Integrand size = 21, antiderivative size = 337 \[ \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {e x^2}{2 c}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} \sqrt [6]{a} c^{4/3}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} \sqrt [6]{a} c^{4/3}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 \sqrt [6]{a} c^{4/3}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 \sqrt [6]{a} c^{4/3}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [6]{a} c^{4/3}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [6]{a} c^{4/3}} \] Output:
-1/2*e*x^2/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^(1/6)/c^(4/3)-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^(1/6)/c^(4/3)-1/6*(c^(1 /2)*d+a^(1/2)*e)*ln(a^(1/6)-c^(1/6)*x)/a^(1/6)/c^(4/3)+1/6*(c^(1/2)*d-a^(1 /2)*e)*ln(a^(1/6)+c^(1/6)*x)/a^(1/6)/c^(4/3)-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^(1/6)/c^(4/3)+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^(1/6)/c^(4/3)
Time = 0.12 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.05 \[ \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx=\frac {-6 \sqrt [6]{a} \sqrt [3]{c} e x^2+2 \sqrt {3} \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )-2 \sqrt {3} \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )-2 \sqrt {c} d \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )-2 \sqrt {a} e \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )+2 \sqrt {c} d \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )-2 \sqrt {a} e \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )-\sqrt {c} d \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {a} e \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {c} d \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {a} e \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [6]{a} c^{4/3}} \] Input:
Integrate[(x^4*(d + e*x^3))/(a - c*x^6),x]
Output:
(-6*a^(1/6)*c^(1/3)*e*x^2 + 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^(1/6)*c ^(4/3))
Time = 0.50 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.85, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1827, 27, 1835, 27, 821, 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.
| \(\displaystyle \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx\) |
| \(\Big \downarrow \) 1827 |
| \(\displaystyle \frac {\int \frac {2 x \left (c d x^3+a e\right )}{a-c x^6}dx}{2 c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x \left (c d x^3+a e\right )}{a-c x^6}dx}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 1835 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right ) \int \frac {x}{\sqrt {c} \left (\sqrt {a}-\sqrt {c} x^3\right )}dx-\frac {1}{2} \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {x}{\sqrt {c} \left (\sqrt {c} x^3+\sqrt {a}\right )}dx}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \int \frac {x}{\sqrt {a}-\sqrt {c} x^3}dx-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {x}{\sqrt {c} x^3+\sqrt {a}}dx}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\frac {\int \frac {1}{\sqrt [6]{a}-\sqrt [6]{c} x}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\int \frac {\sqrt [6]{a}-\sqrt [6]{c} x}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\int \frac {1}{\sqrt [6]{c} x+\sqrt [6]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}\right )}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (-\frac {\int \frac {\sqrt [6]{a}-\sqrt [6]{c} x}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{a}\right )}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx+\frac {\int -\frac {\sqrt [6]{c} \left (\sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{a}\right )}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (\sqrt [6]{a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {1}{2} \int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (-\frac {-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )^2-3}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{c}}-\frac {1}{2} \int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1}{\sqrt {3}}\right )}{\sqrt [6]{c}}-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{a}}{\sqrt [3]{c} x^2+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )}{c}-\frac {e x^2}{2 c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1}{\sqrt {3}}\right )}{\sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\frac {\frac {\log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{a} \sqrt [3]{c}}\right )}{c}-\frac {e x^2}{2 c}\) |
Input:
Int[(x^4*(d + e*x^3))/(a - c*x^6),x]
Output:
-1/2*(e*x^2)/c + (-1/2*((Sqrt[c]*d - Sqrt[a]*e)*(-1/3*Log[a^(1/6) + c^(1/6 )*x]/(a^(1/6)*c^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*c^(1/6)*x)/a^(1/6))/Sq rt[3]])/c^(1/6)) + Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2]/(2*c^(1/ 6)))/(3*a^(1/6)*c^(1/6)))) + ((Sqrt[c]*d + Sqrt[a]*e)*(-1/3*Log[a^(1/6) - c^(1/6)*x]/(a^(1/6)*c^(1/3)) - ((Sqrt[3]*ArcTan[(1 + (2*c^(1/6)*x)/a^(1/6) )/Sqrt[3]])/c^(1/6) - Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2]/(2*c^ (1/6)))/(3*a^(1/6)*c^(1/6))))/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[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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[((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]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (c_.)*(x_)^(n2_)) , x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[-(e/2 + c*(d/(2*q))) Int[(f *x)^m/(q - c*x^n), x], x] + Simp[(e/2 - c*(d/(2*q))) Int[(f*x)^m/(q + c*x ^n), x], x]] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.15
| method | result | size |
| risch | \(-\frac {e \,x^{2}}{2 c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c -a \right )}{\sum }\frac {\left (-\textit {\_R}^{4} c d -a e \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c^{2}}\) | \(52\) |
| default | \(-\frac {e \,x^{2}}{2 c}-\frac {\frac {c \ln \left (\left (\frac {a}{c}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {5}{6}} d}{12 a}-\frac {\ln \left (\left (\frac {a}{c}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {1}{3}} e}{12}+\frac {\sqrt {3}\, d \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{3}} e \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6}-\frac {c \left (\frac {a}{c}\right )^{\frac {4}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} e \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6}-\frac {c d \left (\frac {a}{c}\right )^{\frac {5}{6}} \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {d \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {\ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) a e}{6 c \left (\frac {a}{c}\right )^{\frac {2}{3}}}+\frac {\ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) a e}{6 c \left (\frac {a}{c}\right )^{\frac {2}{3}}}}{c}\) | \(385\) |
Input:
int(x^4*(e*x^3+d)/(-c*x^6+a),x,method=_RETURNVERBOSE)
Output:
-1/2*e*x^2/c+1/6/c^2*sum((-_R^4*c*d-_R*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. 1785 vs. \(2 (235) = 470\).
Time = 0.32 (sec) , antiderivative size = 1785, normalized size of antiderivative = 5.30 \[ \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(e*x^3+d)/(-c*x^6+a),x, algorithm="fricas")
Output:
-1/12*(6*e*x^2 - (sqrt(-3)*c - c)*(-(c^4*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9 *a^2*d^2*e^4)/(a*c^7)) + 3*c*d^2*e + a*e^3)/c^4)^(1/3)*log((c^3*d^7 + a*c^ 2*d^5*e^2 - 5*a^2*c*d^3*e^4 + 3*a^3*d*e^6)*x - 1/2*(2*a*c^4*d^4*e + 6*a^2* c^3*d^2*e^3 + 2*sqrt(-3)*(a*c^4*d^4*e + 3*a^2*c^3*d^2*e^3) - (a*c^7*d^2 + a^2*c^6*e^2 + sqrt(-3)*(a*c^7*d^2 + a^2*c^6*e^2))*sqrt((c^2*d^6 + 6*a*c*d^ 4*e^2 + 9*a^2*d^2*e^4)/(a*c^7)))*(-(c^4*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9* a^2*d^2*e^4)/(a*c^7)) + 3*c*d^2*e + a*e^3)/c^4)^(2/3)) + (sqrt(-3)*c + c)* (-(c^4*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a*c^7)) + 3*c*d^2*e + a*e^3)/c^4)^(1/3)*log((c^3*d^7 + a*c^2*d^5*e^2 - 5*a^2*c*d^3*e^4 + 3*a^ 3*d*e^6)*x - 1/2*(2*a*c^4*d^4*e + 6*a^2*c^3*d^2*e^3 - 2*sqrt(-3)*(a*c^4*d^ 4*e + 3*a^2*c^3*d^2*e^3) - (a*c^7*d^2 + a^2*c^6*e^2 - sqrt(-3)*(a*c^7*d^2 + a^2*c^6*e^2))*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a*c^7)))*( -(c^4*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a*c^7)) + 3*c*d^2*e + a*e^3)/c^4)^(2/3)) - 2*c*(-(c^4*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^ 2*e^4)/(a*c^7)) + 3*c*d^2*e + a*e^3)/c^4)^(1/3)*log((c^3*d^7 + a*c^2*d^5*e ^2 - 5*a^2*c*d^3*e^4 + 3*a^3*d*e^6)*x + (2*a*c^4*d^4*e + 6*a^2*c^3*d^2*e^3 - (a*c^7*d^2 + a^2*c^6*e^2)*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4 )/(a*c^7)))*(-(c^4*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a*c^7)) + 3*c*d^2*e + a*e^3)/c^4)^(2/3)) - (sqrt(-3)*c - c)*((c^4*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a*c^7)) - 3*c*d^2*e - a*e^3)/c^4)^(1/3...
Time = 1.24 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.64 \[ \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx=- \operatorname {RootSum} {\left (46656 t^{6} a c^{8} + t^{3} \left (- 432 a^{2} c^{4} e^{3} - 1296 a c^{5} d^{2} e\right ) + a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 7776 t^{5} a^{2} c^{6} e^{2} - 7776 t^{5} a c^{7} d^{2} + 36 t^{2} a^{3} c^{2} e^{5} + 360 t^{2} a^{2} c^{3} d^{2} e^{3} + 180 t^{2} a c^{4} d^{4} e}{3 a^{3} d e^{6} - 5 a^{2} c d^{3} e^{4} + a c^{2} d^{5} e^{2} + c^{3} d^{7}} \right )} \right )\right )} - \frac {e x^{2}}{2 c} \] Input:
integrate(x**4*(e*x**3+d)/(-c*x**6+a),x)
Output:
-RootSum(46656*_t**6*a*c**8 + _t**3*(-432*a**2*c**4*e**3 - 1296*a*c**5*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, La mbda(_t, _t*log(x + (-7776*_t**5*a**2*c**6*e**2 - 7776*_t**5*a*c**7*d**2 + 36*_t**2*a**3*c**2*e**5 + 360*_t**2*a**2*c**3*d**2*e**3 + 180*_t**2*a*c** 4*d**4*e)/(3*a**3*d*e**6 - 5*a**2*c*d**3*e**4 + a*c**2*d**5*e**2 + c**3*d* *7)))) - e*x**2/(2*c)
Time = 0.13 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {e x^{2}}{2 \, c} - \frac {\frac {2 \, \sqrt {3} {\left (\sqrt {a} c d + a \sqrt {c} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} {\left (\sqrt {a} c d - a \sqrt {c} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}} - \frac {{\left (\sqrt {a} c d + a \sqrt {c} e\right )} \log \left (x^{2} + x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}} + \frac {{\left (\sqrt {a} c d - a \sqrt {c} e\right )} \log \left (x^{2} - x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}} - \frac {2 \, {\left (\sqrt {a} c d - a \sqrt {c} e\right )} \log \left (x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}} + \frac {2 \, {\left (\sqrt {a} c d + a \sqrt {c} e\right )} \log \left (x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{\sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}}{12 \, c} \] Input:
integrate(x^4*(e*x^3+d)/(-c*x^6+a),x, algorithm="maxima")
Output:
-1/2*e*x^2/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*(s qrt(a)/sqrt(c))^(1/3)) + 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*(sqrt(a)/sqrt(c))^(1/3)) - (sqrt(a)*c*d + a*sqrt(c)*e)*log(x^2 + x*(sqrt (a)/sqrt(c))^(1/3) + (sqrt(a)/sqrt(c))^(2/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c)) ^(1/3)) + (sqrt(a)*c*d - a*sqrt(c)*e)*log(x^2 - x*(sqrt(a)/sqrt(c))^(1/3) + (sqrt(a)/sqrt(c))^(2/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(1/3)) - 2*(sqrt(a )*c*d - a*sqrt(c)*e)*log(x + (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/ sqrt(c))^(1/3)) + 2*(sqrt(a)*c*d + a*sqrt(c)*e)*log(x - (sqrt(a)/sqrt(c))^ (1/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(1/3)))/c
Time = 0.16 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.87 \[ \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx=-\frac {e x^{2}}{2 \, c} - \frac {d \arctan \left (\frac {x}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, \left (-a c^{5}\right )^{\frac {1}{6}}} + \frac {{\left (\sqrt {3} a c^{2} e - \sqrt {-a c} c^{2} d\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (-\frac {a}{c}\right )^{\frac {1}{6}}}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, \left (-a c^{5}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a c^{2} e + \sqrt {-a c} c^{2} d\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (-\frac {a}{c}\right )^{\frac {1}{6}}}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, \left (-a c^{5}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} \sqrt {-a c} c^{2} d - a c^{2} e\right )} \log \left (x^{2} + \sqrt {3} x \left (-\frac {a}{c}\right )^{\frac {1}{6}} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, \left (-a c^{5}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} \sqrt {-a c} c^{2} d - a c^{2} e\right )} \log \left (x^{2} - \sqrt {3} x \left (-\frac {a}{c}\right )^{\frac {1}{6}} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, \left (-a c^{5}\right )^{\frac {2}{3}}} + \frac {\left (-a c^{5}\right )^{\frac {1}{3}} e \log \left (x^{2} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, c^{3}} \] Input:
integrate(x^4*(e*x^3+d)/(-c*x^6+a),x, algorithm="giac")
Output:
-1/2*e*x^2/c - 1/3*d*arctan(x/(-a/c)^(1/6))/(-a*c^5)^(1/6) + 1/6*(sqrt(3)* a*c^2*e - sqrt(-a*c)*c^2*d)*arctan((2*x + sqrt(3)*(-a/c)^(1/6))/(-a/c)^(1/ 6))/(-a*c^5)^(2/3) - 1/6*(sqrt(3)*a*c^2*e + sqrt(-a*c)*c^2*d)*arctan((2*x - sqrt(3)*(-a/c)^(1/6))/(-a/c)^(1/6))/(-a*c^5)^(2/3) - 1/12*(sqrt(3)*sqrt( -a*c)*c^2*d - a*c^2*e)*log(x^2 + sqrt(3)*x*(-a/c)^(1/6) + (-a/c)^(1/3))/(- a*c^5)^(2/3) - 1/12*(sqrt(3)*sqrt(-a*c)*c^2*d - a*c^2*e)*log(x^2 - sqrt(3) *x*(-a/c)^(1/6) + (-a/c)^(1/3))/(-a*c^5)^(2/3) + 1/6*(-a*c^5)^(1/3)*e*log( x^2 + (-a/c)^(1/3))/c^3
Time = 22.45 (sec) , antiderivative size = 1273, normalized size of antiderivative = 3.78 \[ \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx =\text {Too large to display} \] Input:
int((x^4*(d + e*x^3))/(a - c*x^6),x)
Output:
log(a*c^7*((c*d^3*(a*c^9)^(1/2) - a^2*c^4*e^3 - 3*a*c^5*d^2*e + 3*a*d*e^2* (a*c^9)^(1/2))/(a*c^8))^(2/3) + a*e^2*x*(a*c^9)^(1/2) + c*d^2*x*(a*c^9)^(1 /2) - 2*a*c^5*d*e*x)*((c*d^3*(a*c^9)^(1/2) - a^2*c^4*e^3 - 3*a*c^5*d^2*e + 3*a*d*e^2*(a*c^9)^(1/2))/(216*a*c^8))^(1/3) + log(a*e^2*x*(a*c^9)^(1/2) - a*c^7*(-(c*d^3*(a*c^9)^(1/2) + a^2*c^4*e^3 + 3*a*c^5*d^2*e + 3*a*d*e^2*(a *c^9)^(1/2))/(a*c^8))^(2/3) + c*d^2*x*(a*c^9)^(1/2) + 2*a*c^5*d*e*x)*(-(c* d^3*(a*c^9)^(1/2) + a^2*c^4*e^3 + 3*a*c^5*d^2*e + 3*a*d*e^2*(a*c^9)^(1/2)) /(216*a*c^8))^(1/3) - (e*x^2)/(2*c) - log((3^(1/2)*a*c^7*((c*d^3*(a*c^9)^( 1/2) - a^2*c^4*e^3 - 3*a*c^5*d^2*e + 3*a*d*e^2*(a*c^9)^(1/2))/(a*c^8))^(2/ 3)*1i)/2 - (a*c^7*((c*d^3*(a*c^9)^(1/2) - a^2*c^4*e^3 - 3*a*c^5*d^2*e + 3* a*d*e^2*(a*c^9)^(1/2))/(a*c^8))^(2/3))/2 + a*e^2*x*(a*c^9)^(1/2) + c*d^2*x *(a*c^9)^(1/2) - 2*a*c^5*d*e*x)*((3^(1/2)*1i)/2 + 1/2)*((c*d^3*(a*c^9)^(1/ 2) - a^2*c^4*e^3 - 3*a*c^5*d^2*e + 3*a*d*e^2*(a*c^9)^(1/2))/(216*a*c^8))^( 1/3) - log(a*c^7*(-(c*d^3*(a*c^9)^(1/2) + a^2*c^4*e^3 + 3*a*c^5*d^2*e + 3* a*d*e^2*(a*c^9)^(1/2))/(a*c^8))^(2/3) - 3^(1/2)*a*c^7*(-(c*d^3*(a*c^9)^(1/ 2) + a^2*c^4*e^3 + 3*a*c^5*d^2*e + 3*a*d*e^2*(a*c^9)^(1/2))/(a*c^8))^(2/3) *1i + 2*a*e^2*x*(a*c^9)^(1/2) + 2*c*d^2*x*(a*c^9)^(1/2) + 4*a*c^5*d*e*x)*( (3^(1/2)*1i)/2 + 1/2)*(-(c*d^3*(a*c^9)^(1/2) + a^2*c^4*e^3 + 3*a*c^5*d^2*e + 3*a*d*e^2*(a*c^9)^(1/2))/(216*a*c^8))^(1/3) + log(a*c^7*(-(c*d^3*(a*c^9 )^(1/2) + a^2*c^4*e^3 + 3*a*c^5*d^2*e + 3*a*d*e^2*(a*c^9)^(1/2))/(a*c^8...
Time = 0.18 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.13 \[ \int \frac {x^4 \left (d+e x^3\right )}{a-c x^6} \, dx=\frac {2 c^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) d -2 c^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) e -2 c^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) d -2 c^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}}\right ) e -c^{\frac {7}{6}} a^{\frac {1}{6}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d +2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) d +c^{\frac {7}{6}} a^{\frac {1}{6}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) d +c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e -2 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) e +c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e -2 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) e -6 a^{\frac {1}{3}} c e \,x^{2}}{12 a^{\frac {1}{3}} c^{2}} \] Input:
int(x^4*(e*x^3+d)/(-c*x^6+a),x)
Output:
(2*c**(1/6)*a**(1/6)*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*c**(2/3)*a**(2/3)*sqrt(3)*atan((c**(1/6)*a **(1/6) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)*sqrt(3)))*e - 2*c**(1/6)*a**(1/ 6)*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*c**(2/3)*a**(2/3)*sqrt(3)*atan((c**(1/6)*a**(1/6) + 2*c**(1/ 3)*x)/(c**(1/6)*a**(1/6)*sqrt(3)))*e - c**(1/6)*a**(1/6)*log( - c**(1/6)*a **(1/6)*x + a**(1/3) + c**(1/3)*x**2)*c*d + 2*c**(1/6)*a**(1/6)*log( - c** (1/6)*a**(1/6) - c**(1/3)*x)*c*d + c**(1/6)*a**(1/6)*log(c**(1/6)*a**(1/6) *x + a**(1/3) + c**(1/3)*x**2)*c*d - 2*c**(1/6)*a**(1/6)*log(c**(1/6)*a**( 1/6) - c**(1/3)*x)*c*d + c**(2/3)*a**(2/3)*log( - c**(1/6)*a**(1/6)*x + a* *(1/3) + c**(1/3)*x**2)*e - 2*c**(2/3)*a**(2/3)*log( - c**(1/6)*a**(1/6) - c**(1/3)*x)*e + c**(2/3)*a**(2/3)*log(c**(1/6)*a**(1/6)*x + a**(1/3) + c* *(1/3)*x**2)*e - 2*c**(2/3)*a**(2/3)*log(c**(1/6)*a**(1/6) - c**(1/3)*x)*e - 6*a**(1/3)*c*e*x**2)/(12*a**(1/3)*c**2)