Integrand size = 20, antiderivative size = 290 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {e x^2}{2 c}+\frac {d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} c^{5/6}}-\frac {d \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} c^{5/6}}+\frac {d \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} c^{5/6}}+\frac {\sqrt [3]{a} e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{c} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} c^{4/3}}-\frac {d \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x}{\sqrt [3]{a}+\sqrt [3]{c} x^2}\right )}{2 \sqrt {3} \sqrt [6]{a} c^{5/6}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 c^{4/3}}+\frac {\sqrt [3]{a} e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{c} x^2+c^{2/3} x^4\right )}{12 c^{4/3}} \] Output:
1/2*e*x^2/c+1/3*d*arctan(c^(1/6)*x/a^(1/6))/a^(1/6)/c^(5/6)+1/6*d*arctan(- 3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(1/6)/c^(5/6)+1/6*d*arctan(3^(1/2)+2*c^(1/6 )*x/a^(1/6))/a^(1/6)/c^(5/6)+1/6*a^(1/3)*e*arctan(1/3*(a^(1/3)-2*c^(1/3)*x ^2)*3^(1/2)/a^(1/3))*3^(1/2)/c^(4/3)-1/6*d*arctanh(3^(1/2)*a^(1/6)*c^(1/6) *x/(a^(1/3)+c^(1/3)*x^2))*3^(1/2)/a^(1/6)/c^(5/6)-1/6*a^(1/3)*e*ln(a^(1/3) +c^(1/3)*x^2)/c^(4/3)+1/12*a^(1/3)*e*ln(a^(2/3)-a^(1/3)*c^(1/3)*x^2+c^(2/3 )*x^4)/c^(4/3)
Time = 0.09 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.20 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {e x^2}{2 c}+\frac {d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} c^{5/6}}+\frac {\left (a^{5/6} c d-\sqrt {3} a^{4/3} \sqrt {c} e\right ) \arctan \left (\frac {-\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{11/6}}+\frac {\left (a^{5/6} c d+\sqrt {3} a^{4/3} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{11/6}}-\frac {\sqrt [3]{a} e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 c^{4/3}}-\frac {\left (-\sqrt {3} a^{5/6} c d-a^{4/3} \sqrt {c} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{11/6}}-\frac {\left (\sqrt {3} a^{5/6} c d-a^{4/3} \sqrt {c} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{11/6}} \] Input:
Integrate[(x^4*(d + e*x^3))/(a + c*x^6),x]
Output:
(e*x^2)/(2*c) + (d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*a^(1/6)*c^(5/6)) + ((a^ (5/6)*c*d - Sqrt[3]*a^(4/3)*Sqrt[c]*e)*ArcTan[(-(Sqrt[3]*a^(1/6)) + 2*c^(1 /6)*x)/a^(1/6)])/(6*a*c^(11/6)) + ((a^(5/6)*c*d + Sqrt[3]*a^(4/3)*Sqrt[c]* e)*ArcTan[(Sqrt[3]*a^(1/6) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(11/6)) - (a^(1 /3)*e*Log[a^(1/3) + c^(1/3)*x^2])/(6*c^(4/3)) - ((-(Sqrt[3]*a^(5/6)*c*d) - a^(4/3)*Sqrt[c]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] )/(12*a*c^(11/6)) - ((Sqrt[3]*a^(5/6)*c*d - a^(4/3)*Sqrt[c]*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(11/6))
Time = 0.58 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\int \frac {2 x \left (a e-c d x^3\right )}{c x^6+a}dx}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\int \frac {x \left (a e-c d x^3\right )}{c x^6+a}dx}{c}\) |
| \(\Big \downarrow \) 1835 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \sqrt {c} \left (\sqrt {-a} e+\sqrt {c} d\right ) \int \frac {x}{\sqrt {c} \left (\sqrt {-a}-\sqrt {c} x^3\right )}dx-\frac {1}{2} \sqrt {c} \left (\sqrt {c} d-\sqrt {-a} e\right ) \int \frac {x}{\sqrt {c} \left (\sqrt {c} x^3+\sqrt {-a}\right )}dx}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \int \frac {x}{\sqrt {-a}-\sqrt {c} x^3}dx-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \int \frac {x}{\sqrt {c} x^3+\sqrt {-a}}dx}{c}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (\frac {\int \frac {1}{\sqrt [6]{-a}-\sqrt [6]{c} x}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\int \frac {\sqrt [6]{-a}-\sqrt [6]{c} x}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\int \frac {1}{\sqrt [6]{c} x+\sqrt [6]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}\right )}{c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\int \frac {\sqrt [6]{-a}-\sqrt [6]{c} x}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{-a}\right )}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx+\frac {\int -\frac {\sqrt [6]{c} \left (\sqrt [6]{-a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{-a}\right )}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (\sqrt [6]{-a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1\right )^2-3}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}\right )^2-3}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}\right )}{\sqrt [6]{c}}-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1}{\sqrt {3}}\right )}{\sqrt [6]{c}}-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}}{\sqrt {3}}\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {e x^2}{2 c}-\frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1}{\sqrt {3}}\right )}{\sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {\log \left (-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}}{\sqrt {3}}\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{c}\) |
Input:
Int[(x^4*(d + e*x^3))/(a + c*x^6),x]
Output:
(e*x^2)/(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))/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)))) + ((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 = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.17
| method | result | size |
| risch | \(\frac {e \,x^{2}}{2 c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (\textit {\_R}^{4} c d -a e \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c^{2}}\) | \(49\) |
| default | \(\frac {e \,x^{2}}{2 c}-\frac {-\frac {c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {5}{6}} d}{12 a}-\frac {c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {4}{3}} e}{12 a}-\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {c \left (\frac {a}{c}\right )^{\frac {4}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{3}+\frac {c \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {5}{6}} d}{12 a}-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {1}{3}} e}{12}-\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6}-\frac {d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}}{c}\) | \(359\) |
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. 1825 vs. \(2 (201) = 402\).
Time = 0.32 (sec) , antiderivative size = 1825, normalized size of antiderivative = 6.29 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+c x^6} \, dx=\text {Too large to display} \] 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^...
Time = 1.22 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+c x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a c^{8} + t^{3} \cdot \left (432 a^{2} c^{4} e^{3} - 1296 a c^{5} d^{2} e\right ) + a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {7776 t^{5} a^{2} c^{6} e^{2} - 7776 t^{5} a c^{7} d^{2} + 36 t^{2} a^{3} c^{2} e^{5} - 360 t^{2} a^{2} c^{3} d^{2} e^{3} + 180 t^{2} a c^{4} d^{4} e}{3 a^{3} d e^{6} + 5 a^{2} c d^{3} e^{4} + a c^{2} d^{5} e^{2} - c^{3} d^{7}} \right )} \right )\right )} + \frac {e x^{2}}{2 c} \] 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, Lamb da(_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 = 0.12 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.03 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {e x^{2}}{2 \, c} - \frac {\frac {2 \, a^{\frac {1}{3}} e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{c^{\frac {1}{3}}} - \frac {4 \, c^{\frac {1}{3}} d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} \sqrt {a} c^{\frac {7}{6}} d - a c^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}} c} - \frac {{\left (\sqrt {3} \sqrt {a} c^{\frac {7}{6}} d + a c^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}} c} - \frac {2 \, {\left (\sqrt {3} a^{\frac {7}{6}} c^{\frac {5}{6}} e + a^{\frac {2}{3}} c^{\frac {4}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} c \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {2 \, {\left (\sqrt {3} a^{\frac {7}{6}} c^{\frac {5}{6}} e - a^{\frac {2}{3}} c^{\frac {4}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} c \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}}{12 \, c} \] 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*a^(1/3)*e*log(c^(1/3)*x^2 + a^(1/3))/c^(1/3) - 4*c^( 1/3)*d*arctan(c^(1/3)*x/sqrt(a^(1/3)*c^(1/3)))/sqrt(a^(1/3)*c^(1/3)) + (sq rt(3)*sqrt(a)*c^(7/6)*d - a*c^(2/3)*e)*log(c^(1/3)*x^2 + sqrt(3)*a^(1/6)*c ^(1/6)*x + a^(1/3))/(a^(2/3)*c) - (sqrt(3)*sqrt(a)*c^(7/6)*d + a*c^(2/3)*e )*log(c^(1/3)*x^2 - sqrt(3)*a^(1/6)*c^(1/6)*x + a^(1/3))/(a^(2/3)*c) - 2*( sqrt(3)*a^(7/6)*c^(5/6)*e + a^(2/3)*c^(4/3)*d)*arctan((2*c^(1/3)*x + sqrt( 3)*a^(1/6)*c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a^(2/3)*c*sqrt(a^(1/3)*c^(1/3) )) + 2*(sqrt(3)*a^(7/6)*c^(5/6)*e - a^(2/3)*c^(4/3)*d)*arctan((2*c^(1/3)*x - sqrt(3)*a^(1/6)*c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a^(2/3)*c*sqrt(a^(1/3) *c^(1/3))))/c
Time = 0.40 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.94 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {e x^{2}}{2 \, c} + \frac {d \left (\frac {a}{c}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, a} + \frac {{\left (\sqrt {3} a c^{2} e + \sqrt {a c} c^{2} d\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, \left (a c^{5}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a c^{2} e - \sqrt {a c} c^{2} d\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, \left (a c^{5}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {3} \sqrt {a c} c^{2} d + a c^{2} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, \left (a c^{5}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {3} \sqrt {a c} c^{2} d + a c^{2} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, \left (a c^{5}\right )^{\frac {2}{3}}} - \frac {\left (a c^{5}\right )^{\frac {1}{3}} e \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, c^{3}} \] 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*(a/c)^(5/6)*arctan(x/(a/c)^(1/6))/a + 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 = 22.52 (sec) , antiderivative size = 1319, normalized size of antiderivative = 4.55 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+c x^6} \, dx =\text {Too large to display} \] 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*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) + (e*x^2)/(2*c) - 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))^(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*...
Time = 0.19 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.11 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+c x^6} \, dx=\frac {-2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d +2 c^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) e +2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d +2 c^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) e +4 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) d +c^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -c^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -2 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e +c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e +c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e +6 a^{\frac {1}{3}} c e \,x^{2}}{12 a^{\frac {1}{3}} c^{2}} \] Input:
int(x^4*(e*x^3+d)/(c*x^6+a),x)
Output:
( - 2*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c **(1/6)*a**(1/6)))*c*d + 2*c**(2/3)*a**(2/3)*sqrt(3)*atan((c**(1/6)*a**(1/ 6)*sqrt(3) - 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*e + 2*c**(1/6)*a**(1/6)*at an((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*d + 2 *c**(2/3)*a**(2/3)*sqrt(3)*atan((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x) /(c**(1/6)*a**(1/6)))*e + 4*c**(1/6)*a**(1/6)*atan((c**(1/3)*x)/(c**(1/6)* a**(1/6)))*c*d + c**(1/6)*a**(1/6)*sqrt(3)*log( - c**(1/6)*a**(1/6)*sqrt(3 )*x + a**(1/3) + c**(1/3)*x**2)*c*d - c**(1/6)*a**(1/6)*sqrt(3)*log(c**(1/ 6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*c*d - 2*c**(2/3)*a**(2/3 )*log(a**(1/3) + c**(1/3)*x**2)*e + c**(2/3)*a**(2/3)*log( - c**(1/6)*a**( 1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*e + c**(2/3)*a**(2/3)*log(c**(1 /6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*e + 6*a**(1/3)*c*e*x**2 )/(12*a**(1/3)*c**2)