\(\int \frac {1}{(3 a b+3 b^2 x+3 b c x^2+c^2 x^3)^2} \, dx\) [91]

Optimal result

Integrand size = 27, antiderivative size = 240 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {b+c x}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+\frac {(b+c x)^3}{c}\right )}+\frac {2 c \arctan \left (\frac {\sqrt [3]{b}+\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {2 c \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} \sqrt [3]{b^2-3 a c} (b+c x)+(b+c x)^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}} \] Output:

-1/3*(c*x+b)/b/(-3*a*c+b^2)/(b*(3*a-b^2/c)+(c*x+b)^3/c)+2/9*c*arctan(1/3*( 
b^(1/3)+2*(c*x+b)/(-3*a*c+b^2)^(1/3))*3^(1/2)/b^(1/3))*3^(1/2)/b^(5/3)/(-3 
*a*c+b^2)^(5/3)-2/9*c*ln(b^(1/3)*(b^(2/3)-(-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)*(-3*a*c+b^ 
2)^(1/3)*(c*x+b)+(c*x+b)^2)/b^(5/3)/(-3*a*c+b^2)^(5/3)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {\frac {3 (b+c x)}{3 a b+x \left (3 b^2+3 b c x+c^2 x^2\right )}+2 c \text {RootSum}\left [3 a b+3 b^2 \text {$\#$1}+3 b c \text {$\#$1}^2+c^2 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1})}{b^2+2 b c \text {$\#$1}+c^2 \text {$\#$1}^2}\&\right ]}{9 \left (b^3-3 a b c\right )} \] Input:

Integrate[(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^(-2),x]
 

Output:

-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)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.12, 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

\(\displaystyle -\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1}{\sqrt {3}}\right )}{c^{2/3}}+\frac {\log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac {b}{c}+x\right )^2\right )}{2 c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\)

Input:

Int[(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^(-2),x]
 

Output:

-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))
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 749

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])
 

rule 750

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 2458

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {\frac {c x}{9 b \left (3 a c -b^{2}\right )}+\frac {1}{27 a c -9 b^{2}}}{\frac {1}{3} c^{2} x^{3}+b c \,x^{2}+b^{2} x +a b}+\frac {2 c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b c \,\textit {\_Z}^{2}+3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}}\right )}{9 b \left (3 a c -b^{2}\right )}\)	\(134\)
risch	\(\frac {\frac {c x}{9 b \left (3 a c -b^{2}\right )}+\frac {1}{27 a c -9 b^{2}}}{\frac {1}{3} c^{2} x^{3}+b c \,x^{2}+b^{2} x +a b}+\frac {2 c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b c \,\textit {\_Z}^{2}+3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\left (3 a c -b^{2}\right ) \left (\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}\right )}\right )}{9 b}\)	\(134\)

Input:

int(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^2,x,method=_RETURNVERBOSE)
 

Output:

(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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (197) = 394\).

Time = 0.09 (sec) , antiderivative size = 700, normalized size of antiderivative = 2.92 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^2,x, algorithm="fricas")
 

Output:

-1/9*(3*b^7 - 18*a*b^5*c + 27*a^2*b^3*c^2 - 6*sqrt(1/3)*(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(sqr 
t(1/3)*(2*(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))/(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)
 

Sympy [A] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\frac {b + c x}{27 a^{2} b^{2} c - 9 a b^{4} + x^{3} \cdot \left (9 a b c^{3} - 3 b^{3} c^{2}\right ) + x^{2} \cdot \left (27 a b^{2} c^{2} - 9 b^{4} c\right ) + x \left (27 a b^{3} c - 9 b^{5}\right )} + \operatorname {RootSum} {\left (t^{3} \cdot \left (177147 a^{5} b^{5} c^{5} - 295245 a^{4} b^{7} c^{4} + 196830 a^{3} b^{9} c^{3} - 65610 a^{2} b^{11} c^{2} + 10935 a b^{13} c - 729 b^{15}\right ) - 8 c^{3}, \left ( t \mapsto t \log {\left (x + \frac {81 t a^{2} b^{2} c^{2} - 54 t a b^{4} c + 9 t b^{6} + 2 b c}{2 c^{2}} \right )} \right )\right )} \] Input:

integrate(1/(c**2*x**3+3*b*c*x**2+3*b**2*x+3*a*b)**2,x)
                                                                                    
                                                                                    
 

Output:

(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))))
 

Maxima [F]

\[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )}^{2}} \,d x } \] Input:

integrate(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^2,x, algorithm="maxima")
 

Output:

-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)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {2 \, \sqrt {3} \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}{c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}\right ) - \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left ({\left | c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}} \right |}\right )}{9 \, {\left (b^{3} - 3 \, a b c\right )}} - \frac {c x + b}{3 \, {\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )} {\left (b^{3} - 3 \, a b c\right )}} \] Input:

integrate(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^2,x, algorithm="giac")
 

Output:

-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))
 

Mupad [B] (verification not implemented)

Time = 13.17 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\frac {\frac {1}{3\,\left (3\,a\,c-b^2\right )}+\frac {c\,x}{3\,b\,\left (3\,a\,c-b^2\right )}}{3\,b^2\,x+3\,b\,c\,x^2+3\,a\,b+c^2\,x^3}+\frac {2\,c\,\ln \left (b+b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+c\,x\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}}-\frac {\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x-\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (c+\sqrt {3}\,c\,1{}\mathrm {i}\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}}-\frac {\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x+\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (c-\sqrt {3}\,c\,1{}\mathrm {i}\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}} \] Input:

int(1/(3*a*b + 3*b^2*x + c^2*x^3 + 3*b*c*x^2)^2,x)
 

Output:

(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))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 956, normalized size of antiderivative = 3.98 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^2,x)
 

Output:

( - 6*b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan((b**(1/3)*(3*a*c - b**2) 
**(1/3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)))*a*b*c - 6 
*b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan((b**(1/3)*(3*a*c - b**2)**(1/ 
3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)))*b**2*c*x - 6*b 
**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan((b**(1/3)*(3*a*c - b**2)**(1/3) 
 - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)))*b*c**2*x**2 - 2* 
b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan((b**(1/3)*(3*a*c - b**2)**(1/3 
) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)))*c**3*x**3 - 3*b 
**(1/3)*(3*a*c - b**2)**(1/3)*log(b**(2/3)*(3*a*c - b**2)**(2/3) - b**(1/3 
)*(3*a*c - b**2)**(1/3)*b - b**(1/3)*(3*a*c - b**2)**(1/3)*c*x + b**2 + 2* 
b*c*x + c**2*x**2)*a*b*c - 3*b**(1/3)*(3*a*c - b**2)**(1/3)*log(b**(2/3)*( 
3*a*c - b**2)**(2/3) - b**(1/3)*(3*a*c - b**2)**(1/3)*b - b**(1/3)*(3*a*c 
- b**2)**(1/3)*c*x + b**2 + 2*b*c*x + c**2*x**2)*b**2*c*x - 3*b**(1/3)*(3* 
a*c - b**2)**(1/3)*log(b**(2/3)*(3*a*c - b**2)**(2/3) - b**(1/3)*(3*a*c - 
b**2)**(1/3)*b - b**(1/3)*(3*a*c - b**2)**(1/3)*c*x + b**2 + 2*b*c*x + c** 
2*x**2)*b*c**2*x**2 - b**(1/3)*(3*a*c - b**2)**(1/3)*log(b**(2/3)*(3*a*c - 
 b**2)**(2/3) - b**(1/3)*(3*a*c - b**2)**(1/3)*b - b**(1/3)*(3*a*c - b**2) 
**(1/3)*c*x + b**2 + 2*b*c*x + c**2*x**2)*c**3*x**3 + 6*b**(1/3)*(3*a*c - 
b**2)**(1/3)*log(b**(1/3)*(3*a*c - b**2)**(1/3) + b + c*x)*a*b*c + 6*b**(1 
/3)*(3*a*c - b**2)**(1/3)*log(b**(1/3)*(3*a*c - b**2)**(1/3) + b + c*x)...