3.39 \(\int \frac{d+\frac{e}{x^3}}{c+\frac{a}{x^6}+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=716 \[ \frac{\left (-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\left (-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}+\frac{\left (-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} c^{4/3} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}+\frac{d x}{c} \]

[Out]

(d*x)/c + ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sq
rt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e + (b^2
*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt
[3]])/(2^(1/3)*Sqrt[3]*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) - ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b
^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(4/3)*(b - Sqrt[b^2 - 4*a*c]
)^(2/3)) - ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1
/3)*c^(1/3)*x])/(3*2^(1/3)*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sq
rt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)
*c^(2/3)*x^2])/(6*2^(1/3)*c^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqr
t[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*
c^(2/3)*x^2])/(6*2^(1/3)*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3))

________________________________________________________________________________________

Rubi [A]  time = 1.63351, antiderivative size = 716, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {1393, 1502, 1422, 200, 31, 634, 617, 204, 628} \[ \frac{\left (-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\left (-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}+\frac{\left (-\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (\frac{-2 a c d+b^2 d-b c e}{\sqrt{b^2-4 a c}}+b d-c e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} c^{4/3} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}+\frac{d x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e/x^3)/(c + a/x^6 + b/x^3),x]

[Out]

(d*x)/c + ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sq
rt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e + (b^2
*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt
[3]])/(2^(1/3)*Sqrt[3]*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) - ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b
^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(4/3)*(b - Sqrt[b^2 - 4*a*c]
)^(2/3)) - ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1
/3)*c^(1/3)*x])/(3*2^(1/3)*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e - (b^2*d - 2*a*c*d - b*c*e)/Sq
rt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)
*c^(2/3)*x^2])/(6*2^(1/3)*c^(4/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + ((b*d - c*e + (b^2*d - 2*a*c*d - b*c*e)/Sqr
t[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*
c^(2/3)*x^2])/(6*2^(1/3)*c^(4/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3))

Rule 1393

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[x^(n*(
2*p + q))*(e + d/x^n)^q*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && Integ
ersQ[p, q] && NegQ[n]

Rule 1502

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[(e*f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(c*(m + n*(2*p + 1) + 1)), x] - Dist[f^n
/(c*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m - n + 1) + (b*e*(m + n*p +
1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2
- 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]

Rule 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[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]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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]

Rubi steps

\begin{align*} \int \frac{d+\frac{e}{x^3}}{c+\frac{a}{x^6}+\frac{b}{x^3}} \, dx &=\int \frac{x^3 \left (e+d x^3\right )}{a+b x^3+c x^6} \, dx\\ &=\frac{d x}{c}-\frac{\int \frac{a d+(b d-c e) x^3}{a+b x^3+c x^6} \, dx}{c}\\ &=\frac{d x}{c}-\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx}{2 c}-\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx}{2 c}\\ &=\frac{d x}{c}-\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} c \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{2^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}-\sqrt [3]{c} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} c \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} c \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{2^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}-\sqrt [3]{c} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} c \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}\\ &=\frac{d x}{c}-\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2\ 2^{2/3} c \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6 \sqrt [3]{2} c^{4/3} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2\ 2^{2/3} c \sqrt [3]{b+\sqrt{b^2-4 a c}}}\\ &=\frac{d x}{c}-\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt [3]{2} c^{4/3} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}\\ &=\frac{d x}{c}+\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} c^{4/3} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (b d-c e-\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (b d-c e+\frac{b^2 d-2 a c d-b c e}{\sqrt{b^2-4 a c}}\right ) \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (b+\sqrt{b^2-4 a c}\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0558162, size = 88, normalized size = 0.12 \[ \frac{d x}{c}-\frac{\text{RootSum}\left [\text{$\#$1}^3 b+\text{$\#$1}^6 c+a\& ,\frac{\text{$\#$1}^3 b d \log (x-\text{$\#$1})-\text{$\#$1}^3 c e \log (x-\text{$\#$1})+a d \log (x-\text{$\#$1})}{\text{$\#$1}^2 b+2 \text{$\#$1}^5 c}\& \right ]}{3 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e/x^3)/(c + a/x^6 + b/x^3),x]

[Out]

(d*x)/c - RootSum[a + b*#1^3 + c*#1^6 & , (a*d*Log[x - #1] + b*d*Log[x - #1]*#1^3 - c*e*Log[x - #1]*#1^3)/(b*#
1^2 + 2*c*#1^5) & ]/(3*c)

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Maple [C]  time = 0.013, size = 67, normalized size = 0.1 \begin{align*}{\frac{dx}{c}}+{\frac{1}{3\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{ \left ( \left ( -bd+ce \right ){{\it \_R}}^{3}-ad \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e/x^3)/(c+a/x^6+b/x^3),x)

[Out]

d*x/c+1/3/c*sum(((-b*d+c*e)*_R^3-a*d)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z^6*c+_Z^3*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{d x}{c} + \frac{-\int \frac{{\left (b d - c e\right )} x^{3} + a d}{c x^{6} + b x^{3} + a}\,{d x}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e/x^3)/(c+a/x^6+b/x^3),x, algorithm="maxima")

[Out]

d*x/c + integrate(-((b*d - c*e)*x^3 + a*d)/(c*x^6 + b*x^3 + a), x)/c

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e/x^3)/(c+a/x^6+b/x^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e/x**3)/(c+a/x**6+b/x**3),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e/x^3)/(c+a/x^6+b/x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError