Integrand size = 23, antiderivative size = 215 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}+\frac {x (d+2 e x)}{18 a b \left (a+b x^3\right )}-\frac {\left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{5/3}}-\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}} \]
1/6*(-e*x^2-d*x-c)/b/(b*x^3+a)^2+1/18*x*(2*e*x+d)/a/b/(b*x^3+a)+1/27*(b^(1 /3)*d-a^(1/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(5/3)-1/54*(d-a^(1/3)*e/b ^(1/3))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/b^(4/3)-1/27*(b^ (1/3)*d+a^(1/3)*e)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(5/ 3)/b^(5/3)*3^(1/2)
Time = 0.19 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {3 b^{2/3} x (d+2 e x)}{a \left (a+b x^3\right )}-\frac {9 b^{2/3} (c+x (d+e x))}{\left (a+b x^3\right )^2}-\frac {2 \sqrt {3} \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {\left (-\sqrt [3]{b} d+\sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{54 b^{5/3}} \]
((3*b^(2/3)*x*(d + 2*e*x))/(a*(a + b*x^3)) - (9*b^(2/3)*(c + x*(d + e*x))) /(a + b*x^3)^2 - (2*Sqrt[3]*(b^(1/3)*d + a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3) *x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*(b^(1/3)*d - a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + ((-(b^(1/3)*d) + a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^ (1/3)*x + b^(2/3)*x^2])/a^(5/3))/(54*b^(5/3))
Time = 0.51 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2363, 2394, 27, 2399, 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.
| \(\displaystyle \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2363 |
| \(\displaystyle \frac {\int \frac {d+2 e x}{\left (b x^3+a\right )^2}dx}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}-\frac {\int -\frac {2 (d+e x)}{b x^3+a}dx}{3 a}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {d+e x}{b x^3+a}dx}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right )-\sqrt [3]{b} \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} d+\sqrt [3]{a} e\right )-\sqrt [3]{b} \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \sqrt [3]{b} \left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {1}{2} \sqrt [3]{b} \left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \left (\frac {\frac {1}{2} \sqrt [3]{b} \left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 \left (\frac {-\frac {1}{2} \left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x (d+2 e x)}{3 a \left (a+b x^3\right )}}{6 b}-\frac {c+d x+e x^2}{6 b \left (a+b x^3\right )^2}\) |
-1/6*(c + d*x + e*x^2)/(b*(a + b*x^3)^2) + ((x*(d + 2*e*x))/(3*a*(a + b*x^ 3)) + (2*(((d - (a^(1/3)*e)/b^(1/3))*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)* b^(1/3)) + (-((Sqrt[3]*(b^(1/3)*d + a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a ^(1/3))/Sqrt[3]])/b^(1/3)) - ((d - (a^(1/3)*e)/b^(1/3))*Log[a^(2/3) - a^(1 /3)*b^(1/3)*x + b^(2/3)*x^2])/2)/(3*a^(2/3)*b^(1/3))))/(3*a))/(6*b)
3.4.51.3.1 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_) + (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[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Pq*(( a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1)) Int[D[Pq, x] *(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x] && E qQ[m - n + 1, 0] && LtQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.54 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\frac {e \,x^{5}}{9 a}+\frac {d \,x^{4}}{18 a}-\frac {e \,x^{2}}{18 b}-\frac {d x}{9 b}-\frac {c}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (e \textit {\_R} +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 a \,b^{2}}\) | \(87\) |
| default | \(\frac {\frac {e \,x^{5}}{9 a}+\frac {d \,x^{4}}{18 a}-\frac {e \,x^{2}}{18 b}-\frac {d x}{9 b}-\frac {c}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 b a}\) | \(246\) |
(1/9/a*e*x^5+1/18*d/a*x^4-1/18*e*x^2/b-1/9*d*x/b-1/6*c/b)/(b*x^3+a)^2+1/27 /a/b^2*sum((_R*e+d)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 2163, normalized size of antiderivative = 10.06 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
1/108*(12*b*e*x^5 + 6*b*d*x^4 - 6*a*e*x^2 - 12*a*d*x - 2*(a*b^3*x^6 + 2*a^ 2*b^2*x^3 + a^3*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/( a^3*b^3*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3)/(a^5*b^5))^(1/3)))*lo g(1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a *e^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^3*b^3*((b*d ^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3)/(a^5*b^5))^(1/3)))^2*a^4*b^3*e - 1 /2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^ 3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^3*b^3*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3)/(a^5*b^5))^(1/3)))*a^2*b^2*d^2 + 2*a* d*e^2 + (b*d^3 + a*e^3)*x) - 18*a*c + ((a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b) *((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3) /(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^3*b^3*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3)/(a^5*b^5))^(1/3))) + 3*sqrt(1/3)*(a*b^3 *x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*d*e* (-I*sqrt(3) + 1)/(a^3*b^3*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3)/(a^ 5*b^5))^(1/3)))^2*a^3*b^3 + 16*d*e)/(a^3*b^3)))*log(-1/4*((1/2)^(1/3)*(I*s qrt(3) + 1)*((b*d^3 + a*e^3)/(a^5*b^5) + (b*d^3 - a*e^3)/(a^5*b^5))^(1/3) - 2*(1/2)^(2/3)*d*e*(-I*sqrt(3) + 1)/(a^3*b^3*((b*d^3 + a*e^3)/(a^5*b^5...
Time = 4.67 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.69 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\operatorname {RootSum} {\left (19683 t^{3} a^{5} b^{5} + 81 t a^{2} b^{2} d e + a e^{3} - b d^{3}, \left ( t \mapsto t \log {\left (x + \frac {729 t^{2} a^{4} b^{3} e + 27 t a^{2} b^{2} d^{2} + 2 a d e^{2}}{a e^{3} + b d^{3}} \right )} \right )\right )} + \frac {- 3 a c - 2 a d x - a e x^{2} + b d x^{4} + 2 b e x^{5}}{18 a^{3} b + 36 a^{2} b^{2} x^{3} + 18 a b^{3} x^{6}} \]
RootSum(19683*_t**3*a**5*b**5 + 81*_t*a**2*b**2*d*e + a*e**3 - b*d**3, Lam bda(_t, _t*log(x + (729*_t**2*a**4*b**3*e + 27*_t*a**2*b**2*d**2 + 2*a*d*e **2)/(a*e**3 + b*d**3)))) + (-3*a*c - 2*a*d*x - a*e*x**2 + b*d*x**4 + 2*b* e*x**5)/(18*a**3*b + 36*a**2*b**2*x**3 + 18*a*b**3*x**6)
Time = 0.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\frac {2 \, b e x^{5} + b d x^{4} - a e x^{2} - 2 \, a d x - 3 \, a c}{18 \, {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )}} + \frac {\sqrt {3} {\left (e \left (\frac {a}{b}\right )^{\frac {1}{3}} + d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (e \left (\frac {a}{b}\right )^{\frac {1}{3}} - d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (e \left (\frac {a}{b}\right )^{\frac {1}{3}} - d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/18*(2*b*e*x^5 + b*d*x^4 - a*e*x^2 - 2*a*d*x - 3*a*c)/(a*b^3*x^6 + 2*a^2* b^2*x^3 + a^3*b) + 1/27*sqrt(3)*(e*(a/b)^(1/3) + d)*arctan(1/3*sqrt(3)*(2* x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^2*(a/b)^(2/3)) + 1/54*(e*(a/b)^(1/3) - d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^2*(a/b)^(2/3)) - 1/27*(e*(a /b)^(1/3) - d)*log(x + (a/b)^(1/3))/(a*b^2*(a/b)^(2/3))
Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {\sqrt {3} {\left (b d - \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (b d + \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b} + \frac {2 \, b e x^{5} + b d x^{4} - a e x^{2} - 2 \, a d x - 3 \, a c}{18 \, {\left (b x^{3} + a\right )}^{2} a b} \]
-1/27*sqrt(3)*(b*d - (-a*b^2)^(1/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1 /3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a*b) - 1/54*(b*d + (-a*b^2)^(1/3)*e)*lo g(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a*b) - 1/27*(e*(-a/ b)^(1/3) + d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b) + 1/18*(2*b* e*x^5 + b*d*x^4 - a*e*x^2 - 2*a*d*x - 3*a*c)/((b*x^3 + a)^2*a*b)
Time = 0.23 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^3} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {d\,e+e^2\,x+{\mathrm {root}\left (19683\,a^5\,b^5\,z^3+81\,a^2\,b^2\,d\,e\,z+a\,e^3-b\,d^3,z,k\right )}^2\,a^3\,b^3\,729+\mathrm {root}\left (19683\,a^5\,b^5\,z^3+81\,a^2\,b^2\,d\,e\,z+a\,e^3-b\,d^3,z,k\right )\,a\,b^2\,d\,x\,27}{a^2\,b\,81}\right )\,\mathrm {root}\left (19683\,a^5\,b^5\,z^3+81\,a^2\,b^2\,d\,e\,z+a\,e^3-b\,d^3,z,k\right )\right )-\frac {\frac {c}{6\,b}-\frac {d\,x^4}{18\,a}-\frac {e\,x^5}{9\,a}+\frac {e\,x^2}{18\,b}+\frac {d\,x}{9\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \]
symsum(log((d*e + e^2*x + 729*root(19683*a^5*b^5*z^3 + 81*a^2*b^2*d*e*z + a*e^3 - b*d^3, z, k)^2*a^3*b^3 + 27*root(19683*a^5*b^5*z^3 + 81*a^2*b^2*d* e*z + a*e^3 - b*d^3, z, k)*a*b^2*d*x)/(81*a^2*b))*root(19683*a^5*b^5*z^3 + 81*a^2*b^2*d*e*z + a*e^3 - b*d^3, z, k), k, 1, 3) - (c/(6*b) - (d*x^4)/(1 8*a) - (e*x^5)/(9*a) + (e*x^2)/(18*b) + (d*x)/(9*b))/(a^2 + b^2*x^6 + 2*a* b*x^3)