Integrand size = 22, antiderivative size = 188 \[ \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx=-\frac {(c d-b e) x}{e^2}+\frac {c x^4}{4 e}-\frac {\left (c d^2-b d e+a e^2\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{\sqrt {3} d^{2/3} e^{7/3}}+\frac {\left (c d^2-b d e+a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac {\left (c d^2-b d e+a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}} \]
-(-b*e+c*d)*x/e^2+1/4*c*x^4/e+1/3*(a*e^2-b*d*e+c*d^2)*ln(d^(1/3)+e^(1/3)*x )/d^(2/3)/e^(7/3)-1/6*(a*e^2-b*d*e+c*d^2)*ln(d^(2/3)-d^(1/3)*e^(1/3)*x+e^( 2/3)*x^2)/d^(2/3)/e^(7/3)-1/3*(a*e^2-b*d*e+c*d^2)*arctan(1/3*(d^(1/3)-2*e^ (1/3)*x)/d^(1/3)*3^(1/2))/d^(2/3)/e^(7/3)*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx=\frac {12 \sqrt [3]{e} (-c d+b e) x+3 c e^{4/3} x^4-\frac {4 \sqrt {3} \left (c d^2+e (-b d+a e)\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{2/3}}+\frac {4 \left (c d^2+e (-b d+a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{2/3}}-\frac {2 \left (c d^2+e (-b d+a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{2/3}}}{12 e^{7/3}} \]
(12*e^(1/3)*(-(c*d) + b*e)*x + 3*c*e^(4/3)*x^4 - (4*Sqrt[3]*(c*d^2 + e*(-( b*d) + a*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(2/3) + (4*(c* d^2 + e*(-(b*d) + a*e))*Log[d^(1/3) + e^(1/3)*x])/d^(2/3) - (2*(c*d^2 + e* (-(b*d) + a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(2/3))/( 12*e^(7/3))
Time = 0.36 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.86, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1741, 27, 913, 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.
| \(\displaystyle \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx\) |
| \(\Big \downarrow \) 1741 |
| \(\displaystyle \frac {\int \frac {4 \left (a e-(c d-b e) x^3\right )}{e x^3+d}dx}{4 e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a e-(c d-b e) x^3}{e x^3+d}dx}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \int \frac {1}{e x^3+d}dx}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{e} x+\sqrt [3]{d}}dx}{3 d^{2/3}}\right )}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\int \frac {2 \sqrt [3]{d}-\sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{e} \left (\sqrt [3]{d}-2 \sqrt [3]{e} x\right )}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {\int \frac {\sqrt [3]{e} \left (\sqrt [3]{d}-2 \sqrt [3]{e} x\right )}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\frac {3}{2} \sqrt [3]{d} \int \frac {1}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{\sqrt [3]{e}}-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{2 \sqrt [3]{e}}}{3 d^{2/3}}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}\right )}{e}-\frac {x (c d-b e)}{e}}{e}+\frac {c x^4}{4 e}\) |
(c*x^4)/(4*e) + (-(((c*d - b*e)*x)/e) + ((c*d^2 - b*d*e + a*e^2)*(Log[d^(1 /3) + e^(1/3)*x]/(3*d^(2/3)*e^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*e^(1/3)* x)/d^(1/3))/Sqrt[3]])/e^(1/3)) - Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3) *x^2]/(2*e^(1/3)))/(3*d^(2/3))))/e)/e
3.1.6.3.1 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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Simp[c*x^(n + 1)*((d + e*x^n)^(q + 1)/(e*(n*(q + 2) + 1))) , x] + Simp[1/(e*(n*(q + 2) + 1)) Int[(d + e*x^n)^q*(a*e*(n*(q + 2) + 1) - (c*d*(n + 1) - b*e*(n*(q + 2) + 1))*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.71 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.36
| method | result | size |
| risch | \(\frac {c \,x^{4}}{4 e}+\frac {b x}{e}-\frac {c d x}{e^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 e^{3}}\) | \(67\) |
| default | \(\frac {\frac {1}{4} c \,x^{4} e +b e x -c d x}{e^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}\) | \(133\) |
1/4*c*x^4/e+1/e*b*x-c*d*x/e^2+1/3/e^3*sum((a*e^2-b*d*e+c*d^2)/_R^2*ln(x-_R ),_R=RootOf(_Z^3*e+d))
Time = 0.28 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.47 \[ \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx=\left [\frac {3 \, c d^{2} e^{2} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} \sqrt {-\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}} \log \left (\frac {2 \, d e x^{3} - 3 \, \left (d^{2} e\right )^{\frac {1}{3}} d x - d^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, d e x^{2} + \left (d^{2} e\right )^{\frac {2}{3}} x - \left (d^{2} e\right )^{\frac {1}{3}} d\right )} \sqrt {-\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}}}{e x^{3} + d}\right ) - 2 \, {\left (c d^{2} - b d e + a e^{2}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x^{2} - \left (d^{2} e\right )^{\frac {2}{3}} x + \left (d^{2} e\right )^{\frac {1}{3}} d\right ) + 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x + \left (d^{2} e\right )^{\frac {2}{3}}\right ) - 12 \, {\left (c d^{3} e - b d^{2} e^{2}\right )} x}{12 \, d^{2} e^{3}}, \frac {3 \, c d^{2} e^{2} x^{4} + 12 \, \sqrt {\frac {1}{3}} {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} \sqrt {\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (d^{2} e\right )^{\frac {2}{3}} x - \left (d^{2} e\right )^{\frac {1}{3}} d\right )} \sqrt {\frac {\left (d^{2} e\right )^{\frac {1}{3}}}{e}}}{d^{2}}\right ) - 2 \, {\left (c d^{2} - b d e + a e^{2}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x^{2} - \left (d^{2} e\right )^{\frac {2}{3}} x + \left (d^{2} e\right )^{\frac {1}{3}} d\right ) + 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} \left (d^{2} e\right )^{\frac {2}{3}} \log \left (d e x + \left (d^{2} e\right )^{\frac {2}{3}}\right ) - 12 \, {\left (c d^{3} e - b d^{2} e^{2}\right )} x}{12 \, d^{2} e^{3}}\right ] \]
[1/12*(3*c*d^2*e^2*x^4 + 6*sqrt(1/3)*(c*d^3*e - b*d^2*e^2 + a*d*e^3)*sqrt( -(d^2*e)^(1/3)/e)*log((2*d*e*x^3 - 3*(d^2*e)^(1/3)*d*x - d^2 + 3*sqrt(1/3) *(2*d*e*x^2 + (d^2*e)^(2/3)*x - (d^2*e)^(1/3)*d)*sqrt(-(d^2*e)^(1/3)/e))/( e*x^3 + d)) - 2*(c*d^2 - b*d*e + a*e^2)*(d^2*e)^(2/3)*log(d*e*x^2 - (d^2*e )^(2/3)*x + (d^2*e)^(1/3)*d) + 4*(c*d^2 - b*d*e + a*e^2)*(d^2*e)^(2/3)*log (d*e*x + (d^2*e)^(2/3)) - 12*(c*d^3*e - b*d^2*e^2)*x)/(d^2*e^3), 1/12*(3*c *d^2*e^2*x^4 + 12*sqrt(1/3)*(c*d^3*e - b*d^2*e^2 + a*d*e^3)*sqrt((d^2*e)^( 1/3)/e)*arctan(sqrt(1/3)*(2*(d^2*e)^(2/3)*x - (d^2*e)^(1/3)*d)*sqrt((d^2*e )^(1/3)/e)/d^2) - 2*(c*d^2 - b*d*e + a*e^2)*(d^2*e)^(2/3)*log(d*e*x^2 - (d ^2*e)^(2/3)*x + (d^2*e)^(1/3)*d) + 4*(c*d^2 - b*d*e + a*e^2)*(d^2*e)^(2/3) *log(d*e*x + (d^2*e)^(2/3)) - 12*(c*d^3*e - b*d^2*e^2)*x)/(d^2*e^3)]
Time = 0.42 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx=\frac {c x^{4}}{4 e} + x \left (\frac {b}{e} - \frac {c d}{e^{2}}\right ) + \operatorname {RootSum} {\left (27 t^{3} d^{2} e^{7} - a^{3} e^{6} + 3 a^{2} b d e^{5} - 3 a^{2} c d^{2} e^{4} - 3 a b^{2} d^{2} e^{4} + 6 a b c d^{3} e^{3} - 3 a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} - 3 b^{2} c d^{4} e^{2} + 3 b c^{2} d^{5} e - c^{3} d^{6}, \left ( t \mapsto t \log {\left (\frac {3 t d e^{2}}{a e^{2} - b d e + c d^{2}} + x \right )} \right )\right )} \]
c*x**4/(4*e) + x*(b/e - c*d/e**2) + RootSum(27*_t**3*d**2*e**7 - a**3*e**6 + 3*a**2*b*d*e**5 - 3*a**2*c*d**2*e**4 - 3*a*b**2*d**2*e**4 + 6*a*b*c*d** 3*e**3 - 3*a*c**2*d**4*e**2 + b**3*d**3*e**3 - 3*b**2*c*d**4*e**2 + 3*b*c* *2*d**5*e - c**3*d**6, Lambda(_t, _t*log(3*_t*d*e**2/(a*e**2 - b*d*e + c*d **2) + x)))
Exception generated. \[ \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.30 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.02 \[ \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx=-\frac {\sqrt {3} {\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-d e^{2}\right )^{\frac {2}{3}} e} - \frac {{\left (c d^{2} - b d e + a e^{2}\right )} \log \left (x^{2} + x \left (-\frac {d}{e}\right )^{\frac {1}{3}} + \left (-\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 \, \left (-d e^{2}\right )^{\frac {2}{3}} e} - \frac {{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} \left (-\frac {d}{e}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {d}{e}\right )^{\frac {1}{3}} \right |}\right )}{3 \, d e^{4}} + \frac {c e^{3} x^{4} - 4 \, c d e^{2} x + 4 \, b e^{3} x}{4 \, e^{4}} \]
-1/3*sqrt(3)*(c*d^2 - b*d*e + a*e^2)*arctan(1/3*sqrt(3)*(2*x + (-d/e)^(1/3 ))/(-d/e)^(1/3))/((-d*e^2)^(2/3)*e) - 1/6*(c*d^2 - b*d*e + a*e^2)*log(x^2 + x*(-d/e)^(1/3) + (-d/e)^(2/3))/((-d*e^2)^(2/3)*e) - 1/3*(c*d^2*e^2 - b*d *e^3 + a*e^4)*(-d/e)^(1/3)*log(abs(x - (-d/e)^(1/3)))/(d*e^4) + 1/4*(c*e^3 *x^4 - 4*c*d*e^2*x + 4*b*e^3*x)/e^4
Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^3+c x^6}{d+e x^3} \, dx=x\,\left (\frac {b}{e}-\frac {c\,d}{e^2}\right )+\frac {c\,x^4}{4\,e}+\frac {\ln \left (e^{1/3}\,x+d^{1/3}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,d^{2/3}\,e^{7/3}}+\frac {\ln \left (2\,e^{1/3}\,x-d^{1/3}+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,d^{2/3}\,e^{7/3}}-\frac {\ln \left (d^{1/3}-2\,e^{1/3}\,x+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,d^{2/3}\,e^{7/3}} \]
x*(b/e - (c*d)/e^2) + (c*x^4)/(4*e) + (log(e^(1/3)*x + d^(1/3))*(a*e^2 + c *d^2 - b*d*e))/(3*d^(2/3)*e^(7/3)) + (log(3^(1/2)*d^(1/3)*1i + 2*e^(1/3)*x - d^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(a*e^2 + c*d^2 - b*d*e))/(3*d^(2/3)*e^( 7/3)) - (log(3^(1/2)*d^(1/3)*1i - 2*e^(1/3)*x + d^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(a*e^2 + c*d^2 - b*d*e))/(3*d^(2/3)*e^(7/3))