Integrand size = 18, antiderivative size = 323 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \arctan \left (\frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} a^{5/6} c^{2/3}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}} \] Output:
-1/6*(d-a^(1/2)*e/c^(1/2))*arctan(1/3*(a^(1/6)-2*c^(1/6)*x)*3^(1/2)/a^(1/6 ))*3^(1/2)/a^(5/6)/c^(1/6)+1/6*(c^(1/2)*d+a^(1/2)*e)*arctan(1/3*(a^(1/6)+2 *c^(1/6)*x)*3^(1/2)/a^(1/6))*3^(1/2)/a^(5/6)/c^(2/3)-1/6*(c^(1/2)*d+a^(1/2 )*e)*ln(a^(1/6)-c^(1/6)*x)/a^(5/6)/c^(2/3)+1/6*(d-a^(1/2)*e/c^(1/2))*ln(a^ (1/6)+c^(1/6)*x)/a^(5/6)/c^(1/6)-1/12*(d-a^(1/2)*e/c^(1/2))*ln(a^(1/3)-a^( 1/6)*c^(1/6)*x+c^(1/3)*x^2)/a^(5/6)/c^(1/6)+1/12*(c^(1/2)*d+a^(1/2)*e)*ln( a^(1/3)+a^(1/6)*c^(1/6)*x+c^(1/3)*x^2)/a^(5/6)/c^(2/3)
Time = 0.12 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.04 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\frac {-2 \sqrt {3} \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )-2 \sqrt {c} d \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )-2 \sqrt {a} e \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )+2 \sqrt {c} d \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )-2 \sqrt {a} e \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )-\sqrt {c} d \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {a} e \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {c} d \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {a} e \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}} \] Input:
Integrate[(d + e*x^3)/(a - c*x^6),x]
Output:
(-2*Sqrt[3]*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(1 - (2*c^(1/6)*x)/a^(1/6))/Sqr t[3]] + 2*Sqrt[3]*(Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(1 + (2*c^(1/6)*x)/a^(1/6 ))/Sqrt[3]] - 2*Sqrt[c]*d*Log[a^(1/6) - c^(1/6)*x] - 2*Sqrt[a]*e*Log[a^(1/ 6) - c^(1/6)*x] + 2*Sqrt[c]*d*Log[a^(1/6) + c^(1/6)*x] - 2*Sqrt[a]*e*Log[a ^(1/6) + c^(1/6)*x] - Sqrt[c]*d*Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)* x^2] + Sqrt[a]*e*Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] + Sqrt[c]* d*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] + Sqrt[a]*e*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3))
Time = 0.46 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1747, 750, 16, 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.
| \(\displaystyle \int \frac {d+e x^3}{a-c x^6} \, dx\) |
| \(\Big \downarrow \) 1747 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \int \frac {1}{a-\sqrt {a} \sqrt {c} x^3}dx+\frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a} \sqrt {c} x^3+a}dx\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a}-\sqrt [6]{c} x\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\int \frac {\sqrt [6]{a} \left (\sqrt [6]{c} x+2 \sqrt [6]{a}\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x}dx}{3 a^{2/3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a}-\sqrt [6]{c} x\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\int \frac {\sqrt [6]{a} \left (\sqrt [6]{c} x+2 \sqrt [6]{a}\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 a^{2/3}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {2 \sqrt [6]{a}-\sqrt [6]{c} x}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+2 \sqrt [6]{a}}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{a} \sqrt [6]{c} \left (\sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{a} \sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{a}\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [6]{c} \left (\sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{a} \sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{a}\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [3]{a}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {\int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [3]{a}}+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}{\sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {\int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}+\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}-\frac {\log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}+\frac {\log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )\) |
Input:
Int[(d + e*x^3)/(a - c*x^6),x]
Output:
((d - (Sqrt[a]*e)/Sqrt[c])*(Log[a^(1/6) + c^(1/6)*x]/(3*a^(5/6)*c^(1/6)) + (-((Sqrt[3]*ArcTan[(1 - (2*c^(1/6)*x)/a^(1/6))/Sqrt[3]])/(a^(1/3)*c^(1/6) )) - Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2]/(2*a^(1/3)*c^(1/6)))/( 3*Sqrt[a])))/2 + ((d + (Sqrt[a]*e)/Sqrt[c])*(-1/3*Log[a^(1/6) - c^(1/6)*x] /(a^(5/6)*c^(1/6)) + ((Sqrt[3]*ArcTan[(1 + (2*c^(1/6)*x)/a^(1/6))/Sqrt[3]] )/(a^(1/3)*c^(1/6)) + Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2]/(2*a^ (1/3)*c^(1/6)))/(3*Sqrt[a])))/2
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_) + (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_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{ q = Rt[-a/c, 2]}, Simp[(d + e*q)/2 Int[1/(a + c*q*x^n), x], x] + Simp[(d - e*q)/2 Int[1/(a - c*q*x^n), x], x]] /; FreeQ[{a, c, d, e, n}, x] && EqQ [n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NegQ[a*c] && IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.11
| method | result | size |
| risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c -a \right )}{\sum }\frac {\left (\textit {\_R}^{3} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c}\) | \(36\) |
| default | \(\frac {e \left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {e \left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 a}+\frac {d \left (\frac {a}{c}\right )^{\frac {1}{6}} \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {d \left (\frac {a}{c}\right )^{\frac {1}{6}} \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 a}-\frac {\ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) e}{6 c \left (\frac {a}{c}\right )^{\frac {1}{3}}}+\frac {\ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) d}{6 c \left (\frac {a}{c}\right )^{\frac {5}{6}}}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (\left (\frac {a}{c}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \ln \left (\left (\frac {a}{c}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) d}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, e \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \sqrt {3}\, d \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 a}-\frac {\ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) e}{6 c \left (\frac {a}{c}\right )^{\frac {1}{3}}}-\frac {\ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) d}{6 c \left (\frac {a}{c}\right )^{\frac {5}{6}}}\) | \(386\) |
Input:
int((e*x^3+d)/(-c*x^6+a),x,method=_RETURNVERBOSE)
Output:
-1/6/c*sum((_R^3*e+d)/_R^5*ln(x-_R),_R=RootOf(_Z^6*c-a))
Leaf count of result is larger than twice the leaf count of optimal. 1613 vs. \(2 (223) = 446\).
Time = 0.15 (sec) , antiderivative size = 1613, normalized size of antiderivative = 4.99 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\text {Too large to display} \] Input:
integrate((e*x^3+d)/(-c*x^6+a),x, algorithm="fricas")
Output:
-1/12*(sqrt(-3) + 1)*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2* e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2*d^5 + 2*a* c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d^4 + 3*a^2*c*d^2*e^2 + sqrt(-3)*( a*c^2*d^4 + 3*a^2*c*d^2*e^2) - (sqrt(-3)*a^4*c^2*e + a^4*c^2*e)*sqrt((c^2* d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2) )^(1/3)) + 1/12*(sqrt(-3) - 1)*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2* d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d^4 + 3*a^2*c*d^2*e^2 - sqrt(-3)*(a*c^2*d^4 + 3*a^2*c*d^2*e^2) + (sqrt(-3)*a^4*c^2*e - a^4*c^2*e)* sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*(-(a^2*c^2*sqrt ((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3) /(a^2*c^2))^(1/3)) - 1/12*(sqrt(-3) + 1)*((a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d ^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3)*l og(-(c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d^4 + 3*a^2*c*d ^2*e^2 + sqrt(-3)*(a*c^2*d^4 + 3*a^2*c*d^2*e^2) + (sqrt(-3)*a^4*c^2*e + a^ 4*c^2*e)*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*((a^2* c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3)) + 1/12*(sqrt(-3) - 1)*((a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e - a*e^3)/(a^2*c^...
Time = 1.85 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.52 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=- \operatorname {RootSum} {\left (46656 t^{6} a^{5} c^{4} + t^{3} \left (- 432 a^{4} c^{2} e^{3} - 1296 a^{3} c^{3} d^{2} e\right ) + a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a^{4} c^{2} e + 6 t a^{3} e^{4} + 36 t a^{2} c d^{2} e^{2} + 6 t a c^{2} d^{4}}{3 a^{2} d e^{4} - 2 a c d^{3} e^{2} - c^{2} d^{5}} \right )} \right )\right )} \] Input:
integrate((e*x**3+d)/(-c*x**6+a),x)
Output:
-RootSum(46656*_t**6*a**5*c**4 + _t**3*(-432*a**4*c**2*e**3 - 1296*a**3*c* *3*d**2*e) + a**3*e**6 - 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 - c**3*d* *6, Lambda(_t, _t*log(x + (-1296*_t**4*a**4*c**2*e + 6*_t*a**3*e**4 + 36*_ t*a**2*c*d**2*e**2 + 6*_t*a*c**2*d**4)/(3*a**2*d*e**4 - 2*a*c*d**3*e**2 - c**2*d**5))))
Time = 0.12 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\frac {\sqrt {3} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} {\left (\sqrt {c} d - \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {c} d + \sqrt {a} e\right )} \log \left (x^{2} + x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{12 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (x^{2} - x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{12 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {c} d + \sqrt {a} e\right )} \log \left (x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} \] Input:
integrate((e*x^3+d)/(-c*x^6+a),x, algorithm="maxima")
Output:
1/6*sqrt(3)*(sqrt(c)*d + sqrt(a)*e)*arctan(1/3*sqrt(3)*(2*x + (sqrt(a)/sqr t(c))^(1/3))/(sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) + 1/6*sqrt(3)*(sqrt(c)*d - sqrt(a)*e)*arctan(1/3*sqrt(3)*(2*x - (sqrt(a)/s qrt(c))^(1/3))/(sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3) ) + 1/12*(sqrt(c)*d + sqrt(a)*e)*log(x^2 + x*(sqrt(a)/sqrt(c))^(1/3) + (sq rt(a)/sqrt(c))^(2/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) - 1/12*(sqrt(c)* d - sqrt(a)*e)*log(x^2 - x*(sqrt(a)/sqrt(c))^(1/3) + (sqrt(a)/sqrt(c))^(2/ 3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) + 1/6*(sqrt(c)*d - sqrt(a)*e)*log( x + (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) - 1/6*(sq rt(c)*d + sqrt(a)*e)*log(x - (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/ sqrt(c))^(2/3))
Time = 0.14 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\frac {e {\left | c \right |} \log \left (x^{2} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (-a c^{5}\right )^{\frac {1}{3}}} + \frac {\left (-a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, a c} + \frac {{\left (\left (-a c^{5}\right )^{\frac {1}{6}} c^{3} d - \sqrt {3} \left (-a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (-\frac {a}{c}\right )^{\frac {1}{6}}}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\left (-a c^{5}\right )^{\frac {1}{6}} c^{3} d + \sqrt {3} \left (-a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (-\frac {a}{c}\right )^{\frac {1}{6}}}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\sqrt {3} \left (-a c^{5}\right )^{\frac {1}{6}} c^{3} d + \left (-a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (-\frac {a}{c}\right )^{\frac {1}{6}} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} - \frac {{\left (\sqrt {3} \left (-a c^{5}\right )^{\frac {1}{6}} c^{3} d - \left (-a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (-\frac {a}{c}\right )^{\frac {1}{6}} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} \] Input:
integrate((e*x^3+d)/(-c*x^6+a),x, algorithm="giac")
Output:
1/6*e*abs(c)*log(x^2 + (-a/c)^(1/3))/(-a*c^5)^(1/3) + 1/3*(-a*c^5)^(1/6)*d *arctan(x/(-a/c)^(1/6))/(a*c) + 1/6*((-a*c^5)^(1/6)*c^3*d - sqrt(3)*(-a*c^ 5)^(2/3)*e)*arctan((2*x + sqrt(3)*(-a/c)^(1/6))/(-a/c)^(1/6))/(a*c^4) + 1/ 6*((-a*c^5)^(1/6)*c^3*d + sqrt(3)*(-a*c^5)^(2/3)*e)*arctan((2*x - sqrt(3)* (-a/c)^(1/6))/(-a/c)^(1/6))/(a*c^4) + 1/12*(sqrt(3)*(-a*c^5)^(1/6)*c^3*d + (-a*c^5)^(2/3)*e)*log(x^2 + sqrt(3)*x*(-a/c)^(1/6) + (-a/c)^(1/3))/(a*c^4 ) - 1/12*(sqrt(3)*(-a*c^5)^(1/6)*c^3*d - (-a*c^5)^(2/3)*e)*log(x^2 - sqrt( 3)*x*(-a/c)^(1/6) + (-a/c)^(1/3))/(a*c^4)
Time = 11.90 (sec) , antiderivative size = 1293, normalized size of antiderivative = 4.00 \[ \int \frac {d+e x^3}{a-c x^6} \, dx =\text {Too large to display} \] Input:
int((d + e*x^3)/(a - c*x^6),x)
Output:
log(a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a *d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) + e*x*(a^5*c^5)^(1/2) + a^2*c^3*d *x)*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*( a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(a^3*c^3*(-(a^4*c^2*e^3 - c*d^3* (a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e - 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^ (1/3) - e*x*(a^5*c^5)^(1/2) + a^2*c^3*d*x)*(-(a^4*c^2*e^3 - c*d^3*(a^5*c^5 )^(1/2) + 3*a^3*c^3*d^2*e - 3*a*d*e^2*(a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3 ) - log(a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) - 2*e*x*(a^5*c^5)^(1/2) + 3^( 1/2)*a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3* a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3)*1i - 2*a^2*c^3*d*x)*((3^(1/2)*1i )/2 + 1/2)*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a* d*e^2*(a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(e*x*(a^5*c^5)^(1/2) - (a ^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^ 2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3))/2 + (3^(1/2)*a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5 *c^4))^(1/3)*1i)/2 + a^2*c^3*d*x)*((3^(1/2)*1i)/2 - 1/2)*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(216* a^5*c^4))^(1/3) + log(a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(a^5*c^5)^(1/2) + 3*a ^3*c^3*d^2*e - 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) + 2*e*x*(a^5...
Time = 0.25 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.02 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) d +2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) a e +2 \sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) d +2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) a e -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) d +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) d +\mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a e -2 \,\mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) a e +\mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) a e -2 \,\mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) a e}{12 c^{\frac {2}{3}} a^{\frac {4}{3}}} \] Input:
int((e*x^3+d)/(-c*x^6+a),x)
Output:
( - 2*sqrt(c)*sqrt(a)*sqrt(3)*atan((c**(1/6)*a**(1/6) - 2*c**(1/3)*x)/(c** (1/6)*a**(1/6)*sqrt(3)))*d + 2*sqrt(3)*atan((c**(1/6)*a**(1/6) - 2*c**(1/3 )*x)/(c**(1/6)*a**(1/6)*sqrt(3)))*a*e + 2*sqrt(c)*sqrt(a)*sqrt(3)*atan((c* *(1/6)*a**(1/6) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)*sqrt(3)))*d + 2*sqrt(3) *atan((c**(1/6)*a**(1/6) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)*sqrt(3)))*a*e - sqrt(c)*sqrt(a)*log( - c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*d + 2*sqrt(c)*sqrt(a)*log( - c**(1/6)*a**(1/6) - c**(1/3)*x)*d + sqrt(c)*sq rt(a)*log(c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*d - 2*sqrt(c)*sq rt(a)*log(c**(1/6)*a**(1/6) - c**(1/3)*x)*d + log( - c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*a*e - 2*log( - c**(1/6)*a**(1/6) - c**(1/3)*x)* a*e + log(c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*a*e - 2*log(c**( 1/6)*a**(1/6) - c**(1/3)*x)*a*e)/(12*c**(2/3)*a**(1/3)*a)