Integrand size = 20, antiderivative size = 177 \[ \int \frac {c+d x+e x^2}{a+b x^3} \, dx=-\frac {\left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{2/3}}+\frac {\left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{2/3}}-\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {e \log \left (a+b x^3\right )}{3 b} \]
1/3*(b^(1/3)*c-a^(1/3)*d)*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(2/3)-1/6*(c-a^( 1/3)*d/b^(1/3))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(1/3)+ 1/3*e*ln(b*x^3+a)/b-1/3*(b^(1/3)*c+a^(1/3)*d)*arctan(1/3*(a^(1/3)-2*b^(1/3 )*x)/a^(1/3)*3^(1/2))/a^(2/3)/b^(2/3)*3^(1/2)
Time = 0.09 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x+e x^2}{a+b x^3} \, dx=\frac {-2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 a e \log \left (a+b x^3\right )}{6 a b} \]
(-2*Sqrt[3]*a^(1/3)*b^(1/3)*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3) *x)/a^(1/3))/Sqrt[3]] + 2*b^(1/3)*(a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(1 /3) + b^(1/3)*x] - b^(1/3)*(a^(1/3)*b^(1/3)*c - a^(2/3)*d)*Log[a^(2/3) - a ^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*a*e*Log[a + b*x^3])/(6*a*b)
Time = 0.41 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2410, 792, 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.
| \(\displaystyle \int \frac {c+d x+e x^2}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2410 |
| \(\displaystyle \int \frac {c+d x}{b x^3+a}dx+e \int \frac {x^2}{b x^3+a}dx\) |
| \(\Big \downarrow \) 792 |
| \(\displaystyle \int \frac {c+d x}{b x^3+a}dx+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} c+\sqrt [3]{a} d\right )-\sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} c+\sqrt [3]{a} d\right )-\sqrt [3]{b} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \sqrt [3]{b} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt [3]{b} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt [3]{b} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {1}{2} \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac {e \log \left (a+b x^3\right )}{3 b}\) |
((c - (a^(1/3)*d)/b^(1/3))*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*(b^(1/3)*c + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sq rt[3]])/b^(1/3)) - ((c - (a^(1/3)*d)/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)) + (e*Log[a + b*x^3])/(3*b)
3.4.40.3.1 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_.)*(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 = 1.52 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.21
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\) | \(37\) |
| default | \(c \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+d \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {e \ln \left (b \,x^{3}+a \right )}{3 b}\) | \(200\) |
Result contains complex when optimal does not.
Time = 1.21 (sec) , antiderivative size = 4671, normalized size of antiderivative = 26.39 \[ \int \frac {c+d x+e x^2}{a+b x^3} \, dx=\text {Too large to display} \]
-1/12*(2*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(e^2/b^2 - (b*c*d + a*e^2)/(a*b^2 ))/(2*e^3/b^3 - 3*(b*c*d + a*e^2)*e/(a*b^3) + (b*c^3 + a*d^3)/(a^2*b^2) + (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^2*b^3))^(1/3) + (1/2)^(1/3)*( I*sqrt(3) + 1)*(2*e^3/b^3 - 3*(b*c*d + a*e^2)*e/(a*b^3) + (b*c^3 + a*d^3)/ (a^2*b^2) + (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^2*b^3))^(1/3) - 2 *e/b)*b*log(1/4*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(e^2/b^2 - (b*c*d + a*e^2) /(a*b^2))/(2*e^3/b^3 - 3*(b*c*d + a*e^2)*e/(a*b^3) + (b*c^3 + a*d^3)/(a^2* b^2) + (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^2*b^3))^(1/3) + (1/2)^ (1/3)*(I*sqrt(3) + 1)*(2*e^3/b^3 - 3*(b*c*d + a*e^2)*e/(a*b^3) + (b*c^3 + a*d^3)/(a^2*b^2) + (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^2*b^3))^(1 /3) - 2*e/b)^2*a^2*b^2*d + 2*a*b*c*d^2 - a*b*c^2*e + a^2*d*e^2 - 1/2*(a*b^ 2*c^2 - 2*a^2*b*d*e)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(e^2/b^2 - (b*c*d + a *e^2)/(a*b^2))/(2*e^3/b^3 - 3*(b*c*d + a*e^2)*e/(a*b^3) + (b*c^3 + a*d^3)/ (a^2*b^2) + (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^2*b^3))^(1/3) + ( 1/2)^(1/3)*(I*sqrt(3) + 1)*(2*e^3/b^3 - 3*(b*c*d + a*e^2)*e/(a*b^3) + (b*c ^3 + a*d^3)/(a^2*b^2) + (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)*a*b)/(a^2*b^3 ))^(1/3) - 2*e/b) + (b^2*c^3 + a*b*d^3)*x) - ((2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(e^2/b^2 - (b*c*d + a*e^2)/(a*b^2))/(2*e^3/b^3 - 3*(b*c*d + a*e^2)*e/( a*b^3) + (b*c^3 + a*d^3)/(a^2*b^2) + (b^2*c^3 + a^2*e^3 - (d^3 - 3*c*d*e)* a*b)/(a^2*b^3))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*e^3/b^3 - 3*(b*c...
Time = 0.71 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x+e x^2}{a+b x^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b^{3} - 27 t^{2} a^{2} b^{2} e + t \left (9 a^{2} b e^{2} + 9 a b^{2} c d\right ) - a^{2} e^{3} - 3 a b c d e + a b d^{3} - b^{2} c^{3}, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} a^{2} b^{2} d - 6 t a^{2} b d e + 3 t a b^{2} c^{2} + a^{2} d e^{2} - a b c^{2} e + 2 a b c d^{2}}{a b d^{3} + b^{2} c^{3}} \right )} \right )\right )} \]
RootSum(27*_t**3*a**2*b**3 - 27*_t**2*a**2*b**2*e + _t*(9*a**2*b*e**2 + 9* a*b**2*c*d) - a**2*e**3 - 3*a*b*c*d*e + a*b*d**3 - b**2*c**3, Lambda(_t, _ t*log(x + (9*_t**2*a**2*b**2*d - 6*_t*a**2*b*d*e + 3*_t*a*b**2*c**2 + a**2 *d*e**2 - a*b*c**2*e + 2*a*b*c*d**2)/(a*b*d**3 + b**2*c**3))))
Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x+e x^2}{a+b x^3} \, dx=\frac {\sqrt {3} {\left (b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + b c \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b} + \frac {{\left (2 \, e \left (\frac {a}{b}\right )^{\frac {2}{3}} + d \left (\frac {a}{b}\right )^{\frac {1}{3}} - c\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (e \left (\frac {a}{b}\right )^{\frac {2}{3}} - d \left (\frac {a}{b}\right )^{\frac {1}{3}} + c\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/3*sqrt(3)*(b*d*(a/b)^(2/3) + b*c*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b) + 1/6*(2*e*(a/b)^(2/3) + d*(a/b)^(1/3) - c )*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b*(a/b)^(2/3)) + 1/3*(e*(a/b)^(2 /3) - d*(a/b)^(1/3) + c)*log(x + (a/b)^(1/3))/(b*(a/b)^(2/3))
Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x+e x^2}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (b c - \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}}} - \frac {{\left (b c + \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}}} + \frac {e \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b} - \frac {{\left (b d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + b c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b} \]
-1/3*sqrt(3)*(b*c - (-a*b^2)^(1/3)*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/ 3))/(-a/b)^(1/3))/(-a*b^2)^(2/3) - 1/6*(b*c + (-a*b^2)^(1/3)*d)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(-a*b^2)^(2/3) + 1/3*e*log(abs(b*x^3 + a))/ b - 1/3*(b*d*(-a/b)^(1/3) + b*c)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/( a*b)
Time = 0.25 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.55 \[ \int \frac {c+d x+e x^2}{a+b x^3} \, dx=\sum _{k=1}^3\ln \left (x\,\left (b\,d^2-b\,c\,e\right )+\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,a^2\,b^2\,e\,z^2+9\,a\,b^2\,c\,d\,z+9\,a^2\,b\,e^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )\,\left (-6\,a\,b\,e+\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,a^2\,b^2\,e\,z^2+9\,a\,b^2\,c\,d\,z+9\,a^2\,b\,e^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right )\,a\,b^2\,9+3\,b^2\,c\,x\right )+a\,e^2+b\,c\,d\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,a^2\,b^2\,e\,z^2+9\,a\,b^2\,c\,d\,z+9\,a^2\,b\,e^2\,z-3\,a\,b\,c\,d\,e+a\,b\,d^3-a^2\,e^3-b^2\,c^3,z,k\right ) \]
symsum(log(x*(b*d^2 - b*c*e) + root(27*a^2*b^3*z^3 - 27*a^2*b^2*e*z^2 + 9* a*b^2*c*d*z + 9*a^2*b*e^2*z - 3*a*b*c*d*e + a*b*d^3 - a^2*e^3 - b^2*c^3, z , k)*(9*root(27*a^2*b^3*z^3 - 27*a^2*b^2*e*z^2 + 9*a*b^2*c*d*z + 9*a^2*b*e ^2*z - 3*a*b*c*d*e + a*b*d^3 - a^2*e^3 - b^2*c^3, z, k)*a*b^2 - 6*a*b*e + 3*b^2*c*x) + a*e^2 + b*c*d)*root(27*a^2*b^3*z^3 - 27*a^2*b^2*e*z^2 + 9*a*b ^2*c*d*z + 9*a^2*b*e^2*z - 3*a*b*c*d*e + a*b*d^3 - a^2*e^3 - b^2*c^3, z, k ), k, 1, 3)