Integrand size = 17, antiderivative size = 210 \[ \int \frac {x^2}{a+b (c+d x)^3} \, dx=\frac {c \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{2/3} d^3}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3} d^3}-\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{2/3} d^3}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b d^3} \]
1/3*c*(2*a^(1/3)+b^(1/3)*c)*ln(a^(1/3)+b^(1/3)*(d*x+c))/a^(2/3)/b^(2/3)/d^ 3-1/6*c*(2*a^(1/3)+b^(1/3)*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*( d*x+c)^2)/a^(2/3)/b^(2/3)/d^3+1/3*ln(a+b*(d*x+c)^3)/b/d^3+1/3*c*(2*a^(1/3) -b^(1/3)*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))/a^(1/3)*3^(1/2))/a^(2/3 )/b^(2/3)/d^3*3^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.39 \[ \int \frac {x^2}{a+b (c+d x)^3} \, dx=\frac {\text {RootSum}\left [a+b c^3+3 b c^2 d \text {$\#$1}+3 b c d^2 \text {$\#$1}^2+b d^3 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^2}{c^2+2 c d \text {$\#$1}+d^2 \text {$\#$1}^2}\&\right ]}{3 b d} \]
RootSum[a + b*c^3 + 3*b*c^2*d*#1 + 3*b*c*d^2*#1^2 + b*d^3*#1^3 & , (Log[x - #1]*#1^2)/(c^2 + 2*c*d*#1 + d^2*#1^2) & ]/(3*b*d)
Time = 0.47 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {896, 2410, 792, 2399, 16, 25, 27, 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.
| \(\displaystyle \int \frac {x^2}{a+b (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {\int \frac {d^2 x^2}{b (c+d x)^3+a}d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 2410 |
| \(\displaystyle \frac {\int \frac {c^2-2 c (c+d x)}{b (c+d x)^3+a}d(c+d x)+\int \frac {(c+d x)^2}{b (c+d x)^3+a}d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 792 |
| \(\displaystyle \frac {\int \frac {c^2-2 c (c+d x)}{b (c+d x)^3+a}d(c+d x)+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {\int -\frac {c \left (2 \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right )+\sqrt [3]{b} \left (\sqrt [3]{b} c+2 \sqrt [3]{a}\right ) (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \int \frac {1}{\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\int -\frac {c \left (2 \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right )+\sqrt [3]{b} \left (\sqrt [3]{b} c+2 \sqrt [3]{a}\right ) (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {c \left (2 \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right )+\sqrt [3]{b} \left (\sqrt [3]{b} c+2 \sqrt [3]{a}\right ) (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {c \int \frac {2 \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right )+\sqrt [3]{b} \left (\sqrt [3]{b} c+2 \sqrt [3]{a}\right ) (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-\frac {c \left (\frac {3}{2} \sqrt [3]{a} \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {1}{2} \left (\frac {2 \sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {c \left (\frac {3}{2} \sqrt [3]{a} \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {1}{2} \left (\frac {2 \sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {c \left (\frac {3}{2} \sqrt [3]{a} \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {1}{2} \sqrt [3]{b} \left (\frac {2 \sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {c \left (\frac {3 \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \sqrt [3]{b} \left (\frac {2 \sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {c \left (-\frac {1}{2} \sqrt [3]{b} \left (\frac {2 \sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\sqrt {3} \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {c \left (\frac {1}{2} \left (\frac {2 \sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac {\sqrt {3} \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}+\frac {\log \left (a+b (c+d x)^3\right )}{3 b}}{d^3}\) |
((c*(2*a^(1/3) + b^(1/3)*c)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(3*a^(2/3)*b ^(2/3)) - (c*(-((Sqrt[3]*(2*a^(1/3) - b^(1/3)*c)*ArcTan[(1 - (2*b^(1/3)*(c + d*x))/a^(1/3))/Sqrt[3]])/b^(1/3)) + (((2*a^(1/3))/b^(1/3) + c)*Log[a^(2 /3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/2))/(3*a^(2/3)*b^( 1/3)) + Log[a + b*(c + d*x)^3]/(3*b))/d^3
3.2.4.3.1 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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveConten t[a + b*x^n, x]]/(b*n), x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n - 1]
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
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[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numer ator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Simp[(-r)*((B*r - A*s)/(3*a *s)) Int[1/(r + s*x), x], x] + Simp[r/(3*a*s) Int[(r*(B*r + 2*A*s) + s* (B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] & & NeQ[a*B^3 - b*A^3, 0] && PosQ[a/b]
Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x, 2]}, Int[(A + B*x)/(a + b*x^3), x] + Si mp[C Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !RationalQ[ a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.35
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 b d}\) | \(74\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 b d}\) | \(74\) |
1/3/b/d*sum(_R^2/(_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))
Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 4759, normalized size of antiderivative = 22.66 \[ \int \frac {x^2}{a+b (c+d x)^3} \, dx=\text {Too large to display} \]
-1/12*(2*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((2*b*c^3 - a)/(a*b^2*d^6) + 1/(b ^2*d^6))/((b*c^3 - 8*a)*c^3/(a^2*b^2*d^9) + 3*(2*b*c^3 - a)/(a*b^3*d^9) + 2/(b^3*d^9) + (b^2*c^6 + 2*a*b*c^3 + a^2)/(a^2*b^3*d^9))^(1/3) + (1/2)^(1/ 3)*(I*sqrt(3) + 1)*((b*c^3 - 8*a)*c^3/(a^2*b^2*d^9) + 3*(2*b*c^3 - a)/(a*b ^3*d^9) + 2/(b^3*d^9) + (b^2*c^6 + 2*a*b*c^3 + a^2)/(a^2*b^3*d^9))^(1/3) - 2/(b*d^3))*b*d^3*log(-1/2*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((2*b*c^3 - a)/ (a*b^2*d^6) + 1/(b^2*d^6))/((b*c^3 - 8*a)*c^3/(a^2*b^2*d^9) + 3*(2*b*c^3 - a)/(a*b^3*d^9) + 2/(b^3*d^9) + (b^2*c^6 + 2*a*b*c^3 + a^2)/(a^2*b^3*d^9)) ^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*((b*c^3 - 8*a)*c^3/(a^2*b^2*d^9) + 3* (2*b*c^3 - a)/(a*b^3*d^9) + 2/(b^3*d^9) + (b^2*c^6 + 2*a*b*c^3 + a^2)/(a^2 *b^3*d^9))^(1/3) - 2/(b*d^3))^2*a^2*b^2*d^6 + b^2*c^6 - a*b*c^3 - 1/2*(a*b ^2*c^3 + 4*a^2*b)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((2*b*c^3 - a)/(a*b^2*d^ 6) + 1/(b^2*d^6))/((b*c^3 - 8*a)*c^3/(a^2*b^2*d^9) + 3*(2*b*c^3 - a)/(a*b^ 3*d^9) + 2/(b^3*d^9) + (b^2*c^6 + 2*a*b*c^3 + a^2)/(a^2*b^3*d^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*((b*c^3 - 8*a)*c^3/(a^2*b^2*d^9) + 3*(2*b*c^3 - a)/(a*b^3*d^9) + 2/(b^3*d^9) + (b^2*c^6 + 2*a*b*c^3 + a^2)/(a^2*b^3*d^9) )^(1/3) - 2/(b*d^3))*d^3 + (b^2*c^5 - 8*a*b*c^2)*d*x - 2*a^2) - ((2*(1/2)^ (2/3)*(-I*sqrt(3) + 1)*((2*b*c^3 - a)/(a*b^2*d^6) + 1/(b^2*d^6))/((b*c^3 - 8*a)*c^3/(a^2*b^2*d^9) + 3*(2*b*c^3 - a)/(a*b^3*d^9) + 2/(b^3*d^9) + (b^2 *c^6 + 2*a*b*c^3 + a^2)/(a^2*b^3*d^9))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) +...
Time = 0.54 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{a+b (c+d x)^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b^{3} d^{9} - 27 t^{2} a^{2} b^{2} d^{6} + t \left (9 a^{2} b d^{3} - 18 a b^{2} c^{3} d^{3}\right ) - a^{2} - 2 a b c^{3} - b^{2} c^{6}, \left ( t \mapsto t \log {\left (x + \frac {18 t^{2} a^{2} b^{2} d^{6} - 12 t a^{2} b d^{3} - 3 t a b^{2} c^{3} d^{3} + 2 a^{2} + a b c^{3} - b^{2} c^{6}}{8 a b c^{2} d - b^{2} c^{5} d} \right )} \right )\right )} \]
RootSum(27*_t**3*a**2*b**3*d**9 - 27*_t**2*a**2*b**2*d**6 + _t*(9*a**2*b*d **3 - 18*a*b**2*c**3*d**3) - a**2 - 2*a*b*c**3 - b**2*c**6, Lambda(_t, _t* log(x + (18*_t**2*a**2*b**2*d**6 - 12*_t*a**2*b*d**3 - 3*_t*a*b**2*c**3*d* *3 + 2*a**2 + a*b*c**3 - b**2*c**6)/(8*a*b*c**2*d - b**2*c**5*d))))
\[ \int \frac {x^2}{a+b (c+d x)^3} \, dx=\int { \frac {x^{2}}{{\left (d x + c\right )}^{3} b + a} \,d x } \]
\[ \int \frac {x^2}{a+b (c+d x)^3} \, dx=\int { \frac {x^{2}}{{\left (d x + c\right )}^{3} b + a} \,d x } \]
Time = 9.86 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.08 \[ \int \frac {x^2}{a+b (c+d x)^3} \, dx=\sum _{k=1}^3\ln \left (a+b\,c^3-\mathrm {root}\left (27\,a^2\,b^3\,d^9\,z^3-27\,a^2\,b^2\,d^6\,z^2-18\,a\,b^2\,c^3\,d^3\,z+9\,a^2\,b\,d^3\,z-2\,a\,b\,c^3-b^2\,c^6-a^2,z,k\right )\,a\,b\,d^3\,6+3\,b\,c^2\,d\,x+{\mathrm {root}\left (27\,a^2\,b^3\,d^9\,z^3-27\,a^2\,b^2\,d^6\,z^2-18\,a\,b^2\,c^3\,d^3\,z+9\,a^2\,b\,d^3\,z-2\,a\,b\,c^3-b^2\,c^6-a^2,z,k\right )}^2\,a\,b^2\,d^6\,9+\mathrm {root}\left (27\,a^2\,b^3\,d^9\,z^3-27\,a^2\,b^2\,d^6\,z^2-18\,a\,b^2\,c^3\,d^3\,z+9\,a^2\,b\,d^3\,z-2\,a\,b\,c^3-b^2\,c^6-a^2,z,k\right )\,b^2\,c^3\,d^3\,3+\mathrm {root}\left (27\,a^2\,b^3\,d^9\,z^3-27\,a^2\,b^2\,d^6\,z^2-18\,a\,b^2\,c^3\,d^3\,z+9\,a^2\,b\,d^3\,z-2\,a\,b\,c^3-b^2\,c^6-a^2,z,k\right )\,b^2\,c^2\,d^4\,x\,3\right )\,\mathrm {root}\left (27\,a^2\,b^3\,d^9\,z^3-27\,a^2\,b^2\,d^6\,z^2-18\,a\,b^2\,c^3\,d^3\,z+9\,a^2\,b\,d^3\,z-2\,a\,b\,c^3-b^2\,c^6-a^2,z,k\right ) \]
symsum(log(a + b*c^3 - 6*root(27*a^2*b^3*d^9*z^3 - 27*a^2*b^2*d^6*z^2 - 18 *a*b^2*c^3*d^3*z + 9*a^2*b*d^3*z - 2*a*b*c^3 - b^2*c^6 - a^2, z, k)*a*b*d^ 3 + 3*b*c^2*d*x + 9*root(27*a^2*b^3*d^9*z^3 - 27*a^2*b^2*d^6*z^2 - 18*a*b^ 2*c^3*d^3*z + 9*a^2*b*d^3*z - 2*a*b*c^3 - b^2*c^6 - a^2, z, k)^2*a*b^2*d^6 + 3*root(27*a^2*b^3*d^9*z^3 - 27*a^2*b^2*d^6*z^2 - 18*a*b^2*c^3*d^3*z + 9 *a^2*b*d^3*z - 2*a*b*c^3 - b^2*c^6 - a^2, z, k)*b^2*c^3*d^3 + 3*root(27*a^ 2*b^3*d^9*z^3 - 27*a^2*b^2*d^6*z^2 - 18*a*b^2*c^3*d^3*z + 9*a^2*b*d^3*z - 2*a*b*c^3 - b^2*c^6 - a^2, z, k)*b^2*c^2*d^4*x)*root(27*a^2*b^3*d^9*z^3 - 27*a^2*b^2*d^6*z^2 - 18*a*b^2*c^3*d^3*z + 9*a^2*b*d^3*z - 2*a*b*c^3 - b^2* c^6 - a^2, z, k), k, 1, 3)