Integrand size = 46, antiderivative size = 563 \[ \int \frac {1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\frac {\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {\left (b-\sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{19/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {(-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {\log (x)}{27 a^3}-\frac {\left (3 \sqrt [3]{a}-\frac {b}{c^{2/3}}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{10/3}}-\frac {\left (b+i \sqrt {3} b+6 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{972 a^{10/3} c^{2/3}}-\frac {\left (3 \sqrt [3]{a}-\frac {(-1)^{2/3} b}{c^{2/3}}\right ) \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{10/3}} \]
1/27*ln(x)/a^3-1/486*(3*a^(1/3)-b/c^(2/3))*ln(3*a+3*a^(2/3)*c^(1/3)*x+b*x^ 2)/a^(10/3)-1/486*(3*a^(1/3)-(-1)^(2/3)*b/c^(2/3))*ln(3*a+3*(-1)^(2/3)*a^( 2/3)*c^(1/3)*x+b*x^2)/a^(10/3)-1/972*ln(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x +b*x^2)*(b+6*a^(1/3)*c^(2/3)+I*b*3^(1/2))/a^(10/3)/c^(2/3)+1/81*(b-a^(1/3) *c^(2/3))*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))/a^(19/6)/c^(1/3)*3^(1/2)/(4*b-3*a^(1/3)*c^(2/3))^(1/2) +1/27*(-1)^(2/3)*((-1)^(2/3)*b-a^(1/3)*c^(2/3))*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))/(1-(-1)^(1/3))/(1+(-1)^(1/3))^2/a^(19/6)/c^(1/3)*3^(1/2)/(4*b+3*(-1) ^(1/3)*a^(1/3)*c^(2/3))^(1/2)+1/27*(b-(-1)^(2/3)*a^(1/3)*c^(2/3))*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))/(1+(-1)^(1/3))^2/a^(19/6)/c^(1/3)*3^(1/2)/(4*b-3*( -1)^(2/3)*a^(1/3)*c^(2/3))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.28 \[ \int \frac {1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=-\frac {-3 \log (x)+\text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {27 a^2 b \log (x-\text {$\#$1})+27 a^2 c \log (x-\text {$\#$1}) \text {$\#$1}+9 a b^2 \log (x-\text {$\#$1}) \text {$\#$1}^2+b^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{18 a^2 b+27 a^2 c \text {$\#$1}+12 a b^2 \text {$\#$1}^2+2 b^3 \text {$\#$1}^4}\&\right ]}{81 a^3} \]
-1/81*(-3*Log[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 + 27*a^2*c*#1 + 1 2*a*b^2*#1^2 + 2*b^3*#1^4) & ])/a^3
Time = 1.26 (sec) , antiderivative size = 548, 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.
| \(\displaystyle \int \frac {1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 19683 a^6 \int \left (\frac {3 a^{2/3} \sqrt [3]{c} \left (2 b-3 \sqrt [3]{a} c^{2/3}\right )+b \left (b-3 \sqrt [3]{a} c^{2/3}\right ) x}{4782969 a^{28/3} c^{2/3} \left (b x^2+3 a^{2/3} \sqrt [3]{c} x+3 a\right )}+\frac {1}{531441 a^9 x}-\frac {3 a^{2/3} \sqrt [3]{c} \left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right )+\sqrt [3]{-1} b \left (\sqrt [3]{-1} b+3 \sqrt [3]{a} c^{2/3}\right ) x}{1594323 \left (1+\sqrt [3]{-1}\right )^2 a^{28/3} c^{2/3} \left (b x^2-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a\right )}+\frac {(-1)^{2/3} \left (3 a^{2/3} \sqrt [3]{c} \left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right )+b \left (b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) x\right )}{4782969 a^{28/3} c^{2/3} \left (b x^2+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 19683 a^6 \left (\frac {\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{177147 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{55/6} \sqrt [3]{c} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac {\left (b-\sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{531441 \sqrt {3} a^{55/6} \sqrt [3]{c} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {(-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{531441 \sqrt {3} a^{55/6} \sqrt [3]{c} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac {\left (3 \sqrt [3]{a}-\frac {b}{c^{2/3}}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{9565938 a^{28/3}}-\frac {\left (6 \sqrt [3]{a} c^{2/3}+i \sqrt {3} b+b\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{19131876 a^{28/3} c^{2/3}}-\frac {\left (3 \sqrt [3]{a}-\frac {(-1)^{2/3} b}{c^{2/3}}\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{9565938 a^{28/3}}+\frac {\log (x)}{531441 a^9}\right )\) |
19683*a^6*(((b - (-1)^(2/3)*a^(1/3)*c^(2/3))*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)] )])/(177147*Sqrt[3]*(1 + (-1)^(1/3))^2*a^(55/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^ (1/3)*c^(2/3)]*c^(1/3)) + ((b - a^(1/3)*c^(2/3))*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)])])/(531441*Sqrt[3 ]*a^(55/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(1/3)) + ((-1)^(2/3)*((-1)^(2/3 )*b - a^(1/3)*c^(2/3))*ArcTan[(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)])])/(531441*Sqrt[3]*a^ (55/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3)]*c^(1/3)) + Log[x]/(531441* a^9) - ((3*a^(1/3) - b/c^(2/3))*Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2])/(9 565938*a^(28/3)) - ((b + I*Sqrt[3]*b + 6*a^(1/3)*c^(2/3))*Log[3*a - 3*(-1) ^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(19131876*a^(28/3)*c^(2/3)) - ((3*a^(1/ 3) - ((-1)^(2/3)*b)/c^(2/3))*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x + b* x^2])/(9565938*a^(28/3)))
3.2.41.3.1 Defintions of rubi rules used
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 = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.24
| method | result | size |
| default | \(\frac {\ln \left (x \right )}{27 a^{3}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\left (\textit {\_R}^{5} b^{3}+9 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +27 a^{2} b \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}}{81 a^{3}}\) | \(134\) |
| risch | \(\frac {\ln \left (-x \right )}{27 a^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (27 a^{21} c^{6}-64 a^{20} b^{3} c^{4}\right ) \textit {\_Z}^{6}+\left (243 a^{18} c^{6}-576 a^{17} b^{3} c^{4}\right ) \textit {\_Z}^{5}+\left (729 a^{15} c^{6}-1755 a^{14} c^{4} b^{3}\right ) \textit {\_Z}^{4}+\left (729 a^{12} c^{6}-1917 a^{11} c^{4} b^{3}+16 a^{10} c^{2} b^{6}\right ) \textit {\_Z}^{3}+\left (-243 b^{3} c^{4} a^{8}-171 a^{7} b^{6} c^{2}\right ) \textit {\_Z}^{2}+27 a^{4} b^{6} c^{2} \textit {\_Z} -b^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-972 a^{18} c^{6}+2286 a^{17} b^{3} c^{4}\right ) \textit {\_R}^{5}+\left (-5832 a^{15} c^{6}+13878 a^{14} c^{4} b^{3}+16 a^{13} c^{2} b^{6}\right ) \textit {\_R}^{4}+\left (-8748 a^{12} c^{6}+22680 a^{11} c^{4} b^{3}-420 a^{10} c^{2} b^{6}\right ) \textit {\_R}^{3}+\left (3645 b^{3} c^{4} a^{8}+2610 a^{7} b^{6} c^{2}\right ) \textit {\_R}^{2}+\left (-486 a^{4} b^{6} c^{2}-2 a^{3} b^{9}\right ) \textit {\_R} +21 b^{9}\right ) x +\left (81 a^{18} b \,c^{5}-240 a^{17} b^{4} c^{3}\right ) \textit {\_R}^{5}+\left (-486 a^{15} b \,c^{5}+1296 a^{14} b^{4} c^{3}\right ) \textit {\_R}^{4}+\left (-2187 a^{12} b \,c^{5}+7290 a^{11} b^{4} c^{3}\right ) \textit {\_R}^{3}+\left (4374 a^{8} b^{4} c^{3}-24 a^{7} b^{7} c \right ) \textit {\_R}^{2}+216 a^{4} b^{7} c \textit {\_R} \right )\right )}{243}\) | \(434\) |
1/27*ln(x)/a^3-1/81/a^3*sum((_R^5*b^3+9*_R^3*a*b^2+27*_R^2*a^2*c+27*_R*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=RootO f(_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))
Exception generated. \[ \int \frac {1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \]
Timed out. \[ \int \frac {1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\int { \frac {1}{{\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )} x} \,d x } \]
-1/27*integrate((b^3*x^5 + 9*a*b^2*x^3 + 27*a^2*c*x^2 + 27*a^2*b*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), x)/a^3 + 1/27*lo g(x)/a^3
\[ \int \frac {1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\int { \frac {1}{{\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )} x} \,d x } \]
Time = 9.07 (sec) , antiderivative size = 4002, normalized size of antiderivative = 7.11 \[ \int \frac {1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Too large to display} \]
log(x)/(27*a^3) + symsum(log(7*root(13177032454057536*a^20*b^3*c^4*z^6 - 5 559060566555523*a^21*c^6*z^6 + 488038239039168*a^17*b^3*c^4*z^5 - 20589113 2094649*a^18*c^6*z^5 + 6119306623755*a^14*b^3*c^4*z^4 - 2541865828329*a^15 *c^6*z^4 + 27506854719*a^11*b^3*c^4*z^3 - 229582512*a^10*b^6*c^2*z^3 - 104 60353203*a^12*c^6*z^3 + 14348907*a^8*b^3*c^4*z^2 + 10097379*a^7*b^6*c^2*z^ 2 - 6561*a^4*b^6*c^2*z + b^9, z, k)*b^18*x - 162*root(13177032454057536*a^ 20*b^3*c^4*z^6 - 5559060566555523*a^21*c^6*z^6 + 488038239039168*a^17*b^3* c^4*z^5 - 205891132094649*a^18*c^6*z^5 + 6119306623755*a^14*b^3*c^4*z^4 - 2541865828329*a^15*c^6*z^4 + 27506854719*a^11*b^3*c^4*z^3 - 229582512*a^10 *b^6*c^2*z^3 - 10460353203*a^12*c^6*z^3 + 14348907*a^8*b^3*c^4*z^2 + 10097 379*a^7*b^6*c^2*z^2 - 6561*a^4*b^6*c^2*z + b^9, z, k)^2*a^3*b^18*x + 86093 442*root(13177032454057536*a^20*b^3*c^4*z^6 - 5559060566555523*a^21*c^6*z^ 6 + 488038239039168*a^17*b^3*c^4*z^5 - 205891132094649*a^18*c^6*z^5 + 6119 306623755*a^14*b^3*c^4*z^4 - 2541865828329*a^15*c^6*z^4 + 27506854719*a^11 *b^3*c^4*z^3 - 229582512*a^10*b^6*c^2*z^3 - 10460353203*a^12*c^6*z^3 + 143 48907*a^8*b^3*c^4*z^2 + 10097379*a^7*b^6*c^2*z^2 - 6561*a^4*b^6*c^2*z + b^ 9, z, k)^3*a^8*b^13*c^3 + 34867844010*root(13177032454057536*a^20*b^3*c^4* z^6 - 5559060566555523*a^21*c^6*z^6 + 488038239039168*a^17*b^3*c^4*z^5 - 2 05891132094649*a^18*c^6*z^5 + 6119306623755*a^14*b^3*c^4*z^4 - 25418658283 29*a^15*c^6*z^4 + 27506854719*a^11*b^3*c^4*z^3 - 229582512*a^10*b^6*c^2...