Integrand size = 21, antiderivative size = 333 \[ \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {e x}{c}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} \sqrt [3]{a} c^{7/6}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 \sqrt [3]{a} c^{7/6}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}} \] Output:
-e*x/c+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^(1/3)/c^(7/6)+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^(1/3)/c^(7/6)-1/6*(c^(1/2)*d+ a^(1/2)*e)*ln(a^(1/6)-c^(1/6)*x)/a^(1/3)/c^(7/6)-1/6*(c^(1/2)*d-a^(1/2)*e) *ln(a^(1/6)+c^(1/6)*x)/a^(1/3)/c^(7/6)+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^(1/3)/c^(7/6)+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^(1/3)/c^(7/6)
Time = 0.09 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.05 \[ \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx=\frac {-12 \sqrt [3]{a} \sqrt [6]{c} e x+2 \sqrt {3} \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )-2 \sqrt {c} d \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )-2 \sqrt {a} e \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )-2 \sqrt {c} d \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )+2 \sqrt {a} e \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )+\sqrt {c} d \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )-\sqrt {a} e \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {c} d \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {a} e \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}} \] Input:
Integrate[(x^3*(d + e*x^3))/(a - c*x^6),x]
Output:
(-12*a^(1/3)*c^(1/6)*e*x + 2*Sqrt[3]*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(1 - ( 2*c^(1/6)*x)/a^(1/6))/Sqrt[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^(1/3)*c^ (7/6))
Time = 0.52 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.91, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {1827, 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.
| \(\displaystyle \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx\) |
| \(\Big \downarrow \) 1827 |
| \(\displaystyle \frac {\int \frac {c d x^3+a e}{a-c x^6}dx}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 1747 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{a-\sqrt {a} \sqrt {c} x^3}dx-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{\sqrt {a} \sqrt {c} x^3+a}dx}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\int \frac {\sqrt [6]{a} \left (\sqrt [6]{c} x+2 \sqrt [6]{a}\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x}dx}{3 a^{2/3}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a}-\sqrt [6]{c} x\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\int \frac {\sqrt [6]{a} \left (\sqrt [6]{c} x+2 \sqrt [6]{a}\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 a^{2/3}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a}-\sqrt [6]{c} x\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+2 \sqrt [6]{a}}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\int \frac {2 \sqrt [6]{a}-\sqrt [6]{c} x}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{a} \sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{a}\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{a} \sqrt [6]{c} \left (\sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{a} \sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{a}\right )}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [6]{c} \left (\sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{a} \sqrt [3]{c} x^2-\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [3]{a}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}{\sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {\int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [3]{a}}+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{a} \sqrt [3]{c} x^2+\sqrt {a} \sqrt [6]{c} x+a^{2/3}}dx+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {\int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}+\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )}{c}-\frac {e x}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}+\frac {\log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}\right )-\frac {1}{2} \sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 a^{5/6} \sqrt [6]{c}}+\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [6]{c}}-\frac {\log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [3]{a} \sqrt [6]{c}}}{3 \sqrt {a}}\right )}{c}-\frac {e x}{c}\) |
Input:
Int[(x^3*(d + e*x^3))/(a - c*x^6),x]
Output:
-((e*x)/c) + (-1/2*(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)*(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))/S qrt[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]))) + (Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)*( -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)/c
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]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^( p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*(m + n*(2*p + 1) + 1)) In t[(f*x)^(m - n)*(a + c*x^(2*n))^p*(a*e*(m - n + 1) - c*d*(m + n*(2*p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.15
| method | result | size |
| risch | \(-\frac {e x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c -a \right )}{\sum }\frac {\left (-\textit {\_R}^{3} c d -a e \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c^{2}}\) | \(49\) |
| default | \(-\frac {e x}{c}+\frac {\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (\left (\frac {a}{c}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) d}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \ln \left (\left (\frac {a}{c}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12}-\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, d \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \sqrt {3}\, e \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6}+\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} e \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 a}+\frac {c d \left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {c d \left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 a}-\frac {\ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{3}}}+\frac {\ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) a e}{6 c \left (\frac {a}{c}\right )^{\frac {5}{6}}}-\frac {\ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{3}}}-\frac {\ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) a e}{6 c \left (\frac {a}{c}\right )^{\frac {5}{6}}}}{c}\) | \(394\) |
Input:
int(x^3*(e*x^3+d)/(-c*x^6+a),x,method=_RETURNVERBOSE)
Output:
-e*x/c+1/6/c^2*sum((-_R^3*c*d-a*e)/_R^5*ln(x-_R),_R=RootOf(_Z^6*c-a))
Leaf count of result is larger than twice the leaf count of optimal. 1596 vs. \(2 (233) = 466\).
Time = 0.11 (sec) , antiderivative size = 1596, normalized size of antiderivative = 4.79 \[ \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(e*x^3+d)/(-c*x^6+a),x, algorithm="fricas")
Output:
1/12*(2*c*(-(a*c^3*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^7)) + c*d^3 + 3*a*d*e^2)/(a*c^3))^(1/3)*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a ^2*e^5)*x - (a*c^5*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^7 )) - 3*a*c^2*d^2*e^2 - a^2*c*e^4)*(-(a*c^3*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2 *e^4 + a^2*e^6)/(a*c^7)) + c*d^3 + 3*a*d*e^2)/(a*c^3))^(1/3)) - (sqrt(-3)* c + c)*(-(a*c^3*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^7)) + c*d^3 + 3*a*d*e^2)/(a*c^3))^(1/3)*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2* e^5)*x - 1/2*(3*a*c^2*d^2*e^2 + a^2*c*e^4 + sqrt(-3)*(3*a*c^2*d^2*e^2 + a^ 2*c*e^4) - (sqrt(-3)*a*c^5*d + a*c^5*d)*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^ 4 + a^2*e^6)/(a*c^7)))*(-(a*c^3*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2* e^6)/(a*c^7)) + c*d^3 + 3*a*d*e^2)/(a*c^3))^(1/3)) + (sqrt(-3)*c - c)*(-(a *c^3*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^7)) + c*d^3 + 3*a *d*e^2)/(a*c^3))^(1/3)*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*x - 1/ 2*(3*a*c^2*d^2*e^2 + a^2*c*e^4 - sqrt(-3)*(3*a*c^2*d^2*e^2 + a^2*c*e^4) + (sqrt(-3)*a*c^5*d - a*c^5*d)*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6 )/(a*c^7)))*(-(a*c^3*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^7 )) + c*d^3 + 3*a*d*e^2)/(a*c^3))^(1/3)) + 2*c*((a*c^3*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^7)) - c*d^3 - 3*a*d*e^2)/(a*c^3))^(1/3)*lo g(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*x + (a*c^5*d*sqrt((9*c^2*d^4*e^ 2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^7)) + 3*a*c^2*d^2*e^2 + a^2*c*e^4)*((...
Time = 1.54 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.51 \[ \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx=- \operatorname {RootSum} {\left (46656 t^{6} a^{2} c^{7} + t^{3} \left (- 1296 a^{2} c^{4} d e^{2} - 432 a c^{5} d^{3}\right ) - a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {1296 t^{4} a c^{5} d - 6 t a^{2} c e^{4} - 36 t a c^{2} d^{2} e^{2} - 6 t c^{3} d^{4}}{a^{2} e^{5} + 2 a c d^{2} e^{3} - 3 c^{2} d^{4} e} \right )} \right )\right )} - \frac {e x}{c} \] Input:
integrate(x**3*(e*x**3+d)/(-c*x**6+a),x)
Output:
-RootSum(46656*_t**6*a**2*c**7 + _t**3*(-1296*a**2*c**4*d*e**2 - 432*a*c** 5*d**3) - 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*c**5*d - 6*_t*a**2*c*e**4 - 36*_t*a*c **2*d**2*e**2 - 6*_t*c**3*d**4)/(a**2*e**5 + 2*a*c*d**2*e**3 - 3*c**2*d**4 *e)))) - e*x/c
Time = 0.12 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.01 \[ \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {e x}{c} + \frac {\frac {2 \, \sqrt {3} {\left (\sqrt {a} c d + a \sqrt {c} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {2 \, \sqrt {3} {\left (\sqrt {a} c d - a \sqrt {c} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {a} c d + a \sqrt {c} e\right )} \log \left (x^{2} + x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {a} c d - a \sqrt {c} e\right )} \log \left (x^{2} - x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (\sqrt {a} c d - a \sqrt {c} e\right )} \log \left (x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (\sqrt {a} c d + a \sqrt {c} e\right )} \log \left (x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}}}{12 \, c} \] Input:
integrate(x^3*(e*x^3+d)/(-c*x^6+a),x, algorithm="maxima")
Output:
-e*x/c + 1/12*(2*sqrt(3)*(sqrt(a)*c*d + a*sqrt(c)*e)*arctan(1/3*sqrt(3)*(2 *x + (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a) /sqrt(c))^(2/3)) - 2*sqrt(3)*(sqrt(a)*c*d - a*sqrt(c)*e)*arctan(1/3*sqrt(3 )*(2*x - (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqr t(a)/sqrt(c))^(2/3)) + (sqrt(a)*c*d + a*sqrt(c)*e)*log(x^2 + x*(sqrt(a)/sq rt(c))^(1/3) + (sqrt(a)/sqrt(c))^(2/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3) ) + (sqrt(a)*c*d - a*sqrt(c)*e)*log(x^2 - x*(sqrt(a)/sqrt(c))^(1/3) + (sqr t(a)/sqrt(c))^(2/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) - 2*(sqrt(a)*c*d - a*sqrt(c)*e)*log(x + (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c ))^(2/3)) - 2*(sqrt(a)*c*d + a*sqrt(c)*e)*log(x - (sqrt(a)/sqrt(c))^(1/3)) /(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)))/c
Time = 0.12 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx=\frac {d {\left | c \right |} \log \left (x^{2} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (-a c^{5}\right )^{\frac {1}{3}}} - \frac {e x}{c} + \frac {\left (-a c^{5}\right )^{\frac {1}{6}} e \arctan \left (\frac {x}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, c^{2}} + \frac {{\left (\left (-a c^{5}\right )^{\frac {1}{6}} a c^{2} e - \sqrt {3} \left (-a c^{5}\right )^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (-\frac {a}{c}\right )^{\frac {1}{6}}}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\left (-a c^{5}\right )^{\frac {1}{6}} a c^{2} e + \sqrt {3} \left (-a c^{5}\right )^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (-\frac {a}{c}\right )^{\frac {1}{6}}}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\sqrt {3} \left (-a c^{5}\right )^{\frac {1}{6}} a c^{2} e + \left (-a c^{5}\right )^{\frac {2}{3}} d\right )} \log \left (x^{2} + \sqrt {3} x \left (-\frac {a}{c}\right )^{\frac {1}{6}} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} - \frac {{\left (\sqrt {3} \left (-a c^{5}\right )^{\frac {1}{6}} a c^{2} e - \left (-a c^{5}\right )^{\frac {2}{3}} d\right )} \log \left (x^{2} - \sqrt {3} x \left (-\frac {a}{c}\right )^{\frac {1}{6}} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} \] Input:
integrate(x^3*(e*x^3+d)/(-c*x^6+a),x, algorithm="giac")
Output:
1/6*d*abs(c)*log(x^2 + (-a/c)^(1/3))/(-a*c^5)^(1/3) - e*x/c + 1/3*(-a*c^5) ^(1/6)*e*arctan(x/(-a/c)^(1/6))/c^2 + 1/6*((-a*c^5)^(1/6)*a*c^2*e - sqrt(3 )*(-a*c^5)^(2/3)*d)*arctan((2*x + sqrt(3)*(-a/c)^(1/6))/(-a/c)^(1/6))/(a*c ^4) + 1/6*((-a*c^5)^(1/6)*a*c^2*e + sqrt(3)*(-a*c^5)^(2/3)*d)*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)*a*c^2*e + (-a*c^5)^(2/3)*d)*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)*a*c^2*e - (-a*c^5)^(2/3)*d)*l og(x^2 - sqrt(3)*x*(-a/c)^(1/6) + (-a/c)^(1/3))/(a*c^4)
Time = 22.24 (sec) , antiderivative size = 1267, normalized size of antiderivative = 3.80 \[ \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx =\text {Too large to display} \] Input:
int((x^3*(d + e*x^3))/(a - c*x^6),x)
Output:
log(d*x*(a^3*c^7)^(1/2) + a^2*c^4*(-(a*c^5*d^3 + a*e^3*(a^3*c^7)^(1/2) + 3 *a^2*c^4*d*e^2 + 3*c*d^2*e*(a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) + a^2*c^3*e*x )*(-(a*c^5*d^3 + a*e^3*(a^3*c^7)^(1/2) + 3*a^2*c^4*d*e^2 + 3*c*d^2*e*(a^3* c^7)^(1/2))/(216*a^2*c^7))^(1/3) + log(a^2*c^4*(-(a*c^5*d^3 - a*e^3*(a^3*c ^7)^(1/2) + 3*a^2*c^4*d*e^2 - 3*c*d^2*e*(a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) - d*x*(a^3*c^7)^(1/2) + a^2*c^3*e*x)*(-(a*c^5*d^3 - a*e^3*(a^3*c^7)^(1/2) + 3*a^2*c^4*d*e^2 - 3*c*d^2*e*(a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3) - log( d*x*(a^3*c^7)^(1/2) - (a^2*c^4*(-(a*c^5*d^3 + a*e^3*(a^3*c^7)^(1/2) + 3*a^ 2*c^4*d*e^2 + 3*c*d^2*e*(a^3*c^7)^(1/2))/(a^2*c^7))^(1/3))/2 - (3^(1/2)*a^ 2*c^4*(-(a*c^5*d^3 + a*e^3*(a^3*c^7)^(1/2) + 3*a^2*c^4*d*e^2 + 3*c*d^2*e*( a^3*c^7)^(1/2))/(a^2*c^7))^(1/3)*1i)/2 + a^2*c^3*e*x)*((3^(1/2)*1i)/2 + 1/ 2)*(-(a*c^5*d^3 + a*e^3*(a^3*c^7)^(1/2) + 3*a^2*c^4*d*e^2 + 3*c*d^2*e*(a^3 *c^7)^(1/2))/(216*a^2*c^7))^(1/3) + log(d*x*(a^3*c^7)^(1/2) - (a^2*c^4*(-( a*c^5*d^3 + a*e^3*(a^3*c^7)^(1/2) + 3*a^2*c^4*d*e^2 + 3*c*d^2*e*(a^3*c^7)^ (1/2))/(a^2*c^7))^(1/3))/2 + (3^(1/2)*a^2*c^4*(-(a*c^5*d^3 + a*e^3*(a^3*c^ 7)^(1/2) + 3*a^2*c^4*d*e^2 + 3*c*d^2*e*(a^3*c^7)^(1/2))/(a^2*c^7))^(1/3)*1 i)/2 + a^2*c^3*e*x)*((3^(1/2)*1i)/2 - 1/2)*(-(a*c^5*d^3 + a*e^3*(a^3*c^7)^ (1/2) + 3*a^2*c^4*d*e^2 + 3*c*d^2*e*(a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3) + log(2*d*x*(a^3*c^7)^(1/2) + a^2*c^4*(-(a*c^5*d^3 - a*e^3*(a^3*c^7)^(1/2) + 3*a^2*c^4*d*e^2 - 3*c*d^2*e*(a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) - 3^(1...
Time = 0.19 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.02 \[ \int \frac {x^3 \left (d+e x^3\right )}{a-c x^6} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) e +2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) c d +2 \sqrt {c}\, \sqrt {a}\, \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) e +2 \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) c d -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e +2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) e +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e -2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) e -12 c^{\frac {2}{3}} a^{\frac {1}{3}} e x +\mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) c d -2 \,\mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) c d +\mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) c d -2 \,\mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) c d}{12 c^{\frac {5}{3}} a^{\frac {1}{3}}} \] Input:
int(x^3*(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)))*e + 2*sqrt(3)*atan((c**(1/6)*a**(1/6) - 2*c**(1/3 )*x)/(c**(1/6)*a**(1/6)*sqrt(3)))*c*d + 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)))*e + 2*sqrt(3) *atan((c**(1/6)*a**(1/6) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)*sqrt(3)))*c*d - sqrt(c)*sqrt(a)*log( - c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*e + 2*sqrt(c)*sqrt(a)*log( - c**(1/6)*a**(1/6) - c**(1/3)*x)*e + sqrt(c)*sq rt(a)*log(c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*e - 2*sqrt(c)*sq rt(a)*log(c**(1/6)*a**(1/6) - c**(1/3)*x)*e - 12*c**(2/3)*a**(1/3)*e*x + l og( - c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*c*d - 2*log( - c**(1 /6)*a**(1/6) - c**(1/3)*x)*c*d + log(c**(1/6)*a**(1/6)*x + a**(1/3) + c**( 1/3)*x**2)*c*d - 2*log(c**(1/6)*a**(1/6) - c**(1/3)*x)*c*d)/(12*c**(2/3)*a **(1/3)*c)