Integrand size = 24, antiderivative size = 237 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {10}{9 a^3 d e^3 (c+d x)^2}+\frac {1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {20 b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} d e^3}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d e^3}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d e^3} \]
-10/9/a^3/d/e^3/(d*x+c)^2+1/6/a/d/e^3/(d*x+c)^2/(a+b*(d*x+c)^3)^2+4/9/a^2/ d/e^3/(d*x+c)^2/(a+b*(d*x+c)^3)-20/27*b^(2/3)*ln(a^(1/3)+b^(1/3)*(d*x+c))/ a^(11/3)/d/e^3+10/27*b^(2/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d *x+c)^2)/a^(11/3)/d/e^3+20/27*b^(2/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c ))/a^(1/3)*3^(1/2))/a^(11/3)/d/e^3*3^(1/2)
Time = 0.12 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {-\frac {27 a^{2/3}}{(c+d x)^2}-\frac {9 a^{5/3} b (c+d x)}{\left (a+b (c+d x)^3\right )^2}-\frac {33 a^{2/3} b (c+d x)}{a+b (c+d x)^3}-40 \sqrt {3} b^{2/3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-40 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )+20 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{11/3} d e^3} \]
((-27*a^(2/3))/(c + d*x)^2 - (9*a^(5/3)*b*(c + d*x))/(a + b*(c + d*x)^3)^2 - (33*a^(2/3)*b*(c + d*x))/(a + b*(c + d*x)^3) - 40*Sqrt[3]*b^(2/3)*ArcTa n[(-a^(1/3) + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))] - 40*b^(2/3)*Log[a^( 1/3) + b^(1/3)*(c + d*x)] + 20*b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(54*a^(11/3)*d*e^3)
Time = 0.36 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {895, 819, 819, 847, 750, 16, 1142, 25, 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.
| \(\displaystyle \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 895 |
| \(\displaystyle \frac {\int \frac {1}{(c+d x)^3 \left (b (c+d x)^3+a\right )^3}d(c+d x)}{d e^3}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\frac {4 \int \frac {1}{(c+d x)^3 \left (b (c+d x)^3+a\right )^2}d(c+d x)}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \int \frac {1}{(c+d x)^3 \left (b (c+d x)^3+a\right )}d(c+d x)}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \int \frac {1}{b (c+d x)^3+a}d(c+d x)}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}d(c+d x)}{3 a^{2/3}}\right )}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a (c+d x)^2}\right )}{3 a}+\frac {1}{3 a (c+d x)^2 \left (a+b (c+d x)^3\right )}\right )}{3 a}+\frac {1}{6 a (c+d x)^2 \left (a+b (c+d x)^3\right )^2}}{d e^3}\) |
(1/(6*a*(c + d*x)^2*(a + b*(c + d*x)^3)^2) + (4*(1/(3*a*(c + d*x)^2*(a + b *(c + d*x)^3)) + (5*(-1/2*1/(a*(c + d*x)^2) - (b*(Log[a^(1/3) + b^(1/3)*(c + d*x)]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c + d*x ))/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(2*b^(1/3)))/(3*a^(2/3))))/a))/(3*a)))/(3*a))/(d*e^3 )
3.30.8.3.1 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_)^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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff icient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ {a, b, m, n, p}, x] && LinearPairQ[u, v, 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.06 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.81
| method | result | size |
| default | \(\frac {-\frac {1}{2 a^{3} d \left (d x +c \right )^{2}}-\frac {b \left (\frac {\frac {11 b \,d^{3} x^{4}}{18}+\frac {22 b c \,d^{2} x^{3}}{9}+\frac {11 b \,c^{2} d \,x^{2}}{3}+\left (\frac {22 c^{3} b}{9}+\frac {7 a}{9}\right ) x +\frac {c \left (11 c^{3} b +14 a \right )}{18 d}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {20 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b \,d^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{27 b d}\right )}{a^{3}}}{e^{3}}\) | \(192\) |
| risch | \(\frac {-\frac {10 b^{2} d^{5} x^{6}}{9 a^{3}}-\frac {20 b^{2} c \,d^{4} x^{5}}{3 a^{3}}-\frac {50 b^{2} c^{2} d^{3} x^{4}}{3 a^{3}}-\frac {8 b \,d^{2} \left (25 c^{3} b +2 a \right ) x^{3}}{9 a^{3}}-\frac {2 b c d \left (25 c^{3} b +8 a \right ) x^{2}}{3 a^{3}}-\frac {4 b \,c^{2} \left (5 c^{3} b +4 a \right ) x}{3 a^{3}}-\frac {20 b^{2} c^{6}+32 a b \,c^{3}+9 a^{2}}{18 d \,a^{3}}}{e^{3} \left (d x +c \right )^{2} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {20 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} d^{3} e^{9} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{11} d^{4} e^{9} \textit {\_R}^{3}-3 b^{2} d \right ) x -4 a^{11} c \,d^{3} e^{9} \textit {\_R}^{3}-a^{4} b d \,e^{3} \textit {\_R} -3 b^{2} c \right )\right )}{27}\) | \(269\) |
1/e^3*(-1/2/a^3/d/(d*x+c)^2-1/a^3*b*((11/18*b*d^3*x^4+22/9*b*c*d^2*x^3+11/ 3*b*c^2*d*x^2+(22/9*c^3*b+7/9*a)*x+1/18*c/d*(11*b*c^3+14*a))/(b*d^3*x^3+3* b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2+20/27/b/d*sum(1/(_R^2*d^2+2*_R*c*d+c^2) *ln(x-_R),_R=RootOf(_Z^3*b*d^3+3*_Z^2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a))))
Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (194) = 388\).
Time = 0.32 (sec) , antiderivative size = 1051, normalized size of antiderivative = 4.43 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {60 \, b^{2} d^{6} x^{6} + 360 \, b^{2} c d^{5} x^{5} + 900 \, b^{2} c^{2} d^{4} x^{4} + 60 \, b^{2} c^{6} + 48 \, {\left (25 \, b^{2} c^{3} + 2 \, a b\right )} d^{3} x^{3} + 96 \, a b c^{3} + 36 \, {\left (25 \, b^{2} c^{4} + 8 \, a b c\right )} d^{2} x^{2} + 72 \, {\left (5 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d x - 40 \, \sqrt {3} {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (a d x + a c\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 20 \, {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + {\left (a b d x + a b c\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 40 \, {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b d x + b c - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 27 \, a^{2}}{54 \, {\left (a^{3} b^{2} d^{9} e^{3} x^{8} + 8 \, a^{3} b^{2} c d^{8} e^{3} x^{7} + 28 \, a^{3} b^{2} c^{2} d^{7} e^{3} x^{6} + 2 \, {\left (28 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{6} e^{3} x^{5} + 10 \, {\left (7 \, a^{3} b^{2} c^{4} + a^{4} b c\right )} d^{5} e^{3} x^{4} + 4 \, {\left (14 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{4} e^{3} x^{3} + {\left (28 \, a^{3} b^{2} c^{6} + 20 \, a^{4} b c^{3} + a^{5}\right )} d^{3} e^{3} x^{2} + 2 \, {\left (4 \, a^{3} b^{2} c^{7} + 5 \, a^{4} b c^{4} + a^{5} c\right )} d^{2} e^{3} x + {\left (a^{3} b^{2} c^{8} + 2 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d e^{3}\right )}} \]
-1/54*(60*b^2*d^6*x^6 + 360*b^2*c*d^5*x^5 + 900*b^2*c^2*d^4*x^4 + 60*b^2*c ^6 + 48*(25*b^2*c^3 + 2*a*b)*d^3*x^3 + 96*a*b*c^3 + 36*(25*b^2*c^4 + 8*a*b *c)*d^2*x^2 + 72*(5*b^2*c^5 + 4*a*b*c^2)*d*x - 40*sqrt(3)*(b^2*d^8*x^8 + 8 *b^2*c*d^7*x^7 + 28*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x^5 + b^2*c ^8 + 10*(7*b^2*c^4 + a*b*c)*d^4*x^4 + 2*a*b*c^5 + 4*(14*b^2*c^5 + 5*a*b*c^ 2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a^2)*d^2*x^2 + a^2*c^2 + 2*(4*b^2* c^7 + 5*a*b*c^4 + a^2*c)*d*x)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*(a*d* x + a*c)*(-b^2/a^2)^(2/3) - sqrt(3)*b)/b) + 20*(b^2*d^8*x^8 + 8*b^2*c*d^7* x^7 + 28*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x^5 + b^2*c^8 + 10*(7* b^2*c^4 + a*b*c)*d^4*x^4 + 2*a*b*c^5 + 4*(14*b^2*c^5 + 5*a*b*c^2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a^2)*d^2*x^2 + a^2*c^2 + 2*(4*b^2*c^7 + 5*a*b *c^4 + a^2*c)*d*x)*(-b^2/a^2)^(1/3)*log(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^ 2 + a^2*(-b^2/a^2)^(2/3) + (a*b*d*x + a*b*c)*(-b^2/a^2)^(1/3)) - 40*(b^2*d ^8*x^8 + 8*b^2*c*d^7*x^7 + 28*b^2*c^2*d^6*x^6 + 2*(28*b^2*c^3 + a*b)*d^5*x ^5 + b^2*c^8 + 10*(7*b^2*c^4 + a*b*c)*d^4*x^4 + 2*a*b*c^5 + 4*(14*b^2*c^5 + 5*a*b*c^2)*d^3*x^3 + (28*b^2*c^6 + 20*a*b*c^3 + a^2)*d^2*x^2 + a^2*c^2 + 2*(4*b^2*c^7 + 5*a*b*c^4 + a^2*c)*d*x)*(-b^2/a^2)^(1/3)*log(b*d*x + b*c - a*(-b^2/a^2)^(1/3)) + 27*a^2)/(a^3*b^2*d^9*e^3*x^8 + 8*a^3*b^2*c*d^8*e^3* x^7 + 28*a^3*b^2*c^2*d^7*e^3*x^6 + 2*(28*a^3*b^2*c^3 + a^4*b)*d^6*e^3*x^5 + 10*(7*a^3*b^2*c^4 + a^4*b*c)*d^5*e^3*x^4 + 4*(14*a^3*b^2*c^5 + 5*a^4*...
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (219) = 438\).
Time = 2.67 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.11 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {- 9 a^{2} - 32 a b c^{3} - 20 b^{2} c^{6} - 300 b^{2} c^{2} d^{4} x^{4} - 120 b^{2} c d^{5} x^{5} - 20 b^{2} d^{6} x^{6} + x^{3} \left (- 32 a b d^{3} - 400 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 96 a b c d^{2} - 300 b^{2} c^{4} d^{2}\right ) + x \left (- 96 a b c^{2} d - 120 b^{2} c^{5} d\right )}{18 a^{5} c^{2} d e^{3} + 36 a^{4} b c^{5} d e^{3} + 18 a^{3} b^{2} c^{8} d e^{3} + 504 a^{3} b^{2} c^{2} d^{7} e^{3} x^{6} + 144 a^{3} b^{2} c d^{8} e^{3} x^{7} + 18 a^{3} b^{2} d^{9} e^{3} x^{8} + x^{5} \cdot \left (36 a^{4} b d^{6} e^{3} + 1008 a^{3} b^{2} c^{3} d^{6} e^{3}\right ) + x^{4} \cdot \left (180 a^{4} b c d^{5} e^{3} + 1260 a^{3} b^{2} c^{4} d^{5} e^{3}\right ) + x^{3} \cdot \left (360 a^{4} b c^{2} d^{4} e^{3} + 1008 a^{3} b^{2} c^{5} d^{4} e^{3}\right ) + x^{2} \cdot \left (18 a^{5} d^{3} e^{3} + 360 a^{4} b c^{3} d^{3} e^{3} + 504 a^{3} b^{2} c^{6} d^{3} e^{3}\right ) + x \left (36 a^{5} c d^{2} e^{3} + 180 a^{4} b c^{4} d^{2} e^{3} + 144 a^{3} b^{2} c^{7} d^{2} e^{3}\right )} + \frac {\operatorname {RootSum} {\left (19683 t^{3} a^{11} + 8000 b^{2}, \left ( t \mapsto t \log {\left (x + \frac {- 27 t a^{4} + 20 b c}{20 b d} \right )} \right )\right )}}{d e^{3}} \]
(-9*a**2 - 32*a*b*c**3 - 20*b**2*c**6 - 300*b**2*c**2*d**4*x**4 - 120*b**2 *c*d**5*x**5 - 20*b**2*d**6*x**6 + x**3*(-32*a*b*d**3 - 400*b**2*c**3*d**3 ) + x**2*(-96*a*b*c*d**2 - 300*b**2*c**4*d**2) + x*(-96*a*b*c**2*d - 120*b **2*c**5*d))/(18*a**5*c**2*d*e**3 + 36*a**4*b*c**5*d*e**3 + 18*a**3*b**2*c **8*d*e**3 + 504*a**3*b**2*c**2*d**7*e**3*x**6 + 144*a**3*b**2*c*d**8*e**3 *x**7 + 18*a**3*b**2*d**9*e**3*x**8 + x**5*(36*a**4*b*d**6*e**3 + 1008*a** 3*b**2*c**3*d**6*e**3) + x**4*(180*a**4*b*c*d**5*e**3 + 1260*a**3*b**2*c** 4*d**5*e**3) + x**3*(360*a**4*b*c**2*d**4*e**3 + 1008*a**3*b**2*c**5*d**4* e**3) + x**2*(18*a**5*d**3*e**3 + 360*a**4*b*c**3*d**3*e**3 + 504*a**3*b** 2*c**6*d**3*e**3) + x*(36*a**5*c*d**2*e**3 + 180*a**4*b*c**4*d**2*e**3 + 1 44*a**3*b**2*c**7*d**2*e**3)) + RootSum(19683*_t**3*a**11 + 8000*b**2, Lam bda(_t, _t*log(x + (-27*_t*a**4 + 20*b*c)/(20*b*d))))/(d*e**3)
\[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{3} {\left (d e x + c e\right )}^{3}} \,d x } \]
-1/18*(20*b^2*d^6*x^6 + 120*b^2*c*d^5*x^5 + 300*b^2*c^2*d^4*x^4 + 20*b^2*c ^6 + 16*(25*b^2*c^3 + 2*a*b)*d^3*x^3 + 32*a*b*c^3 + 12*(25*b^2*c^4 + 8*a*b *c)*d^2*x^2 + 24*(5*b^2*c^5 + 4*a*b*c^2)*d*x + 9*a^2)/(a^3*b^2*d^9*e^3*x^8 + 8*a^3*b^2*c*d^8*e^3*x^7 + 28*a^3*b^2*c^2*d^7*e^3*x^6 + 2*(28*a^3*b^2*c^ 3 + a^4*b)*d^6*e^3*x^5 + 10*(7*a^3*b^2*c^4 + a^4*b*c)*d^5*e^3*x^4 + 4*(14* a^3*b^2*c^5 + 5*a^4*b*c^2)*d^4*e^3*x^3 + (28*a^3*b^2*c^6 + 20*a^4*b*c^3 + a^5)*d^3*e^3*x^2 + 2*(4*a^3*b^2*c^7 + 5*a^4*b*c^4 + a^5*c)*d^2*e^3*x + (a^ 3*b^2*c^8 + 2*a^4*b*c^5 + a^5*c^2)*d*e^3) - 20/9*b*integrate(1/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/(a^3*e^3)
Time = 0.32 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {10 \, {\left (2 \, \sqrt {3} \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | -b d x - b c + \left (-a b^{2}\right )^{\frac {1}{3}} \right |}\right )\right )}}{27 \, a^{3} e^{3}} - \frac {20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 400 \, b^{2} c^{3} d^{3} x^{3} + 300 \, b^{2} c^{4} d^{2} x^{2} + 120 \, b^{2} c^{5} d x + 20 \, b^{2} c^{6} + 32 \, a b d^{3} x^{3} + 96 \, a b c d^{2} x^{2} + 96 \, a b c^{2} d x + 32 \, a b c^{3} + 9 \, a^{2}}{18 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a d x + a c\right )}^{2} a^{3} d e^{3}} \]
10/27*(2*sqrt(3)*(-b^2/(a^2*d^3))^(1/3)*arctan(-(b*d*x + b*c - (-a*b^2)^(1 /3))/(sqrt(3)*b*d*x + sqrt(3)*b*c + sqrt(3)*(-a*b^2)^(1/3))) - (-b^2/(a^2* d^3))^(1/3)*log(4*(sqrt(3)*b*d*x + sqrt(3)*b*c + sqrt(3)*(-a*b^2)^(1/3))^2 + 4*(b*d*x + b*c - (-a*b^2)^(1/3))^2) + 2*(-b^2/(a^2*d^3))^(1/3)*log(abs( -b*d*x - b*c + (-a*b^2)^(1/3))))/(a^3*e^3) - 1/18*(20*b^2*d^6*x^6 + 120*b^ 2*c*d^5*x^5 + 300*b^2*c^2*d^4*x^4 + 400*b^2*c^3*d^3*x^3 + 300*b^2*c^4*d^2* x^2 + 120*b^2*c^5*d*x + 20*b^2*c^6 + 32*a*b*d^3*x^3 + 96*a*b*c*d^2*x^2 + 9 6*a*b*c^2*d*x + 32*a*b*c^3 + 9*a^2)/((b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2* d^2*x^2 + 4*b*c^3*d*x + b*c^4 + a*d*x + a*c)^2*a^3*d*e^3)
Time = 8.15 (sec) , antiderivative size = 563, normalized size of antiderivative = 2.38 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {20\,b^{2/3}\,\ln \left (a^6\,b^{1/3}\,c-{\left (-a\right )}^{19/3}+a^6\,b^{1/3}\,d\,x\right )}{27\,{\left (-a\right )}^{11/3}\,d\,e^3}-\frac {\frac {9\,a^2+32\,a\,b\,c^3+20\,b^2\,c^6}{18\,a^3\,d}+\frac {2\,x^2\,\left (25\,d\,b^2\,c^4+8\,a\,d\,b\,c\right )}{3\,a^3}+\frac {4\,x\,\left (5\,b^2\,c^5+4\,a\,b\,c^2\right )}{3\,a^3}+\frac {8\,x^3\,\left (25\,b^2\,c^3\,d^2+2\,a\,b\,d^2\right )}{9\,a^3}+\frac {10\,b^2\,d^5\,x^6}{9\,a^3}+\frac {50\,b^2\,c^2\,d^3\,x^4}{3\,a^3}+\frac {20\,b^2\,c\,d^4\,x^5}{3\,a^3}}{x^4\,\left (70\,b^2\,c^4\,d^4\,e^3+10\,a\,b\,c\,d^4\,e^3\right )+x^2\,\left (a^2\,d^2\,e^3+20\,a\,b\,c^3\,d^2\,e^3+28\,b^2\,c^6\,d^2\,e^3\right )+x^3\,\left (56\,b^2\,c^5\,d^3\,e^3+20\,a\,b\,c^2\,d^3\,e^3\right )+x^5\,\left (56\,b^2\,c^3\,d^5\,e^3+2\,a\,b\,d^5\,e^3\right )+x\,\left (2\,d\,a^2\,c\,e^3+10\,d\,a\,b\,c^4\,e^3+8\,d\,b^2\,c^7\,e^3\right )+a^2\,c^2\,e^3+b^2\,c^8\,e^3+b^2\,d^8\,e^3\,x^8+2\,a\,b\,c^5\,e^3+28\,b^2\,c^2\,d^6\,e^3\,x^6+8\,b^2\,c\,d^7\,e^3\,x^7}+\frac {20\,b^{2/3}\,\ln \left ({\left (-a\right )}^{19/3}+2\,a^6\,b^{1/3}\,c+2\,a^6\,b^{1/3}\,d\,x-\sqrt {3}\,{\left (-a\right )}^{19/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,{\left (-a\right )}^{11/3}\,d\,e^3}-\frac {20\,b^{2/3}\,\ln \left ({\left (-a\right )}^{19/3}+2\,a^6\,b^{1/3}\,c+2\,a^6\,b^{1/3}\,d\,x+\sqrt {3}\,{\left (-a\right )}^{19/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,{\left (-a\right )}^{11/3}\,d\,e^3} \]
(20*b^(2/3)*log(a^6*b^(1/3)*c - (-a)^(19/3) + a^6*b^(1/3)*d*x))/(27*(-a)^( 11/3)*d*e^3) - ((9*a^2 + 20*b^2*c^6 + 32*a*b*c^3)/(18*a^3*d) + (2*x^2*(25* b^2*c^4*d + 8*a*b*c*d))/(3*a^3) + (4*x*(5*b^2*c^5 + 4*a*b*c^2))/(3*a^3) + (8*x^3*(25*b^2*c^3*d^2 + 2*a*b*d^2))/(9*a^3) + (10*b^2*d^5*x^6)/(9*a^3) + (50*b^2*c^2*d^3*x^4)/(3*a^3) + (20*b^2*c*d^4*x^5)/(3*a^3))/(x^4*(70*b^2*c^ 4*d^4*e^3 + 10*a*b*c*d^4*e^3) + x^2*(a^2*d^2*e^3 + 28*b^2*c^6*d^2*e^3 + 20 *a*b*c^3*d^2*e^3) + x^3*(56*b^2*c^5*d^3*e^3 + 20*a*b*c^2*d^3*e^3) + x^5*(5 6*b^2*c^3*d^5*e^3 + 2*a*b*d^5*e^3) + x*(8*b^2*c^7*d*e^3 + 2*a^2*c*d*e^3 + 10*a*b*c^4*d*e^3) + a^2*c^2*e^3 + b^2*c^8*e^3 + b^2*d^8*e^3*x^8 + 2*a*b*c^ 5*e^3 + 28*b^2*c^2*d^6*e^3*x^6 + 8*b^2*c*d^7*e^3*x^7) + (20*b^(2/3)*log((- a)^(19/3) - 3^(1/2)*(-a)^(19/3)*1i + 2*a^6*b^(1/3)*c + 2*a^6*b^(1/3)*d*x)* ((3^(1/2)*1i)/2 - 1/2))/(27*(-a)^(11/3)*d*e^3) - (20*b^(2/3)*log((-a)^(19/ 3) + 3^(1/2)*(-a)^(19/3)*1i + 2*a^6*b^(1/3)*c + 2*a^6*b^(1/3)*d*x)*((3^(1/ 2)*1i)/2 + 1/2))/(27*(-a)^(11/3)*d*e^3)