\) | \(444\) |
Input:
int(1/x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x,method= _RETURNVERBOSE)
Output:
-1/27/a^3/x+1/81/a^3*sum((-_R^4*b^3-9*_R^2*a*b^2-27*_R*a^2*c-27*a^2*b)/(2* _R^5*b^3+12*_R^3*a*b^2+27*_R^2*a^2*c+18*_R*a^2*b)*ln(x-_R),_R=RootOf(_Z^6* b^3+9*_Z^4*a*b^2+27*_Z^3*a^2*c+27*_Z^2*a^2*b+27*a^3))
Timed out. \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(b**3*x**6+9*a*b**2*x**4+27*a**2*c*x**3+27*a**2*b*x**2+27 *a**3),x)
Output:
Timed out
\[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\int { \frac {1}{{\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="maxima")
Output:
-1/27*integrate((b^3*x^4 + 9*a*b^2*x^2 + 27*a^2*c*x + 27*a^2*b)/(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3), x)/a^3 - 1/27/(a^3*x )
\[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\int { \frac {1}{{\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="giac")
Output:
integrate(1/((b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3 )*x^2), x)
Time = 22.51 (sec) , antiderivative size = 2663, normalized size of antiderivative = 4.13 \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*(27*a^3 + b^3*x^6 + 27*a^2*b*x^2 + 9*a*b^2*x^4 + 27*a^2*c*x^3)) ,x)
Output:
symsum(log(-282429536481*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094 635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 10930023061814 7*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6 *c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 1004423 49*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)*a^23*b^9*(2*b^10*x + 2541865828329*root(355779876259553472*a^23*b^3*c^4*z^6 - 1500946352969991 21*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4 *c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8 *c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^4*a^17*c^5 - 45*a*b^8*c + 3874 20489*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c ^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 28242953 6481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^2*a^10*c^6*x - 401769396*root(355779876 259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909 922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13* b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^2*a^9*b^4*c^3 - 2066242608*root(355779876259553472*a^23*...
\[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\frac {-\left (\int \frac {x^{4}}{b^{3} x^{6}+9 a \,b^{2} x^{4}+27 a^{2} c \,x^{3}+27 a^{2} b \,x^{2}+27 a^{3}}d x \right ) b^{3} x -9 \left (\int \frac {x^{2}}{b^{3} x^{6}+9 a \,b^{2} x^{4}+27 a^{2} c \,x^{3}+27 a^{2} b \,x^{2}+27 a^{3}}d x \right ) a \,b^{2} x -27 \left (\int \frac {x}{b^{3} x^{6}+9 a \,b^{2} x^{4}+27 a^{2} c \,x^{3}+27 a^{2} b \,x^{2}+27 a^{3}}d x \right ) a^{2} c x -27 \left (\int \frac {1}{b^{3} x^{6}+9 a \,b^{2} x^{4}+27 a^{2} c \,x^{3}+27 a^{2} b \,x^{2}+27 a^{3}}d x \right ) a^{2} b x -1}{27 a^{3} x} \] Input:
int(1/x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x)
Output:
( - int(x**4/(27*a**3 + 27*a**2*b*x**2 + 27*a**2*c*x**3 + 9*a*b**2*x**4 + b**3*x**6),x)*b**3*x - 9*int(x**2/(27*a**3 + 27*a**2*b*x**2 + 27*a**2*c*x* *3 + 9*a*b**2*x**4 + b**3*x**6),x)*a*b**2*x - 27*int(x/(27*a**3 + 27*a**2* b*x**2 + 27*a**2*c*x**3 + 9*a*b**2*x**4 + b**3*x**6),x)*a**2*c*x - 27*int( 1/(27*a**3 + 27*a**2*b*x**2 + 27*a**2*c*x**3 + 9*a*b**2*x**4 + b**3*x**6), x)*a**2*b*x - 1)/(27*a**3*x)