Integrand size = 27, antiderivative size = 245 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {2 c \arctan \left (\frac {\sqrt [3]{b}+\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {2 c \log \left (b-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+c^2 \left (\frac {b}{c}+x\right )^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}} \]
-1/3*c*(b/c+x)/b/(-3*a*c+b^2)/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)-2/9*c*ln(b -b^(1/3)*(-3*a*c+b^2)^(1/3)+c*x)/b^(5/3)/(-3*a*c+b^2)^(5/3)+1/9*c*ln(b^(2/ 3)*(-3*a*c+b^2)^(2/3)+b^(1/3)*c*(-3*a*c+b^2)^(1/3)*(b/c+x)+c^2*(b/c+x)^2)/ b^(5/3)/(-3*a*c+b^2)^(5/3)+2/9*c*arctan(1/3*(b^(1/3)+2*(c*x+b)/(-3*a*c+b^2 )^(1/3))/b^(1/3)*3^(1/2))/b^(5/3)/(-3*a*c+b^2)^(5/3)*3^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {\frac {3 (b+c x)}{3 a b+x \left (3 b^2+3 b c x+c^2 x^2\right )}+2 c \text {RootSum}\left [3 a b+3 b^2 \text {$\#$1}+3 b c \text {$\#$1}^2+c^2 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1})}{b^2+2 b c \text {$\#$1}+c^2 \text {$\#$1}^2}\&\right ]}{9 \left (b^3-3 a b c\right )} \]
-1/9*((3*(b + c*x))/(3*a*b + x*(3*b^2 + 3*b*c*x + c^2*x^2)) + 2*c*RootSum[ 3*a*b + 3*b^2*#1 + 3*b*c*#1^2 + c^2*#1^3 & , Log[x - #1]/(b^2 + 2*b*c*#1 + c^2*#1^2) & ])/(b^3 - 3*a*b*c)
Time = 0.49 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2458, 749, 750, 16, 25, 1142, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{\left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}d\left (\frac {b}{c}+x\right )\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {2 c \int \frac {1}{c^2 \left (\frac {b}{c}+x\right )^3+b \left (3 a-\frac {b^2}{c}\right )}d\left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {2 c \left (\frac {c^{2/3} \int \frac {1}{c^{2/3} \left (\frac {b}{c}+x\right )-\frac {\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {c^{2/3} \int -\frac {c \left (\frac {b}{c}+x\right )+2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c} \left (c^{4/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}\right )}d\left (\frac {b}{c}+x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {2 c \left (\frac {c^{2/3} \int -\frac {c \left (\frac {b}{c}+x\right )+2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c} \left (c^{4/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}\right )}d\left (\frac {b}{c}+x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \int \frac {c \left (\frac {b}{c}+x\right )+2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {3}{2} \sqrt [3]{b} \sqrt [3]{b^2-3 a c} \int \frac {1}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )+\frac {\int \frac {c^{2/3} \left (2 c \left (\frac {b}{c}+x\right )+\sqrt [3]{b} \sqrt [3]{b^2-3 a c}\right )}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )}{2 c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {3}{2} \sqrt [3]{b} \sqrt [3]{b^2-3 a c} \int \frac {1}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )+\frac {1}{2} \int \frac {2 c \left (\frac {b}{c}+x\right )+\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {1}{2} \int \frac {2 c \left (\frac {b}{c}+x\right )+\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )-\frac {3 \int \frac {1}{-\left (\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1\right )^2-3}d\left (\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1\right )}{c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {1}{2} \int \frac {2 c \left (\frac {b}{c}+x\right )+\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1}{\sqrt {3}}\right )}{c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1}{\sqrt {3}}\right )}{c^{2/3}}+\frac {\log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac {b}{c}+x\right )^2\right )}{2 c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\) |
-1/3*(c*(b/c + x))/(b*(b^2 - 3*a*c)*(b*(3*a - b^2/c) + c^2*(b/c + x)^3)) - (2*c*(Log[b^(1/3)*(b^2 - 3*a*c)^(1/3) - c*(b/c + x)]/(3*b^(2/3)*(b^2 - 3* a*c)^(2/3)) - (c^(2/3)*((Sqrt[3]*ArcTan[(1 + (2*c*(b/c + x))/(b^(1/3)*(b^2 - 3*a*c)^(1/3)))/Sqrt[3]])/c^(2/3) + Log[b^(2/3)*(b^2 - 3*a*c)^(2/3) + b^ (1/3)*c*(b^2 - 3*a*c)^(1/3)*(b/c + x) + c^2*(b/c + x)^2]/(2*c^(2/3))))/(3* b^(2/3)*(b^2 - 3*a*c)^(2/3))))/(3*b*(b^2 - 3*a*c))
3.1.13.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.55
| method | result | size |
| default | \(\frac {\frac {c x}{9 b \left (3 a c -b^{2}\right )}+\frac {1}{27 a c -9 b^{2}}}{\frac {1}{3} c^{2} x^{3}+b c \,x^{2}+b^{2} x +a b}+\frac {2 c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b \,\textit {\_Z}^{2} c +3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}}\right )}{9 b \left (3 a c -b^{2}\right )}\) | \(134\) |
| risch | \(\frac {\frac {c x}{9 b \left (3 a c -b^{2}\right )}+\frac {1}{27 a c -9 b^{2}}}{\frac {1}{3} c^{2} x^{3}+b c \,x^{2}+b^{2} x +a b}+\frac {2 c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b \,\textit {\_Z}^{2} c +3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\left (3 a c -b^{2}\right ) \left (\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}\right )}\right )}{9 b}\) | \(134\) |
(1/9*c/b/(3*a*c-b^2)*x+1/9/(3*a*c-b^2))/(1/3*c^2*x^3+b*c*x^2+b^2*x+a*b)+2/ 9*c/b/(3*a*c-b^2)*sum(1/(_R^2*c^2+2*_R*b*c+b^2)*ln(x-_R),_R=RootOf(_Z^3*c^ 2+3*_Z^2*b*c+3*_Z*b^2+3*a*b))
Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (204) = 408\).
Time = 0.28 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.87 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {3 \, b^{7} - 18 \, a b^{5} c + 27 \, a^{2} b^{3} c^{2} - 2 \, \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{6}} {\left (3 \, a b^{4} c - 9 \, a^{2} b^{2} c^{2} + {\left (b^{3} c^{3} - 3 \, a b c^{4}\right )} x^{3} + 3 \, {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} x^{2} + 3 \, {\left (b^{5} c - 3 \, a b^{3} c^{2}\right )} x\right )} \arctan \left (\frac {2 \, \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c x + b\right )} + \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}} {\left (b^{3} - 3 \, a b c\right )}}{3 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {5}{6}}}\right ) - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + 3 \, a b c\right )} \log \left (-b^{5} + 3 \, a b^{3} c - {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} x^{2} - 2 \, {\left (b^{4} c - 3 \, a b^{2} c^{2}\right )} x - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c x + b\right )} - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}} {\left (b^{3} - 3 \, a b c\right )}\right ) + 2 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + 3 \, a b c\right )} \log \left (-b^{4} + 3 \, a b^{2} c - {\left (b^{3} c - 3 \, a b c^{2}\right )} x + {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}}\right ) + 3 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 9 \, a^{2} b^{2} c^{3}\right )} x}{9 \, {\left (3 \, a b^{10} - 27 \, a^{2} b^{8} c + 81 \, a^{3} b^{6} c^{2} - 81 \, a^{4} b^{4} c^{3} + {\left (b^{9} c^{2} - 9 \, a b^{7} c^{3} + 27 \, a^{2} b^{5} c^{4} - 27 \, a^{3} b^{3} c^{5}\right )} x^{3} + 3 \, {\left (b^{10} c - 9 \, a b^{8} c^{2} + 27 \, a^{2} b^{6} c^{3} - 27 \, a^{3} b^{4} c^{4}\right )} x^{2} + 3 \, {\left (b^{11} - 9 \, a b^{9} c + 27 \, a^{2} b^{7} c^{2} - 27 \, a^{3} b^{5} c^{3}\right )} x\right )}} \]
-1/9*(3*b^7 - 18*a*b^5*c + 27*a^2*b^3*c^2 - 2*sqrt(3)*(b^6 - 6*a*b^4*c + 9 *a^2*b^2*c^2)^(1/6)*(3*a*b^4*c - 9*a^2*b^2*c^2 + (b^3*c^3 - 3*a*b*c^4)*x^3 + 3*(b^4*c^2 - 3*a*b^2*c^3)*x^2 + 3*(b^5*c - 3*a*b^3*c^2)*x)*arctan(1/3*( 2*sqrt(3)*(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(2/3)*(c*x + b) + sqrt(3)*(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(1/3)*(b^3 - 3*a*b*c))/(b^6 - 6*a*b^4*c + 9* a^2*b^2*c^2)^(5/6)) - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(2/3)*(c^3*x^3 + 3 *b*c^2*x^2 + 3*b^2*c*x + 3*a*b*c)*log(-b^5 + 3*a*b^3*c - (b^3*c^2 - 3*a*b* c^3)*x^2 - 2*(b^4*c - 3*a*b^2*c^2)*x - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^( 2/3)*(c*x + b) - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(1/3)*(b^3 - 3*a*b*c)) + 2*(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(2/3)*(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2 *c*x + 3*a*b*c)*log(-b^4 + 3*a*b^2*c - (b^3*c - 3*a*b*c^2)*x + (b^6 - 6*a* b^4*c + 9*a^2*b^2*c^2)^(2/3)) + 3*(b^6*c - 6*a*b^4*c^2 + 9*a^2*b^2*c^3)*x) /(3*a*b^10 - 27*a^2*b^8*c + 81*a^3*b^6*c^2 - 81*a^4*b^4*c^3 + (b^9*c^2 - 9 *a*b^7*c^3 + 27*a^2*b^5*c^4 - 27*a^3*b^3*c^5)*x^3 + 3*(b^10*c - 9*a*b^8*c^ 2 + 27*a^2*b^6*c^3 - 27*a^3*b^4*c^4)*x^2 + 3*(b^11 - 9*a*b^9*c + 27*a^2*b^ 7*c^2 - 27*a^3*b^5*c^3)*x)
Time = 0.66 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\frac {b + c x}{27 a^{2} b^{2} c - 9 a b^{4} + x^{3} \cdot \left (9 a b c^{3} - 3 b^{3} c^{2}\right ) + x^{2} \cdot \left (27 a b^{2} c^{2} - 9 b^{4} c\right ) + x \left (27 a b^{3} c - 9 b^{5}\right )} + \operatorname {RootSum} {\left (t^{3} \cdot \left (177147 a^{5} b^{5} c^{5} - 295245 a^{4} b^{7} c^{4} + 196830 a^{3} b^{9} c^{3} - 65610 a^{2} b^{11} c^{2} + 10935 a b^{13} c - 729 b^{15}\right ) - 8 c^{3}, \left ( t \mapsto t \log {\left (x + \frac {81 t a^{2} b^{2} c^{2} - 54 t a b^{4} c + 9 t b^{6} + 2 b c}{2 c^{2}} \right )} \right )\right )} \]
(b + c*x)/(27*a**2*b**2*c - 9*a*b**4 + x**3*(9*a*b*c**3 - 3*b**3*c**2) + x **2*(27*a*b**2*c**2 - 9*b**4*c) + x*(27*a*b**3*c - 9*b**5)) + RootSum(_t** 3*(177147*a**5*b**5*c**5 - 295245*a**4*b**7*c**4 + 196830*a**3*b**9*c**3 - 65610*a**2*b**11*c**2 + 10935*a*b**13*c - 729*b**15) - 8*c**3, Lambda(_t, _t*log(x + (81*_t*a**2*b**2*c**2 - 54*_t*a*b**4*c + 9*_t*b**6 + 2*b*c)/(2 *c**2))))
\[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )}^{2}} \,d x } \]
-2/3*c*integrate(1/(c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b), x)/(b^3 - 3*a* b*c) - 1/3*(c*x + b)/(3*a*b^4 - 9*a^2*b^2*c + (b^3*c^2 - 3*a*b*c^3)*x^3 + 3*(b^4*c - 3*a*b^2*c^2)*x^2 + 3*(b^5 - 3*a*b^3*c)*x)
Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {2 \, \sqrt {3} \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}{c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}\right ) - \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left ({\left | c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}} \right |}\right )}{9 \, {\left (b^{3} - 3 \, a b c\right )}} - \frac {c x + b}{3 \, {\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )} {\left (b^{3} - 3 \, a b c\right )}} \]
-1/9*(2*sqrt(3)*(c^3/(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2))^(1/3)*arctan((sqrt (3)*c*x + sqrt(3)*b - sqrt(3)*(-b^3 + 3*a*b*c)^(1/3))/(c*x + b + (-b^3 + 3 *a*b*c)^(1/3))) - (c^3/(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2))^(1/3)*log(4*(sqr t(3)*c*x + sqrt(3)*b - sqrt(3)*(-b^3 + 3*a*b*c)^(1/3))^2 + 4*(c*x + b + (- b^3 + 3*a*b*c)^(1/3))^2) + 2*(c^3/(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2))^(1/3) *log(abs(c*x + b + (-b^3 + 3*a*b*c)^(1/3))))/(b^3 - 3*a*b*c) - 1/3*(c*x + b)/((c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b)*(b^3 - 3*a*b*c))
Time = 10.17 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\frac {\frac {1}{3\,\left (3\,a\,c-b^2\right )}+\frac {c\,x}{3\,b\,\left (3\,a\,c-b^2\right )}}{3\,b^2\,x+3\,b\,c\,x^2+3\,a\,b+c^2\,x^3}+\frac {2\,c\,\ln \left (b+b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+c\,x\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}}-\frac {\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x-\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (c+\sqrt {3}\,c\,1{}\mathrm {i}\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}}-\frac {\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x+\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (c-\sqrt {3}\,c\,1{}\mathrm {i}\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}} \]
(1/(3*(3*a*c - b^2)) + (c*x)/(3*b*(3*a*c - b^2)))/(3*a*b + 3*b^2*x + c^2*x ^3 + 3*b*c*x^2) + (2*c*log(b + b^(1/3)*(3*a*c - b^2)^(1/3) + c*x))/(9*b^(5 /3)*(3*a*c - b^2)^(5/3)) - (log(2*b - b^(1/3)*(3*a*c - b^2)^(1/3) + 2*c*x - 3^(1/2)*b^(1/3)*(3*a*c - b^2)^(1/3)*1i)*(c + 3^(1/2)*c*1i))/(9*b^(5/3)*( 3*a*c - b^2)^(5/3)) - (log(2*b - b^(1/3)*(3*a*c - b^2)^(1/3) + 2*c*x + 3^( 1/2)*b^(1/3)*(3*a*c - b^2)^(1/3)*1i)*(c - 3^(1/2)*c*1i))/(9*b^(5/3)*(3*a*c - b^2)^(5/3))