Integrand size = 42, antiderivative size = 522 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\sqrt [3]{-1} \left (2 \sqrt [3]{-1} b+3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} a^{17/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \] Output:
-1/81*(-1)^(1/3)*(2*(-1)^(1/3)*b+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))*3^(1/2)/(1+(-1)^(1/3))^2/a^(17/6)/(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2 /3))^(1/2)/c^(2/3)-1/243*(2*b-3*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))*3^(1/2)/a^(17/6 )/(4*b-3*a^(1/3)*c^(2/3))^(1/2)/c^(2/3)-1/81*(2*(-1)^(2/3)*b-3*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))*3^(1/2)/(1-(-1)^(1/3))/(1+(-1)^(1/3) )^2/a^(17/6)/(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2)/c^(2/3)+1/162*ln(3*a +3*a^(2/3)*c^(1/3)*x+b*x^2)/a^(8/3)/c^(1/3)-1/54*ln(3*a-3*(-1)^(1/3)*a^(2/ 3)*c^(1/3)*x+b*x^2)/(1+(-1)^(1/3))^2/a^(8/3)/c^(1/3)-1/162*(-1)^(1/3)*ln(3 *a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/a^(8/3)/c^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.19 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {1}{3} \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{18 a^2 b \text {$\#$1}+27 a^2 c \text {$\#$1}^2+12 a b^2 \text {$\#$1}^3+2 b^3 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(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)^( -1),x]
Output:
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*#1 + 27*a^2*c*#1^2 + 12*a*b^2*#1^3 + 2*b^3*#1^5) & ]/3
Time = 1.67 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, 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.
| \(\displaystyle \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 19683 a^6 \int \left (-\frac {\sqrt [3]{a} \left (b-3 \sqrt [3]{a} c^{2/3}\right )-b \sqrt [3]{c} x}{1594323 a^{26/3} c^{2/3} \left (b x^2+3 a^{2/3} \sqrt [3]{c} x+3 a\right )}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \left (\sqrt [3]{-1} b+3 \sqrt [3]{a} c^{2/3}\right )-b \sqrt [3]{c} x}{531441 \left (1+\sqrt [3]{-1}\right )^2 a^{26/3} c^{2/3} \left (b x^2-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a\right )}-\frac {\sqrt [3]{a} \left ((-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right )+\sqrt [3]{-1} b \sqrt [3]{c} x}{1594323 a^{26/3} c^{2/3} \left (b x^2+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 19683 a^6 \left (-\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{a} c^{2/3}+2 \sqrt [3]{-1} b\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{531441 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{53/6} c^{2/3} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{1594323 \sqrt {3} a^{53/6} c^{2/3} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{1594323 \sqrt {3} a^{53/6} c^{2/3} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac {\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{3188646 a^{26/3} \sqrt [3]{c}}-\frac {\log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{1062882 \left (1+\sqrt [3]{-1}\right )^2 a^{26/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{3188646 a^{26/3} \sqrt [3]{c}}\right )\) |
Input:
Int[(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)^(-1),x]
Output:
19683*a^6*(-1/531441*((-1)^(1/3)*(2*(-1)^(1/3)*b + 3*a^(1/3)*c^(2/3))*ArcT an[(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^(53/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(2/3)) - ((2*b - 3*a^(1/3)*c^(2/3))*Arc Tan[(3*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*a^(1/3)*c^(2 /3)])])/(1594323*Sqrt[3]*a^(53/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(2/3)) - ((2*(-1)^(2/3)*b - 3*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)])])/(1 594323*Sqrt[3]*a^(53/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3)]*c^(2/3)) + Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2]/(3188646*a^(26/3)*c^(1/3)) - Log[ 3*a - 3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2]/(1062882*(1 + (-1)^(1/3))^2* a^(26/3)*c^(1/3)) - ((-1)^(1/3)*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(3188646*a^(26/3)*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 = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.17
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(90\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(90\) |
Input:
int(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,method=_RET URNVERBOSE)
Output:
1/3*sum(1/(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. \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Exception raised: RuntimeError} \] Input:
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, algo rithm="fricas")
Output:
Exception raised: RuntimeError >> no explicit roots found
Timed out. \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Timed out} \] Input:
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)
Output:
Timed out
\[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {1}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \] Input:
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, algo rithm="maxima")
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)
\[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {1}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \] Input:
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, algo rithm="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)
Time = 22.68 (sec) , antiderivative size = 1394, normalized size of antiderivative = 2.67 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Too large to display} \] Input:
int(1/(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(6561*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^ 18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 38 7420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z , k)^2*a^4*b^12*c^2 - 6*root(488038239039168*a^17*b^3*c^4*z^6 - 2058911320 94649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3* z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)*b^15*x - 4782969*root(488038239039168*a^17*b^3*c^4*z^6 - 2058 91132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^ 3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^ 5*c*z + b^6, z, k)^3*a^7*b^11*c^3 - 229582512*root(488038239039168*a^17*b^ 3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z ^2 - 2916*a^3*b^5*c*z + b^6, z, k)^4*a^9*b^13*c^2 - 387420489*root(4880382 39039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^1 2*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 26572 05*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^4*a^10*b^10*c^4 + 16736 5651248*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z ^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489* a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^5*a ^12*b^12*c^3 - 94143178827*root(488038239039168*a^17*b^3*c^4*z^6 - 2058...
\[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int \frac {1}{b^{3} x^{6}+9 a \,b^{2} x^{4}+27 a^{2} c \,x^{3}+27 a^{2} b \,x^{2}+27 a^{3}}d x \] Input:
int(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)
Output:
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)