3.1.28 \(\int (d+e x)^2 (a+b \text {ArcTan}(c x^3)) \, dx\) [28]

Optimal. Leaf size=315 \[ -\frac {b d e \text {ArcTan}\left (\sqrt [3]{c} x\right )}{c^{2/3}}-\frac {b d^3 \text {ArcTan}\left (c x^3\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \text {ArcTan}\left (c x^3\right )\right )}{3 e}+\frac {b d e \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \text {ArcTan}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {\sqrt {3} b d^2 \text {ArcTan}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c} \]

[Out]

-b*d*e*arctan(c^(1/3)*x)/c^(2/3)-1/3*b*d^3*arctan(c*x^3)/e+1/3*(e*x+d)^3*(a+b*arctan(c*x^3))/e-1/2*b*d*e*arcta
n(2*c^(1/3)*x-3^(1/2))/c^(2/3)-1/2*b*d*e*arctan(2*c^(1/3)*x+3^(1/2))/c^(2/3)+1/2*b*d^2*ln(1+c^(2/3)*x^2)/c^(1/
3)-1/4*b*d^2*ln(1-c^(2/3)*x^2+c^(4/3)*x^4)/c^(1/3)-1/6*b*e^2*ln(c^2*x^6+1)/c+1/2*b*d^2*arctan(1/3*(1-2*c^(2/3)
*x^2)*3^(1/2))*3^(1/2)/c^(1/3)-1/4*b*d*e*ln(1+c^(2/3)*x^2-c^(1/3)*x*3^(1/2))*3^(1/2)/c^(2/3)+1/4*b*d*e*ln(1+c^
(2/3)*x^2+c^(1/3)*x*3^(1/2))*3^(1/2)/c^(2/3)

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Rubi [A]
time = 0.50, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 15, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4980, 1845, 281, 298, 31, 648, 631, 210, 642, 301, 632, 209, 1483, 649, 266} \begin {gather*} \frac {(d+e x)^3 \left (a+b \text {ArcTan}\left (c x^3\right )\right )}{3 e}+\frac {\sqrt {3} b d^2 \text {ArcTan}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {b d e \text {ArcTan}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+\frac {b d e \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \text {ArcTan}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{2 c^{2/3}}-\frac {b d^3 \text {ArcTan}\left (c x^3\right )}{3 e}+\frac {b d^2 \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac {b d^2 \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}-\frac {b e^2 \log \left (c^2 x^6+1\right )}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + b*ArcTan[c*x^3]),x]

[Out]

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

Rule 31

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

Rule 209

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

Rule 210

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])

Rule 266

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 298

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 301

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(
2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 +
2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m))*Int[1/(r^2 + s^2*x^2), x] +
Dist[2*(r^(m + 1)/(a*n*s^m)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rule 631

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 632

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

Rule 642

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]

Rule 648

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 649

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

Rule 1483

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Sim
plify[m - n + 1], 0]

Rule 1845

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(c*x)^(m + ii)*((Coeff[Pq,
x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; Fr
eeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n

Rule 4980

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a
+ b*ArcTan[c*x^n])/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[x^(n - 1)*((d + e*x)^(m + 1)/(1 + c^2*x^(
2*n))), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\int \left (a (d+e x)^2+b (d+e x)^2 \tan ^{-1}\left (c x^3\right )\right ) \, dx\\ &=\frac {a (d+e x)^3}{3 e}+b \int (d+e x)^2 \tan ^{-1}\left (c x^3\right ) \, dx\\ &=\frac {a (d+e x)^3}{3 e}+b \int \left (d^2 \tan ^{-1}\left (c x^3\right )+2 d e x \tan ^{-1}\left (c x^3\right )+e^2 x^2 \tan ^{-1}\left (c x^3\right )\right ) \, dx\\ &=\frac {a (d+e x)^3}{3 e}+\left (b d^2\right ) \int \tan ^{-1}\left (c x^3\right ) \, dx+(2 b d e) \int x \tan ^{-1}\left (c x^3\right ) \, dx+\left (b e^2\right ) \int x^2 \tan ^{-1}\left (c x^3\right ) \, dx\\ &=\frac {a (d+e x)^3}{3 e}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )-\left (3 b c d^2\right ) \int \frac {x^3}{1+c^2 x^6} \, dx-(3 b c d e) \int \frac {x^4}{1+c^2 x^6} \, dx-\left (b c e^2\right ) \int \frac {x^5}{1+c^2 x^6} \, dx\\ &=\frac {a (d+e x)^3}{3 e}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}-\frac {1}{2} \left (3 b c d^2\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x^3} \, dx,x,x^2\right )-\frac {(b d e) \int \frac {1}{1+c^{2/3} x^2} \, dx}{\sqrt [3]{c}}-\frac {(b d e) \int \frac {-\frac {1}{2}+\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{\sqrt [3]{c}}-\frac {(b d e) \int \frac {-\frac {1}{2}-\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{\sqrt [3]{c}}\\ &=\frac {a (d+e x)^3}{3 e}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}+\frac {1}{2} \left (b \sqrt [3]{c} d^2\right ) \text {Subst}\left (\int \frac {1}{1+c^{2/3} x} \, dx,x,x^2\right )-\frac {1}{2} \left (b \sqrt [3]{c} d^2\right ) \text {Subst}\left (\int \frac {1+c^{2/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac {\left (\sqrt {3} b d e\right ) \int \frac {-\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 c^{2/3}}+\frac {\left (\sqrt {3} b d e\right ) \int \frac {\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 c^{2/3}}-\frac {(b d e) \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}-\frac {(b d e) \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}\\ &=\frac {a (d+e x)^3}{3 e}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {-c^{2/3}+2 c^{4/3} x}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{c}}-\frac {1}{4} \left (3 b \sqrt [3]{c} d^2\right ) \text {Subst}\left (\int \frac {1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac {(b d e) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \sqrt {3} c^{2/3}}+\frac {(b d e) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \sqrt {3} c^{2/3}}\\ &=\frac {a (d+e x)^3}{3 e}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )+\frac {b d e \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}-\frac {\left (3 b d^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 c^{2/3} x^2\right )}{2 \sqrt [3]{c}}\\ &=\frac {a (d+e x)^3}{3 e}-\frac {b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac {1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right )+\frac {b d e \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac {b d e \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {\sqrt {3} b d^2 \tan ^{-1}\left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {b d^2 \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} b d e \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}+\frac {\sqrt {3} b d e \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{4 c^{2/3}}-\frac {b d^2 \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac {b e^2 \log \left (1+c^2 x^6\right )}{6 c}\\ \end {align*}

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Mathematica [A]
time = 102.00, size = 297, normalized size = 0.94 \begin {gather*} \frac {12 a c d^2 x+12 a c d e x^2+4 a c e^2 x^3-12 b \sqrt [3]{c} d e \text {ArcTan}\left (\sqrt [3]{c} x\right )+4 b c x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {ArcTan}\left (c x^3\right )+6 b \sqrt [3]{c} d \left (\sqrt {3} \sqrt [3]{c} d+e\right ) \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )+6 b \sqrt [3]{c} d \left (\sqrt {3} \sqrt [3]{c} d-e\right ) \text {ArcTan}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )+6 b c^{2/3} d^2 \log \left (1+c^{2/3} x^2\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d+\sqrt {3} e\right ) \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d-\sqrt {3} e\right ) \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-2 b e^2 \log \left (1+c^2 x^6\right )}{12 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(a + b*ArcTan[c*x^3]),x]

[Out]

(12*a*c*d^2*x + 12*a*c*d*e*x^2 + 4*a*c*e^2*x^3 - 12*b*c^(1/3)*d*e*ArcTan[c^(1/3)*x] + 4*b*c*x*(3*d^2 + 3*d*e*x
+ e^2*x^2)*ArcTan[c*x^3] + 6*b*c^(1/3)*d*(Sqrt[3]*c^(1/3)*d + e)*ArcTan[Sqrt[3] - 2*c^(1/3)*x] + 6*b*c^(1/3)*
d*(Sqrt[3]*c^(1/3)*d - e)*ArcTan[Sqrt[3] + 2*c^(1/3)*x] + 6*b*c^(2/3)*d^2*Log[1 + c^(2/3)*x^2] - 3*b*c^(1/3)*d
*(c^(1/3)*d + Sqrt[3]*e)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] - 3*b*c^(1/3)*d*(c^(1/3)*d - Sqrt[3]*e)*Log[
1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] - 2*b*e^2*Log[1 + c^2*x^6])/(12*c)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(517\) vs. \(2(244)=488\).
time = 9.23, size = 518, normalized size = 1.64

method	result	size
default	\(\frac {\left (e x +d \right )^{3} a}{3 e}+\frac {b \,e^{2} \arctan \left (c \,x^{3}\right ) x^{3}}{3}+b e \arctan \left (c \,x^{3}\right ) x^{2} d +b \arctan \left (c \,x^{3}\right ) x \,d^{2}+\frac {b \,d^{3} \arctan \left (c \,x^{3}\right )}{3 e}-\frac {b e c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} d}{4}-\frac {b c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2}}{4}-\frac {b \,e^{2} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{6 c}-\frac {b e \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {b c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, d^{2}}{2}-\frac {b c \sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d^{3}}{3 e}+\frac {b e c \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} d}{4}-\frac {b c \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2}}{4}-\frac {b \,e^{2} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{6 c}-\frac {b e \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {b c \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, d^{2}}{2}-\frac {b c \sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d^{3}}{3 e}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right ) \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}} d^{2}}{2}-\frac {b \,e^{2} \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{6 c}+\frac {b c \sqrt {\frac {1}{c^{2}}}\, \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d^{3}}{3 e}-\frac {b e \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right ) d}{c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\)	\(518\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a+b*arctan(c*x^3)),x,method=_RETURNVERBOSE)

[Out]

1/3*(e*x+d)^3*a/e+1/3*b*e^2*arctan(c*x^3)*x^3+b*e*arctan(c*x^3)*x^2*d+b*arctan(c*x^3)*x*d^2+1/3*b*d^3*arctan(c
*x^3)/e-1/4*b*e*c*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(5/6)*d-1/4*b*c*ln(x^2-3^(1/2)
*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d^2-1/6*b*e^2/c*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))-1/
2*b*e/c/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*d-1/2*b*c*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)-3^(1/
2))*3^(1/2)*d^2-1/3*b/e*c*(1/c^2)^(1/2)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))*d^3+1/4*b*e*c*ln(x^2+3^(1/2)*(1/c^2)
^(1/6)*x+(1/c^2)^(1/3))*3^(1/2)*(1/c^2)^(5/6)*d-1/4*b*c*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))*(1/c^2)^
(2/3)*d^2-1/6*b*e^2/c*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))-1/2*b*e/c/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)
^(1/6)+3^(1/2))*d+1/2*b*c*(1/c^2)^(2/3)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*3^(1/2)*d^2-1/3*b/e*c*(1/c^2)^(1/2)*
arctan(2*x/(1/c^2)^(1/6)+3^(1/2))*d^3+1/2*b*c*ln(x^2+(1/c^2)^(1/3))*(1/c^2)^(2/3)*d^2-1/6*b*e^2/c*ln(x^2+(1/c^
2)^(1/3))+1/3*b/e*c*(1/c^2)^(1/2)*arctan(x/(1/c^2)^(1/6))*d^3-b*e/c/(1/c^2)^(1/6)*arctan(x/(1/c^2)^(1/6))*d

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Maxima [A]
time = 0.57, size = 280, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e - \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b d^{2} + a d^{2} x + \frac {1}{4} \, {\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b d e + \frac {{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b e^{2}}{6 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*arctan(c*x^3)),x, algorithm="maxima")

[Out]

1/3*a*x^3*e^2 + a*d*x^2*e - 1/4*(c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(4/3)*x^2 - c^(2/3))/c^(2/3))/c^(4/3) +
log(c^(4/3)*x^4 - c^(2/3)*x^2 + 1)/c^(4/3) - 2*log((c^(2/3)*x^2 + 1)/c^(2/3))/c^(4/3)) - 4*x*arctan(c*x^3))*b*
d^2 + a*d^2*x + 1/4*(4*x^2*arctan(c*x^3) + c*(sqrt(3)*log(c^(2/3)*x^2 + sqrt(3)*c^(1/3)*x + 1)/c^(5/3) - sqrt(
3)*log(c^(2/3)*x^2 - sqrt(3)*c^(1/3)*x + 1)/c^(5/3) - 4*arctan(c^(1/3)*x)/c^(5/3) - 2*arctan((2*c^(2/3)*x + sq
rt(3)*c^(1/3))/c^(1/3))/c^(5/3) - 2*arctan((2*c^(2/3)*x - sqrt(3)*c^(1/3))/c^(1/3))/c^(5/3)))*b*d*e + 1/6*(2*c
*x^3*arctan(c*x^3) - log(c^2*x^6 + 1))*b*e^2/c

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*arctan(c*x^3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]
time = 28.54, size = 151, normalized size = 0.48 \begin {gather*} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} - 3 b c d^{2} \operatorname {RootSum} {\left (216 t^{3} c^{4} + 1, \left ( t \mapsto t \log {\left (36 t^{2} c^{2} + x^{2} \right )} \right )\right )} - 3 b c d e \operatorname {RootSum} {\left (46656 t^{6} c^{10} + 1, \left ( t \mapsto t \log {\left (7776 t^{5} c^{8} + x \right )} \right )\right )} + b d^{2} x \operatorname {atan}{\left (c x^{3} \right )} + b d e x^{2} \operatorname {atan}{\left (c x^{3} \right )} + b e^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {x^{3} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {\log {\left (c^{2} x^{6} + 1 \right )}}{6 c} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(a+b*atan(c*x**3)),x)

[Out]

a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 - 3*b*c*d**2*RootSum(216*_t**3*c**4 + 1, Lambda(_t, _t*log(36*_t**2*c**2
+ x**2))) - 3*b*c*d*e*RootSum(46656*_t**6*c**10 + 1, Lambda(_t, _t*log(7776*_t**5*c**8 + x))) + b*d**2*x*atan
(c*x**3) + b*d*e*x**2*atan(c*x**3) + b*e**2*Piecewise((0, Eq(c, 0)), (x**3*atan(c*x**3)/3 - log(c**2*x**6 + 1)
/(6*c), True))

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Giac [A]
time = 9.91, size = 318, normalized size = 1.01 \begin {gather*} \frac {1}{3} \, b e^{2} x^{3} \arctan \left (c x^{3}\right ) + \frac {1}{3} \, a e^{2} x^{3} + b d e x^{2} \arctan \left (c x^{3}\right ) + a d e x^{2} + b d^{2} x \arctan \left (c x^{3}\right ) + a d^{2} x - \frac {b c d e \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {5}{3}}} + \frac {{\left (\sqrt {3} b c d^{2} {\left | c \right |}^{\frac {2}{3}} - b c d e {\left | c \right |}^{\frac {1}{3}}\right )} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, c^{2}} - \frac {{\left (\sqrt {3} b c d^{2} {\left | c \right |}^{\frac {2}{3}} + b c d e {\left | c \right |}^{\frac {1}{3}}\right )} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{2 \, c^{2}} + \frac {{\left (3 \, \sqrt {3} b c d e {\left | c \right |}^{\frac {1}{3}} - 3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} - 2 \, b c e^{2}\right )} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{12 \, c^{2}} - \frac {{\left (3 \, \sqrt {3} b c d e {\left | c \right |}^{\frac {1}{3}} + 3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} + 2 \, b c e^{2}\right )} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{12 \, c^{2}} + \frac {{\left (3 \, b c d^{2} {\left | c \right |}^{\frac {2}{3}} - b c e^{2}\right )} \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{6 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*arctan(c*x^3)),x, algorithm="giac")

[Out]

1/3*b*e^2*x^3*arctan(c*x^3) + 1/3*a*e^2*x^3 + b*d*e*x^2*arctan(c*x^3) + a*d*e*x^2 + b*d^2*x*arctan(c*x^3) + a*
d^2*x - b*c*d*e*arctan(x*abs(c)^(1/3))/abs(c)^(5/3) + 1/2*(sqrt(3)*b*c*d^2*abs(c)^(2/3) - b*c*d*e*abs(c)^(1/3)
)*arctan((2*x + sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/c^2 - 1/2*(sqrt(3)*b*c*d^2*abs(c)^(2/3) + b*c*d*e*abs(c)^(
1/3))*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/c^2 + 1/12*(3*sqrt(3)*b*c*d*e*abs(c)^(1/3) - 3*b*c*d^2
*abs(c)^(2/3) - 2*b*c*e^2)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 - 1/12*(3*sqrt(3)*b*c*d*e*ab
s(c)^(1/3) + 3*b*c*d^2*abs(c)^(2/3) + 2*b*c*e^2)*log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 + 1/6*
(3*b*c*d^2*abs(c)^(2/3) - b*c*e^2)*log(x^2 + 1/abs(c)^(2/3))/c^2

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Mupad [B]
time = 0.52, size = 988, normalized size = 3.14 \begin {gather*} \mathrm {atan}\left (c\,x^3\right )\,\left (b\,d^2\,x+b\,d\,e\,x^2+\frac {b\,e^2\,x^3}{3}\right )+\left (\sum _{k=1}^6\ln \left (x\,\left (6\,b^5\,c^7\,d^2\,e^8-162\,b^5\,c^9\,d^8\,e^2\right )+\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (x\,\left (486\,b^4\,c^{10}\,d^8+90\,b^4\,c^8\,d^2\,e^6\right )-\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,\left (3888\,b^2\,c^{10}\,d^3\,e+\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\,b\,c^{11}\,d^2\,x\,3888+648\,b^2\,c^{10}\,d^2\,e^2\,x\right )+972\,b^3\,c^9\,d^3\,e^3-324\,b^3\,c^9\,d^2\,e^4\,x\right )\right )-243\,b^5\,c^9\,d^9\,e+9\,b^5\,c^7\,d^3\,e^7\right )\,\mathrm {root}\left (46656\,a^6\,c^6+46656\,a^5\,b\,c^5\,e^2+19440\,a^4\,b^2\,c^4\,e^4+4320\,a^3\,b^3\,c^3\,e^6-11664\,a^3\,b^3\,c^5\,d^6+20412\,a^2\,b^4\,c^4\,d^6\,e^2+540\,a^2\,b^4\,c^2\,e^8-972\,a\,b^5\,c^3\,d^6\,e^4+36\,a\,b^5\,c\,e^{10}-54\,b^6\,c^2\,d^6\,e^6+729\,b^6\,c^4\,d^{12}+b^6\,e^{12},a,k\right )\right )+\frac {a\,e^2\,x^3}{3}+a\,d^2\,x+a\,d\,e\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x^3))*(d + e*x)^2,x)

[Out]

atan(c*x^3)*((b*e^2*x^3)/3 + b*d^2*x + b*d*e*x^2) + symsum(log(x*(6*b^5*c^7*d^2*e^8 - 162*b^5*c^9*d^8*e^2) + r
oot(46656*a^6*c^6 + 46656*a^5*b*c^5*e^2 + 19440*a^4*b^2*c^4*e^4 + 4320*a^3*b^3*c^3*e^6 - 11664*a^3*b^3*c^5*d^6
+ 20412*a^2*b^4*c^4*d^6*e^2 + 540*a^2*b^4*c^2*e^8 - 972*a*b^5*c^3*d^6*e^4 + 36*a*b^5*c*e^10 - 54*b^6*c^2*d^6*
e^6 + 729*b^6*c^4*d^12 + b^6*e^12, a, k)*(x*(486*b^4*c^10*d^8 + 90*b^4*c^8*d^2*e^6) - root(46656*a^6*c^6 + 466
56*a^5*b*c^5*e^2 + 19440*a^4*b^2*c^4*e^4 + 4320*a^3*b^3*c^3*e^6 - 11664*a^3*b^3*c^5*d^6 + 20412*a^2*b^4*c^4*d^
6*e^2 + 540*a^2*b^4*c^2*e^8 - 972*a*b^5*c^3*d^6*e^4 + 36*a*b^5*c*e^10 - 54*b^6*c^2*d^6*e^6 + 729*b^6*c^4*d^12
+ b^6*e^12, a, k)*(root(46656*a^6*c^6 + 46656*a^5*b*c^5*e^2 + 19440*a^4*b^2*c^4*e^4 + 4320*a^3*b^3*c^3*e^6 - 1
1664*a^3*b^3*c^5*d^6 + 20412*a^2*b^4*c^4*d^6*e^2 + 540*a^2*b^4*c^2*e^8 - 972*a*b^5*c^3*d^6*e^4 + 36*a*b^5*c*e^
10 - 54*b^6*c^2*d^6*e^6 + 729*b^6*c^4*d^12 + b^6*e^12, a, k)*(3888*b^2*c^10*d^3*e + 3888*root(46656*a^6*c^6 +
46656*a^5*b*c^5*e^2 + 19440*a^4*b^2*c^4*e^4 + 4320*a^3*b^3*c^3*e^6 - 11664*a^3*b^3*c^5*d^6 + 20412*a^2*b^4*c^4
*d^6*e^2 + 540*a^2*b^4*c^2*e^8 - 972*a*b^5*c^3*d^6*e^4 + 36*a*b^5*c*e^10 - 54*b^6*c^2*d^6*e^6 + 729*b^6*c^4*d^
12 + b^6*e^12, a, k)*b*c^11*d^2*x + 648*b^2*c^10*d^2*e^2*x) + 972*b^3*c^9*d^3*e^3 - 324*b^3*c^9*d^2*e^4*x)) -
243*b^5*c^9*d^9*e + 9*b^5*c^7*d^3*e^7)*root(46656*a^6*c^6 + 46656*a^5*b*c^5*e^2 + 19440*a^4*b^2*c^4*e^4 + 4320
*a^3*b^3*c^3*e^6 - 11664*a^3*b^3*c^5*d^6 + 20412*a^2*b^4*c^4*d^6*e^2 + 540*a^2*b^4*c^2*e^8 - 972*a*b^5*c^3*d^6
*e^4 + 36*a*b^5*c*e^10 - 54*b^6*c^2*d^6*e^6 + 729*b^6*c^4*d^12 + b^6*e^12, a, k), k, 1, 6) + (a*e^2*x^3)/3 + a
*d^2*x + a*d*e*x^2

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