Integrand size = 20, antiderivative size = 288 \[ \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx=-\frac {d}{a x}-\frac {\sqrt [6]{c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{c} d \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{c} d \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{c} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3} \sqrt [3]{c}}+\frac {\sqrt [6]{c} d \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x}{\sqrt [3]{a}+\sqrt [3]{c} x^2}\right )}{2 \sqrt {3} a^{7/6}}+\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 a^{2/3} \sqrt [3]{c}}-\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{c} x^2+c^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{c}} \] Output:
-d/a/x-1/3*c^(1/6)*d*arctan(c^(1/6)*x/a^(1/6))/a^(7/6)-1/6*c^(1/6)*d*arcta n(-3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(7/6)-1/6*c^(1/6)*d*arctan(3^(1/2)+2*c^( 1/6)*x/a^(1/6))/a^(7/6)-1/6*e*arctan(1/3*(a^(1/3)-2*c^(1/3)*x^2)*3^(1/2)/a ^(1/3))*3^(1/2)/a^(2/3)/c^(1/3)+1/6*c^(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^(7/6)+1/6*e*ln(a^(1/3)+c^(1/3)*x^2) /a^(2/3)/c^(1/3)-1/12*e*ln(a^(2/3)-a^(1/3)*c^(1/3)*x^2+c^(2/3)*x^4)/a^(2/3 )/c^(1/3)
Time = 0.28 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.05 \[ \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx=\frac {-12 a c^{5/6} d-4 a^{5/6} c d x \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )-2 \left (-a^{5/6} c d+\sqrt {3} a^{4/3} \sqrt {c} e\right ) x \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )-2 \left (a^{5/6} c d+\sqrt {3} a^{4/3} \sqrt {c} e\right ) x \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+2 a^{4/3} \sqrt {c} e x \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )-\left (\sqrt {3} a^{5/6} c d+a^{4/3} \sqrt {c} e\right ) x \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\left (\sqrt {3} a^{5/6} c d-a^{4/3} \sqrt {c} e\right ) x \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^2 c^{5/6} x} \] Input:
Integrate[(d + e*x^3)/(x^2*(a + c*x^6)),x]
Output:
(-12*a*c^(5/6)*d - 4*a^(5/6)*c*d*x*ArcTan[(c^(1/6)*x)/a^(1/6)] - 2*(-(a^(5 /6)*c*d) + Sqrt[3]*a^(4/3)*Sqrt[c]*e)*x*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^( 1/6)] - 2*(a^(5/6)*c*d + Sqrt[3]*a^(4/3)*Sqrt[c]*e)*x*ArcTan[Sqrt[3] + (2* c^(1/6)*x)/a^(1/6)] + 2*a^(4/3)*Sqrt[c]*e*x*Log[a^(1/3) + c^(1/3)*x^2] - ( Sqrt[3]*a^(5/6)*c*d + a^(4/3)*Sqrt[c]*e)*x*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c ^(1/6)*x + c^(1/3)*x^2] + (Sqrt[3]*a^(5/6)*c*d - a^(4/3)*Sqrt[c]*e)*x*Log[ a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^2*c^(5/6)*x)
Time = 0.55 (sec) , antiderivative size = 313, 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 = {1829, 25, 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.
| \(\displaystyle \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx\) |
| \(\Big \downarrow \) 1829 |
| \(\displaystyle -\frac {\int -\frac {x \left (a e-c d x^3\right )}{c x^6+a}dx}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x \left (a e-c d x^3\right )}{c x^6+a}dx}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 1835 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {c} \left (\sqrt {-a} e+\sqrt {c} d\right ) \int \frac {x}{\sqrt {c} \left (\sqrt {-a}-\sqrt {c} x^3\right )}dx-\frac {1}{2} \sqrt {c} \left (\sqrt {c} d-\sqrt {-a} e\right ) \int \frac {x}{\sqrt {c} \left (\sqrt {c} x^3+\sqrt {-a}\right )}dx}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \int \frac {x}{\sqrt {-a}-\sqrt {c} x^3}dx-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \int \frac {x}{\sqrt {c} x^3+\sqrt {-a}}dx}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (\frac {\int \frac {1}{\sqrt [6]{-a}-\sqrt [6]{c} x}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\int \frac {\sqrt [6]{-a}-\sqrt [6]{c} x}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\int \frac {1}{\sqrt [6]{c} x+\sqrt [6]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\int \frac {\sqrt [6]{-a}-\sqrt [6]{c} x}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\int \frac {\sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{-a}\right )}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx+\frac {\int -\frac {\sqrt [6]{c} \left (\sqrt [6]{-a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (2 \sqrt [6]{c} x+\sqrt [6]{-a}\right )}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\int \frac {\sqrt [6]{c} \left (\sqrt [6]{-a}-2 \sqrt [6]{c} x\right )}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {3}{2} \sqrt [6]{-a} \int \frac {1}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1\right )^2-3}d\left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}\right )^2-3}d\left (1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}\right )}{\sqrt [6]{c}}-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1}{\sqrt {3}}\right )}{\sqrt [6]{c}}-\frac {1}{2} \int \frac {2 \sqrt [6]{c} x+\sqrt [6]{-a}}{\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [6]{-a}-2 \sqrt [6]{c} x}{\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}}{\sqrt {3}}\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {1}{2} \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1}{\sqrt {3}}\right )}{\sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )-\frac {1}{2} \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\frac {\frac {\log \left (-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}+\sqrt [3]{c} x^2\right )}{2 \sqrt [6]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}}{\sqrt {3}}\right )}{\sqrt [6]{c}}}{3 \sqrt [6]{-a} \sqrt [6]{c}}-\frac {\log \left (\sqrt [6]{-a}+\sqrt [6]{c} x\right )}{3 \sqrt [6]{-a} \sqrt [3]{c}}\right )}{a}-\frac {d}{a x}\) |
Input:
Int[(d + e*x^3)/(x^2*(a + c*x^6)),x]
Output:
-(d/(a*x)) + (-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)/a
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[d*(f*x)^(m + 1)*((a + c*x^(2*n))^(p + 1)/(a*f*(m + 1 ))), x] + Simp[1/(a*f^n*(m + 1)) Int[(f*x)^(m + n)*(a + c*x^(2*n))^p*(a*e *(m + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && LtQ[m, -1] && 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 = 214, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {d}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} c^{2} \textit {\_Z}^{6}+\left (-2 a^{5} c \,e^{3}+6 a^{4} c^{2} d^{2} e \right ) \textit {\_Z}^{3}+a^{3} e^{6}+3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}+c^{3} d^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (7 \textit {\_R}^{6} a^{7} c^{2}+\left (-13 a^{5} c \,e^{3}+39 a^{4} c^{2} d^{2} e \right ) \textit {\_R}^{3}+6 a^{3} e^{6}+18 a^{2} c \,d^{2} e^{4}+18 a \,c^{2} d^{4} e^{2}+6 c^{3} d^{6}\right ) x +a^{6} c^{2} d \,\textit {\_R}^{5}+\left (2 a^{4} c d \,e^{3}+2 a^{3} c^{2} d^{3} e \right ) \textit {\_R}^{2}\right )\right )}{6}\) | \(214\) |
| default | \(-\frac {d}{a x}+\frac {-\frac {c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {5}{6}} d}{12 a}-\frac {c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {4}{3}} e}{12 a}-\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {c \left (\frac {a}{c}\right )^{\frac {4}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{3}+\frac {c \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {5}{6}} d}{12 a}-\frac {\ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {1}{3}} e}{12}-\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6}-\frac {d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}}{a}\) | \(358\) |
Input:
int((e*x^3+d)/x^2/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
-d/a/x+1/6*sum(_R*ln((7*_R^6*a^7*c^2+(-13*a^5*c*e^3+39*a^4*c^2*d^2*e)*_R^3 +6*a^3*e^6+18*a^2*c*d^2*e^4+18*a*c^2*d^4*e^2+6*c^3*d^6)*x+a^6*c^2*d*_R^5+( 2*a^4*c*d*e^3+2*a^3*c^2*d^3*e)*_R^2),_R=RootOf(a^7*c^2*_Z^6+(-2*a^5*c*e^3+ 6*a^4*c^2*d^2*e)*_Z^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))
Leaf count of result is larger than twice the leaf count of optimal. 1885 vs. \(2 (201) = 402\).
Time = 0.31 (sec) , antiderivative size = 1885, normalized size of antiderivative = 6.55 \[ \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x^3+d)/x^2/(c*x^6+a),x, algorithm="fricas")
Output:
1/12*(2*a*x*(-(a^3*c*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^7* c)) + 3*c*d^2*e - a*e^3)/(a^3*c))^(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^3*c^2*d^4*e - 6*a^4*c*d^2*e^3 - (a^6 *c^2*d^2 - a^7*c*e^2)*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^7 *c)))*(-(a^3*c*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^7*c)) + 3*c*d^2*e - a*e^3)/(a^3*c))^(2/3)) + 2*a*x*((a^3*c*sqrt(-(c^2*d^6 - 6*a*c* d^4*e^2 + 9*a^2*d^2*e^4)/(a^7*c)) - 3*c*d^2*e + a*e^3)/(a^3*c))^(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^3*c^2* d^4*e - 6*a^4*c*d^2*e^3 + (a^6*c^2*d^2 - a^7*c*e^2)*sqrt(-(c^2*d^6 - 6*a*c *d^4*e^2 + 9*a^2*d^2*e^4)/(a^7*c)))*((a^3*c*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^7*c)) - 3*c*d^2*e + a*e^3)/(a^3*c))^(2/3)) + (sqrt(-3 )*a*x - a*x)*(-(a^3*c*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^7 *c)) + 3*c*d^2*e - a*e^3)/(a^3*c))^(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^3*c^2*d^4*e - 6*a^4*c*d^2*e^3 + 2*sqrt(-3)*(a^3*c^2*d^4*e - 3*a^4*c*d^2*e^3) - (a^6*c^2*d^2 - a^7*c*e^2 + sqrt(-3)*(a^6*c^2*d^2 - a^7*c*e^2))*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^ 2*d^2*e^4)/(a^7*c)))*(-(a^3*c*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e ^4)/(a^7*c)) + 3*c*d^2*e - a*e^3)/(a^3*c))^(2/3)) - (sqrt(-3)*a*x + a*x)*( -(a^3*c*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^7*c)) + 3*c*d^2 *e - a*e^3)/(a^3*c))^(1/3)*log(-(c^3*d^7 - a*c^2*d^5*e^2 - 5*a^2*c*d^3*...
Time = 1.25 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.72 \[ \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{7} c^{2} + t^{3} \left (- 432 a^{5} c e^{3} + 1296 a^{4} c^{2} d^{2} e\right ) + a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 7776 t^{5} a^{7} c e^{2} + 7776 t^{5} a^{6} c^{2} d^{2} + 36 t^{2} a^{5} e^{5} - 360 t^{2} a^{4} c d^{2} e^{3} + 180 t^{2} a^{3} c^{2} d^{4} e}{3 a^{3} d e^{6} + 5 a^{2} c d^{3} e^{4} + a c^{2} d^{5} e^{2} - c^{3} d^{7}} \right )} \right )\right )} - \frac {d}{a x} \] Input:
integrate((e*x**3+d)/x**2/(c*x**6+a),x)
Output:
RootSum(46656*_t**6*a**7*c**2 + _t**3*(-432*a**5*c*e**3 + 1296*a**4*c**2*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 + (-7776*_t**5*a**7*c*e**2 + 7776*_t**5*a**6*c**2*d**2 + 36*_t**2*a**5*e**5 - 360*_t**2*a**4*c*d**2*e**3 + 180*_t**2*a**3*c**2*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) ))) - d/(a*x)
Time = 0.12 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx=\frac {\frac {2 \, a^{\frac {1}{3}} e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{c^{\frac {1}{3}}} - \frac {4 \, c^{\frac {1}{3}} d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} \sqrt {a} c^{\frac {7}{6}} d - a c^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}} c} - \frac {{\left (\sqrt {3} \sqrt {a} c^{\frac {7}{6}} d + a c^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}} c} - \frac {2 \, {\left (\sqrt {3} a^{\frac {7}{6}} c^{\frac {5}{6}} e + a^{\frac {2}{3}} c^{\frac {4}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} c \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {2 \, {\left (\sqrt {3} a^{\frac {7}{6}} c^{\frac {5}{6}} e - a^{\frac {2}{3}} c^{\frac {4}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} c \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}}{12 \, a} - \frac {d}{a x} \] Input:
integrate((e*x^3+d)/x^2/(c*x^6+a),x, algorithm="maxima")
Output:
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)) + (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) - (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))))/a - d/(a*x)
Time = 0.40 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx=-\frac {c d \left (\frac {a}{c}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, a^{2}} + \frac {\left (a c^{5}\right )^{\frac {1}{3}} e \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, a c^{2}} - \frac {d}{a x} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{3}} a c^{2} e + \left (a c^{5}\right )^{\frac {5}{6}} d\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2} c^{4}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{3}} a c^{2} e - \left (a c^{5}\right )^{\frac {5}{6}} d\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2} c^{4}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{3}} a c^{2} e - \sqrt {3} \left (a c^{5}\right )^{\frac {5}{6}} d\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a^{2} c^{4}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{3}} a c^{2} e + \sqrt {3} \left (a c^{5}\right )^{\frac {5}{6}} d\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a^{2} c^{4}} \] Input:
integrate((e*x^3+d)/x^2/(c*x^6+a),x, algorithm="giac")
Output:
-1/3*c*d*(a/c)^(5/6)*arctan(x/(a/c)^(1/6))/a^2 + 1/6*(a*c^5)^(1/3)*e*log(x ^2 + (a/c)^(1/3))/(a*c^2) - d/(a*x) - 1/6*(sqrt(3)*(a*c^5)^(1/3)*a*c^2*e + (a*c^5)^(5/6)*d)*arctan((2*x + sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^4 ) + 1/6*(sqrt(3)*(a*c^5)^(1/3)*a*c^2*e - (a*c^5)^(5/6)*d)*arctan((2*x - sq rt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^4) - 1/12*((a*c^5)^(1/3)*a*c^2*e - sqrt(3)*(a*c^5)^(5/6)*d)*log(x^2 + sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a ^2*c^4) - 1/12*((a*c^5)^(1/3)*a*c^2*e + sqrt(3)*(a*c^5)^(5/6)*d)*log(x^2 - sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^4)
Time = 23.64 (sec) , antiderivative size = 1526, normalized size of antiderivative = 5.30 \[ \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx=\text {Too large to display} \] Input:
int((d + e*x^3)/(x^2*(a + c*x^6)),x)
Output:
log(a^6*c^2*((a^5*c*e^3 + c*d^3*(-a^7*c^3)^(1/2) - 3*a^4*c^2*d^2*e - 3*a*d *e^2*(-a^7*c^3)^(1/2))/(a^7*c^2))^(2/3) - a*e^2*x*(-a^7*c^3)^(1/2) + c*d^2 *x*(-a^7*c^3)^(1/2) - 2*a^4*c^2*d*e*x)*((a^5*c*e^3 + c*d^3*(-a^7*c^3)^(1/2 ) - 3*a^4*c^2*d^2*e - 3*a*d*e^2*(-a^7*c^3)^(1/2))/(216*a^7*c^2))^(1/3) + l og(a^6*c^2*((a^5*c*e^3 - c*d^3*(-a^7*c^3)^(1/2) - 3*a^4*c^2*d^2*e + 3*a*d* e^2*(-a^7*c^3)^(1/2))/(a^7*c^2))^(2/3) + a*e^2*x*(-a^7*c^3)^(1/2) - c*d^2* x*(-a^7*c^3)^(1/2) - 2*a^4*c^2*d*e*x)*((a^5*c*e^3 - c*d^3*(-a^7*c^3)^(1/2) - 3*a^4*c^2*d^2*e + 3*a*d*e^2*(-a^7*c^3)^(1/2))/(216*a^7*c^2))^(1/3) - d/ (a*x) - log((((3^(1/2)*1i)/2 - 1/2)*((a^5*c*e^3 + c*d^3*(-a^7*c^3)^(1/2) - 3*a^4*c^2*d^2*e - 3*a*d*e^2*(-a^7*c^3)^(1/2))/(a^7*c^2))^(2/3)*(36*a^9*c^ 6*d^3 - 108*a^10*c^5*d*e^2 + 36*a^10*c^5*x*((3^(1/2)*1i)/2 + 1/2)*(a*e^2 - c*d^2)*((a^5*c*e^3 + c*d^3*(-a^7*c^3)^(1/2) - 3*a^4*c^2*d^2*e - 3*a*d*e^2 *(-a^7*c^3)^(1/2))/(a^7*c^2))^(1/3)))/36 + a^7*c^4*e*x*(a*e^2 + c*d^2)^2)* ((3^(1/2)*1i)/2 + 1/2)*((a^5*c*e^3 + c*d^3*(-a^7*c^3)^(1/2) - 3*a^4*c^2*d^ 2*e - 3*a*d*e^2*(-a^7*c^3)^(1/2))/(216*a^7*c^2))^(1/3) + log((((3^(1/2)*1i )/2 + 1/2)*((a^5*c*e^3 + c*d^3*(-a^7*c^3)^(1/2) - 3*a^4*c^2*d^2*e - 3*a*d* e^2*(-a^7*c^3)^(1/2))/(a^7*c^2))^(2/3)*(108*a^10*c^5*d*e^2 - 36*a^9*c^6*d^ 3 + 36*a^10*c^5*x*((3^(1/2)*1i)/2 - 1/2)*(a*e^2 - c*d^2)*((a^5*c*e^3 + c*d ^3*(-a^7*c^3)^(1/2) - 3*a^4*c^2*d^2*e - 3*a*d*e^2*(-a^7*c^3)^(1/2))/(a^7*c ^2))^(1/3)))/36 + a^7*c^4*e*x*(a*e^2 + c*d^2)^2)*((3^(1/2)*1i)/2 - 1/2)...
Time = 0.19 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x^3}{x^2 \left (a+c x^6\right )} \, dx=\frac {2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d x -2 c^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) e x -2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d x -2 c^{\frac {2}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) e x -4 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) d x -c^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d x +c^{\frac {7}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d x +2 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e x -c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e x -c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e x -12 a^{\frac {1}{3}} c d}{12 a^{\frac {4}{3}} c x} \] Input:
int((e*x^3+d)/x^2/(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*x - 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*x - 2*c**(1/6)*a**(1/6)*a tan((c**(1/6)*a**(1/6)*sqrt(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*d*x - 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*x - 4*c**(1/6)*a**(1/6)*atan((c**(1/3)*x)/(c**( 1/6)*a**(1/6)))*c*d*x - 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*x + c**(1/6)*a**(1/6)*sqrt(3)*l og(c**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*c*d*x + 2*c**(2 /3)*a**(2/3)*log(a**(1/3) + c**(1/3)*x**2)*e*x - 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*x - 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*x - 12 *a**(1/3)*c*d)/(12*a**(1/3)*a*c*x)