Integrand size = 44, antiderivative size = 469 \[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{13/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}} \]
-1/162*ln(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/a^(7/3)/c^(2/3)+1/54*(-1)^(2/3)*l n(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(1+(-1)^(1/3))^2/a^(7/3)/c^(2/ 3)-1/162*(-1)^(2/3)*ln(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/a^(7/3)/c ^(2/3)-1/81*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^(13/6)/c^(1/3)*3^(1/2)/(4*b-3*a^(1/3)*c^(2/3))^(1/ 2)+1/27*(-1)^(1/3)*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^(13/6)/c^(1/3)*3^(1/2)/(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2) -1/27*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^(13/6)/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 = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.20 \[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {1}{3} \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{18 a^2 b+27 a^2 c \text {$\#$1}+12 a b^2 \text {$\#$1}^2+2 b^3 \text {$\#$1}^4}\&\right ] \]
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 & , Log[x - #1]/(18*a^2*b + 27*a^2*c*#1 + 12*a*b^2*#1^2 + 2*b^3*#1^4) & ]/3
Time = 0.86 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, 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.
| \(\displaystyle \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 19683 a^6 \int \left (-\frac {b x+3 a^{2/3} \sqrt [3]{c}}{1594323 a^{25/3} c^{2/3} \left (b x^2+3 a^{2/3} \sqrt [3]{c} x+3 a\right )}+\frac {(-1)^{2/3} b x+3 a^{2/3} \sqrt [3]{c}}{531441 \left (1+\sqrt [3]{-1}\right )^2 a^{25/3} c^{2/3} \left (b x^2-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a\right )}-\frac {(-1)^{2/3} \left (b x+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}\right )}{1594323 a^{25/3} c^{2/3} \left (b x^2+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 19683 a^6 \left (-\frac {\arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{177147 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{49/6} \sqrt [3]{c} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac {\arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{531441 \sqrt {3} a^{49/6} \sqrt [3]{c} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{531441 \sqrt {3} a^{49/6} \sqrt [3]{c} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac {\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{3188646 a^{25/3} c^{2/3}}+\frac {(-1)^{2/3} \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{1062882 \left (1+\sqrt [3]{-1}\right )^2 a^{25/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{3188646 a^{25/3} c^{2/3}}\right )\) |
19683*a^6*(-1/177147*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)])]/(Sqrt[3]*(1 + (-1)^(1 /3))^2*a^(49/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(1/3)) - 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^(49/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(1/3)) + ((-1 )^(1/3)*ArcTan[(3*(-1)^(2/3)*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqr t[4*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3)])])/(531441*Sqrt[3]*a^(49/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3)]*c^(1/3)) - Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2]/(3188646*a^(25/3)*c^(2/3)) + ((-1)^(2/3)*Log[3*a - 3*(-1)^(1/3)*a ^(2/3)*c^(1/3)*x + b*x^2])/(1062882*(1 + (-1)^(1/3))^2*a^(25/3)*c^(2/3)) - ((-1)^(2/3)*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(3188646*a ^(25/3)*c^(2/3)))
3.2.39.3.1 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 = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.19
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(91\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(91\) |
1/3*sum(_R/(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))
Exception generated. \[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Exception raised: RuntimeError} \]
Timed out. \[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Timed out} \]
\[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {x}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \]
\[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {x}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \]
Time = 9.61 (sec) , antiderivative size = 1057, normalized size of antiderivative = 2.25 \[ \int \frac {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} \, dx=\sum _{k=1}^6\ln \left (b^{12}\,x+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^{10}\,b^{11}\,c^3\,1033121304+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{12}\,b^{12}\,c^3\,167365651248-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{13}\,b^9\,c^5\,94143178827+\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )\,a^2\,b^{13}\,x\,54+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^2\,a^5\,b^{11}\,c^2\,x\,177147+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^3\,a^7\,b^{12}\,c^2\,x\,17006112-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^3\,a^8\,b^9\,c^4\,x\,14348907+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^9\,b^{13}\,c^2\,x\,229582512+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^{10}\,b^{10}\,c^4\,x\,387420489-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{12}\,b^{11}\,c^4\,x\,20920706406\right )\,\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right ) \]
symsum(log(b^12*x + 1033121304*root(18075490334784*a^14*b^3*c^4*z^6 - 7625 597484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z ^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^4*a^10*b^1 1*c^3 + 167365651248*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987* a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348 907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^5*a^12*b^12*c^3 - 94 143178827*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^ 6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4 *z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^5*a^13*b^9*c^5 + 54*root(180754 90334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 + 1162261467*a^10*b *c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2 *c^2*z^2 + b^3, z, k)*a^2*b^13*x + 177147*root(18075490334784*a^14*b^3*c^4 *z^6 - 7625597484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^ 7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k) ^2*a^5*b^11*c^2*x + 17006112*root(18075490334784*a^14*b^3*c^4*z^6 - 762559 7484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^3*a^7*b^12*c ^2*x - 14348907*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15* c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a ^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^3*a^8*b^9*c^4*x + 2295...