### 3.16 $$\int \frac{x (d+e x^3)}{a+b x^3+c x^6} \, dx$$

Optimal. Leaf size=634 $\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}$

[Out]

-(((e + (2*c*d - b*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^(2/3)*Sqrt[3]*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3))) - ((e - (2*c*d - b*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^(2/3)*Sqrt[3]*c^(2/3)*(b + Sqrt[b^2 -
4*a*c])^(1/3)) - ((e + (2*c*d - b*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^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) - ((e - (2*c*d - b*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^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)) + ((e + (2*c*d - b*e)
/Sqrt[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^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) + ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*L
og[(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^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3))

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Rubi [A]  time = 0.727749, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.304, Rules used = {1510, 292, 31, 634, 617, 204, 628} $\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(((e + (2*c*d - b*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^(2/3)*Sqrt[3]*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3))) - ((e - (2*c*d - b*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^(2/3)*Sqrt[3]*c^(2/3)*(b + Sqrt[b^2 -
4*a*c])^(1/3)) - ((e + (2*c*d - b*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^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) - ((e - (2*c*d - b*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^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)) + ((e + (2*c*d - b*e)
/Sqrt[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^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) + ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*L
og[(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^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3))

Rule 1510

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

Rule 292

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

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

Mathematica [C]  time = 0.0312528, size = 59, normalized size = 0.09 $\frac{1}{3} \text{RootSum}\left [\text{\#1}^3 b+\text{\#1}^6 c+a\& ,\frac{\text{\#1}^3 e \log (x-\text{\#1})+d \log (x-\text{\#1})}{2 \text{\#1}^4 c+\text{\#1} b}\& \right ]$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

RootSum[a + b*#1^3 + c*#1^6 & , (d*Log[x - #1] + e*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ]/3

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Maple [C]  time = 0.002, size = 49, normalized size = 0.1 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{ \left ({{\it \_R}}^{4}e+{\it \_R}\,d \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(x*(e*x^3+d)/(c*x^6+b*x^3+a),x)

[Out]

1/3*sum((_R^4*e+_R*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*} \int \frac{{\left (e x^{3} + d\right )} x}{c x^{6} + b x^{3} + a}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Sympy [A]  time = 133.311, size = 920, normalized size = 1.45 \begin{align*} \operatorname{RootSum}{\left (t^{6} \left (46656 a^{4} c^{5} - 34992 a^{3} b^{2} c^{4} + 8748 a^{2} b^{4} c^{3} - 729 a b^{6} c^{2}\right ) + t^{3} \left (- 432 a^{3} b c^{2} e^{3} + 1296 a^{3} c^{3} d e^{2} + 216 a^{2} b^{3} c e^{3} - 648 a^{2} b^{2} c^{2} d e^{2} - 432 a^{2} c^{4} d^{3} - 27 a b^{5} e^{3} + 81 a b^{4} c d e^{2} + 216 a b^{2} c^{3} d^{3} - 27 b^{4} c^{2} d^{3}\right ) + a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}, \left ( t \mapsto t \log{\left (x + \frac{15552 t^{5} a^{5} c^{5} e^{2} - 11664 t^{5} a^{4} b^{2} c^{4} e^{2} - 15552 t^{5} a^{4} c^{6} d^{2} + 2916 t^{5} a^{3} b^{4} c^{3} e^{2} + 11664 t^{5} a^{3} b^{2} c^{5} d^{2} - 243 t^{5} a^{2} b^{6} c^{2} e^{2} - 2916 t^{5} a^{2} b^{4} c^{4} d^{2} + 243 t^{5} a b^{6} c^{3} d^{2} - 108 t^{2} a^{4} b c^{2} e^{5} + 360 t^{2} a^{4} c^{3} d e^{4} + 63 t^{2} a^{3} b^{3} c e^{5} - 270 t^{2} a^{3} b^{2} c^{2} d e^{4} + 360 t^{2} a^{3} b c^{3} d^{2} e^{3} - 720 t^{2} a^{3} c^{4} d^{3} e^{2} - 9 t^{2} a^{2} b^{5} e^{5} + 45 t^{2} a^{2} b^{4} c d e^{4} - 90 t^{2} a^{2} b^{3} c^{2} d^{2} e^{3} + 180 t^{2} a^{2} b^{2} c^{3} d^{3} e^{2} + 180 t^{2} a^{2} b c^{4} d^{4} e + 72 t^{2} a^{2} c^{5} d^{5} - 45 t^{2} a b^{3} c^{3} d^{4} e - 54 t^{2} a b^{2} c^{4} d^{5} + 9 t^{2} b^{4} c^{3} d^{5}}{2 a^{4} c e^{7} - a^{3} b^{2} e^{7} - a^{3} b c d e^{6} - 2 a^{3} c^{2} d^{2} e^{5} + 2 a^{2} b^{3} d e^{6} - 6 a^{2} b^{2} c d^{2} e^{5} + 15 a^{2} b c^{2} d^{3} e^{4} - 10 a^{2} c^{3} d^{4} e^{3} - a b^{4} d^{2} e^{5} + 5 a b^{3} c d^{3} e^{4} - 15 a b^{2} c^{2} d^{4} e^{3} + 17 a b c^{3} d^{5} e^{2} - 6 a c^{4} d^{6} e + b^{3} c^{2} d^{5} e^{2} - 2 b^{2} c^{3} d^{6} e + b c^{4} d^{7}} \right )} \right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**6*(46656*a**4*c**5 - 34992*a**3*b**2*c**4 + 8748*a**2*b**4*c**3 - 729*a*b**6*c**2) + _t**3*(-432*a
**3*b*c**2*e**3 + 1296*a**3*c**3*d*e**2 + 216*a**2*b**3*c*e**3 - 648*a**2*b**2*c**2*d*e**2 - 432*a**2*c**4*d**
3 - 27*a*b**5*e**3 + 81*a*b**4*c*d*e**2 + 216*a*b**2*c**3*d**3 - 27*b**4*c**2*d**3) + a**3*e**6 - 3*a**2*b*d*e
**5 + 3*a**2*c*d**2*e**4 + 3*a*b**2*d**2*e**4 - 6*a*b*c*d**3*e**3 + 3*a*c**2*d**4*e**2 - b**3*d**3*e**3 + 3*b*
*2*c*d**4*e**2 - 3*b*c**2*d**5*e + c**3*d**6, Lambda(_t, _t*log(x + (15552*_t**5*a**5*c**5*e**2 - 11664*_t**5*
a**4*b**2*c**4*e**2 - 15552*_t**5*a**4*c**6*d**2 + 2916*_t**5*a**3*b**4*c**3*e**2 + 11664*_t**5*a**3*b**2*c**5
*d**2 - 243*_t**5*a**2*b**6*c**2*e**2 - 2916*_t**5*a**2*b**4*c**4*d**2 + 243*_t**5*a*b**6*c**3*d**2 - 108*_t**
2*a**4*b*c**2*e**5 + 360*_t**2*a**4*c**3*d*e**4 + 63*_t**2*a**3*b**3*c*e**5 - 270*_t**2*a**3*b**2*c**2*d*e**4
+ 360*_t**2*a**3*b*c**3*d**2*e**3 - 720*_t**2*a**3*c**4*d**3*e**2 - 9*_t**2*a**2*b**5*e**5 + 45*_t**2*a**2*b**
4*c*d*e**4 - 90*_t**2*a**2*b**3*c**2*d**2*e**3 + 180*_t**2*a**2*b**2*c**3*d**3*e**2 + 180*_t**2*a**2*b*c**4*d*
*4*e + 72*_t**2*a**2*c**5*d**5 - 45*_t**2*a*b**3*c**3*d**4*e - 54*_t**2*a*b**2*c**4*d**5 + 9*_t**2*b**4*c**3*d
**5)/(2*a**4*c*e**7 - a**3*b**2*e**7 - a**3*b*c*d*e**6 - 2*a**3*c**2*d**2*e**5 + 2*a**2*b**3*d*e**6 - 6*a**2*b
**2*c*d**2*e**5 + 15*a**2*b*c**2*d**3*e**4 - 10*a**2*c**3*d**4*e**3 - a*b**4*d**2*e**5 + 5*a*b**3*c*d**3*e**4
- 15*a*b**2*c**2*d**4*e**3 + 17*a*b*c**3*d**5*e**2 - 6*a*c**4*d**6*e + b**3*c**2*d**5*e**2 - 2*b**2*c**3*d**6*
e + b*c**4*d**7))))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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