Integrand size = 46, antiderivative size = 487 \[ \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{3 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{7/6} b \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} a^{7/6} b \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{3 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{7/6} b \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 a^{4/3} b c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{18 \left (1+\sqrt [3]{-1}\right )^2 a^{4/3} b c^{2/3}}+\frac {(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 a^{4/3} b c^{2/3}} \] Output:
-1/9*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^(7/6)/b/(4 *b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))^(1/2)/c^(1/3)-1/27*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^(7 /6)/b/(4*b-3*a^(1/3)*c^(2/3))^(1/2)/c^(1/3)+1/9*(-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))*3^(1/2)/(1-(-1)^(1/3))/(1+(-1)^(1/3))^2/a^(7/6)/b/(4*b+3* (-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2)/c^(1/3)+1/54*ln(3*a+3*a^(2/3)*c^(1/3)*x+ b*x^2)/a^(4/3)/b/c^(2/3)-1/18*(-1)^(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^(4/3)/b/c^(2/3)+1/54*(-1)^(2/3)*ln(3*a+3*(- 1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/a^(4/3)/b/c^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.20 \[ \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {1}{3} \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^2}{18 a^2 b+27 a^2 c \text {$\#$1}+12 a b^2 \text {$\#$1}^2+2 b^3 \text {$\#$1}^4}\&\right ] \] Input:
Integrate[x^3/(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:
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]*#1^2)/(18*a^2*b + 27*a^2*c*#1 + 12*a*b^2*#1^2 + 2*b^3*#1^4 ) & ]/3
Time = 1.32 (sec) , antiderivative size = 472, 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.
| \(\displaystyle \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 19683 a^6 \int \left (\frac {x}{531441 a^{22/3} c^{2/3} \left (b x^2+3 a^{2/3} \sqrt [3]{c} x+3 a\right )}-\frac {(-1)^{2/3} x}{177147 \left (1+\sqrt [3]{-1}\right )^2 a^{22/3} c^{2/3} \left (b x^2-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a\right )}+\frac {(-1)^{2/3} x}{531441 a^{22/3} c^{2/3} \left (b x^2+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 19683 a^6 \left (-\frac {\arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{59049 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{43/6} b \sqrt [3]{c} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac {\arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{177147 \sqrt {3} a^{43/6} b \sqrt [3]{c} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{177147 \sqrt {3} a^{43/6} b \sqrt [3]{c} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac {\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{1062882 a^{22/3} b c^{2/3}}-\frac {(-1)^{2/3} \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{354294 \left (1+\sqrt [3]{-1}\right )^2 a^{22/3} b c^{2/3}}+\frac {(-1)^{2/3} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{1062882 a^{22/3} b c^{2/3}}\right )\) |
Input:
Int[x^3/(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*(-1/59049*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^(43/6)*b*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(1/3)) - ArcTa n[(3*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*a^(1/3)*c^(2/3 )])]/(177147*Sqrt[3]*a^(43/6)*b*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]* Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3)])])/(177147*Sqrt[3]*a^(43/6)*b*Sqr t[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]/(1062882*a^(22/3)*b*c^(2/3)) - ((-1)^(2/3)*Log[3*a - 3*(-1)^ (1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(354294*(1 + (-1)^(1/3))^2*a^(22/3)*b*c^ (2/3)) + ((-1)^(2/3)*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(1 062882*a^(22/3)*b*c^(2/3)))
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 = 93, normalized size of antiderivative = 0.19
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(93\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(93\) |
Input:
int(x^3/(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=_R ETURNVERBOSE)
Output:
1/3*sum(_R^3/(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. \[ \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^3/(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, al gorithm="fricas")
Output:
Exception raised: RuntimeError >> no explicit roots found
Timed out. \[ \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Timed out} \] Input:
integrate(x**3/(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
\[ \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {x^{3}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \] Input:
integrate(x^3/(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, al gorithm="maxima")
Output:
integrate(x^3/(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)
\[ \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {x^{3}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \] Input:
integrate(x^3/(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, al gorithm="giac")
Output:
integrate(x^3/(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 = 0.88 (sec) , antiderivative size = 1354, normalized size of antiderivative = 2.78 \[ \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Too large to display} \] Input:
int(x^3/(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(4782969*root(10460353203*a^9*b^3*c^6*z^6 - 24794911296*a^8*b^6* c^4*z^6 - 14348907*a^6*b^2*c^4*z^4 + 314928*a^4*b^3*c^2*z^3 - 531441*a^5*c ^4*z^3 - 19683*a^3*b*c^2*z^2 - 1, z, k)^2*a^9*b^6*c^3 - 729*a^5*b^7*x + 12 9140163*root(10460353203*a^9*b^3*c^6*z^6 - 24794911296*a^8*b^6*c^4*z^6 - 1 4348907*a^6*b^2*c^4*z^4 + 314928*a^4*b^3*c^2*z^3 - 531441*a^5*c^4*z^3 - 19 683*a^3*b*c^2*z^2 - 1, z, k)^3*a^10*b^8*c^3 + 1549681956*root(10460353203* a^9*b^3*c^6*z^6 - 24794911296*a^8*b^6*c^4*z^6 - 14348907*a^6*b^2*c^4*z^4 + 314928*a^4*b^3*c^2*z^3 - 531441*a^5*c^4*z^3 - 19683*a^3*b*c^2*z^2 - 1, z, k)^4*a^11*b^10*c^3 + 167365651248*root(10460353203*a^9*b^3*c^6*z^6 - 2479 4911296*a^8*b^6*c^4*z^6 - 14348907*a^6*b^2*c^4*z^4 + 314928*a^4*b^3*c^2*z^ 3 - 531441*a^5*c^4*z^3 - 19683*a^3*b*c^2*z^2 - 1, z, k)^5*a^12*b^12*c^3 - 94143178827*root(10460353203*a^9*b^3*c^6*z^6 - 24794911296*a^8*b^6*c^4*z^6 - 14348907*a^6*b^2*c^4*z^4 + 314928*a^4*b^3*c^2*z^3 - 531441*a^5*c^4*z^3 - 19683*a^3*b*c^2*z^2 - 1, z, k)^5*a^13*b^9*c^5 + 98415*root(10460353203*a ^9*b^3*c^6*z^6 - 24794911296*a^8*b^6*c^4*z^6 - 14348907*a^6*b^2*c^4*z^4 + 314928*a^4*b^3*c^2*z^3 - 531441*a^5*c^4*z^3 - 19683*a^3*b*c^2*z^2 - 1, z, k)*a^7*b^7*c + 4374*root(10460353203*a^9*b^3*c^6*z^6 - 24794911296*a^8*b^6 *c^4*z^6 - 14348907*a^6*b^2*c^4*z^4 + 314928*a^4*b^3*c^2*z^3 - 531441*a^5* c^4*z^3 - 19683*a^3*b*c^2*z^2 - 1, z, k)*a^6*b^9*x - 2125764*root(10460353 203*a^9*b^3*c^6*z^6 - 24794911296*a^8*b^6*c^4*z^6 - 14348907*a^6*b^2*c^...
\[ \int \frac {x^3}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int \frac {x^{3}}{b^{3} x^{6}+9 a \,b^{2} x^{4}+27 a^{2} c \,x^{3}+27 a^{2} b \,x^{2}+27 a^{3}}d x \] Input:
int(x^3/(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**3/(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)