Integrand size = 23, antiderivative size = 190 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}-\frac {\left (\sqrt [3]{b} d+2 \sqrt [3]{a} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} b^{5/3}}+\frac {\left (\sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{5/3}}-\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}} \]
1/3*(-e*x^2-d*x-c)/b/(b*x^3+a)+1/9*(b^(1/3)*d-2*a^(1/3)*e)*ln(a^(1/3)+b^(1 /3)*x)/a^(2/3)/b^(5/3)-1/18*(d-2*a^(1/3)*e/b^(1/3))*ln(a^(2/3)-a^(1/3)*b^( 1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(4/3)-1/9*(b^(1/3)*d+2*a^(1/3)*e)*arctan(1/3 *(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(2/3)/b^(5/3)*3^(1/2)
Time = 0.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^2} \, dx=\frac {-\frac {6 b^{2/3} (c+x (d+e x))}{a+b x^3}-\frac {2 \sqrt {3} \left (\sqrt [3]{b} d+2 \sqrt [3]{a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {2 \left (\sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac {\left (-\sqrt [3]{b} d+2 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}}{18 b^{5/3}} \]
((-6*b^(2/3)*(c + x*(d + e*x)))/(a + b*x^3) - (2*Sqrt[3]*(b^(1/3)*d + 2*a^ (1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*(b^(1/3 )*d - 2*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + ((-(b^(1/3)*d) + 2* a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(18*b^ (5/3))
Time = 0.44 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2363, 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.
| \(\displaystyle \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2363 |
| \(\displaystyle \frac {\int \frac {d+2 e x}{b x^3+a}dx}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {\int \frac {2 \sqrt [3]{a} \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right )-\sqrt [3]{b} \left (\sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\int \frac {2 \sqrt [3]{a} \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right )-\sqrt [3]{b} \left (\sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \sqrt [3]{b} \left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt [3]{b} \left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt [3]{b} \left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} \left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (2 \sqrt [3]{a} e+\sqrt [3]{b} d\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (d-\frac {2 \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 b}-\frac {c+d x+e x^2}{3 b \left (a+b x^3\right )}\) |
-1/3*(c + d*x + e*x^2)/(b*(a + b*x^3)) + (((d - (2*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 + 2*a^ (1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - ((d - (2* 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*b)
3.4.44.3.1 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[((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 = 1.58 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(\frac {-\frac {e \,x^{2}}{3 b}-\frac {d x}{3 b}-\frac {c}{3 b}}{b \,x^{3}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (2 e \textit {\_R} +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{2}}\) | \(67\) |
| default | \(\frac {-\frac {e \,x^{2}}{3 b}-\frac {d x}{3 b}-\frac {c}{3 b}}{b \,x^{3}+a}+\frac {d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+2 e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 b}\) | \(226\) |
(-1/3*e*x^2/b-1/3*d*x/b-1/3*c/b)/(b*x^3+a)+1/9/b^2*sum((2*_R*e+d)/_R^2*ln( x-_R),_R=RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 2077, normalized size of antiderivative = 10.93 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]
-1/36*(12*e*x^2 + 2*(b^2*x^3 + a*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + 8*a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2*b^5))^(1/3) + 4*(1/2)^(2/3)*d *e*(I*sqrt(3) - 1)/(a*b^3*((b*d^3 + 8*a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3) /(a^2*b^5))^(1/3)))*log(1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + 8*a*e^3 )/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2*b^5))^(1/3) + 4*(1/2)^(2/3)*d*e*(I*sq rt(3) - 1)/(a*b^3*((b*d^3 + 8*a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2*b^ 5))^(1/3)))^2*a^2*b^3*e - 1/2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + 8*a*e ^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2*b^5))^(1/3) + 4*(1/2)^(2/3)*d*e*(I* sqrt(3) - 1)/(a*b^3*((b*d^3 + 8*a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2* b^5))^(1/3)))*a*b^2*d^2 + 8*a*d*e^2 + (b*d^3 + 8*a*e^3)*x) + 12*d*x - ((b^ 2*x^3 + a*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + 8*a*e^3)/(a^2*b^5) + ( b*d^3 - 8*a*e^3)/(a^2*b^5))^(1/3) + 4*(1/2)^(2/3)*d*e*(I*sqrt(3) - 1)/(a*b ^3*((b*d^3 + 8*a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2*b^5))^(1/3))) + 3 *sqrt(1/3)*(b^2*x^3 + a*b)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((b*d^3 + 8 *a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2*b^5))^(1/3) + 4*(1/2)^(2/3)*d*e *(I*sqrt(3) - 1)/(a*b^3*((b*d^3 + 8*a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/( a^2*b^5))^(1/3)))^2*a*b^3 + 32*d*e)/(a*b^3)))*log(-1/2*((1/2)^(1/3)*(I*sqr t(3) + 1)*((b*d^3 + 8*a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2*b^5))^(1/3 ) + 4*(1/2)^(2/3)*d*e*(I*sqrt(3) - 1)/(a*b^3*((b*d^3 + 8*a*e^3)/(a^2*b^5) + (b*d^3 - 8*a*e^3)/(a^2*b^5))^(1/3)))^2*a^2*b^3*e + 1/2*((1/2)^(1/3)*(...
Time = 1.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.58 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^2} \, dx=\operatorname {RootSum} {\left (729 t^{3} a^{2} b^{5} + 54 t a b^{2} d e + 8 a e^{3} - b d^{3}, \left ( t \mapsto t \log {\left (x + \frac {162 t^{2} a^{2} b^{3} e + 9 t a b^{2} d^{2} + 8 a d e^{2}}{8 a e^{3} + b d^{3}} \right )} \right )\right )} + \frac {- c - d x - e x^{2}}{3 a b + 3 b^{2} x^{3}} \]
RootSum(729*_t**3*a**2*b**5 + 54*_t*a*b**2*d*e + 8*a*e**3 - b*d**3, Lambda (_t, _t*log(x + (162*_t**2*a**2*b**3*e + 9*_t*a*b**2*d**2 + 8*a*d*e**2)/(8 *a*e**3 + b*d**3)))) + (-c - d*x - e*x**2)/(3*a*b + 3*b**2*x**3)
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {e x^{2} + d x + c}{3 \, {\left (b^{2} x^{3} + a b\right )}} + \frac {\sqrt {3} {\left (2 \, e \left (\frac {a}{b}\right )^{\frac {1}{3}} + d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, e \left (\frac {a}{b}\right )^{\frac {1}{3}} - d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, e \left (\frac {a}{b}\right )^{\frac {1}{3}} - d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/3*(e*x^2 + d*x + c)/(b^2*x^3 + a*b) + 1/9*sqrt(3)*(2*e*(a/b)^(1/3) + d) *arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^2*(a/b)^(2/3)) + 1 /18*(2*e*(a/b)^(1/3) - d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^2*(a/b )^(2/3)) - 1/9*(2*e*(a/b)^(1/3) - d)*log(x + (a/b)^(1/3))/(b^2*(a/b)^(2/3) )
Time = 0.28 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {\sqrt {3} {\left (b d - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b d + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (2 \, e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b} - \frac {e x^{2} + d x + c}{3 \, {\left (b x^{3} + a\right )} b} \]
-1/9*sqrt(3)*(b*d - 2*(-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)*b) - 1/18*(b*d + 2*(-a*b^2)^(1/3)*e)*l og(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*b) - 1/9*(2*e*(-a/ b)^(1/3) + d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b) - 1/3*(e*x^2 + d*x + c)/((b*x^3 + a)*b)
Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 \left (c+d x+e x^2\right )}{\left (a+b x^3\right )^2} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {2\,d\,e+4\,e^2\,x+{\mathrm {root}\left (729\,a^2\,b^5\,z^3+54\,a\,b^2\,d\,e\,z+8\,a\,e^3-b\,d^3,z,k\right )}^2\,a\,b^3\,81+\mathrm {root}\left (729\,a^2\,b^5\,z^3+54\,a\,b^2\,d\,e\,z+8\,a\,e^3-b\,d^3,z,k\right )\,b^2\,d\,x\,9}{b\,9}\right )\,\mathrm {root}\left (729\,a^2\,b^5\,z^3+54\,a\,b^2\,d\,e\,z+8\,a\,e^3-b\,d^3,z,k\right )\right )-\frac {\frac {c}{3\,b}+\frac {e\,x^2}{3\,b}+\frac {d\,x}{3\,b}}{b\,x^3+a} \]
symsum(log((2*d*e + 4*e^2*x + 81*root(729*a^2*b^5*z^3 + 54*a*b^2*d*e*z + 8 *a*e^3 - b*d^3, z, k)^2*a*b^3 + 9*root(729*a^2*b^5*z^3 + 54*a*b^2*d*e*z + 8*a*e^3 - b*d^3, z, k)*b^2*d*x)/(9*b))*root(729*a^2*b^5*z^3 + 54*a*b^2*d*e *z + 8*a*e^3 - b*d^3, z, k), k, 1, 3) - (c/(3*b) + (e*x^2)/(3*b) + (d*x)/( 3*b))/(a + b*x^3)