3.28 \(\int (d+e x)^2 (a+b \tan ^{-1}(c x^3)) \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.709488, antiderivative size = 331, normalized size of antiderivative = 1.05, number of steps used = 25, number of rules used = 14, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {6742, 5027, 275, 292, 31, 634, 617, 204, 628, 5033, 295, 618, 203, 260} \[ \frac{a (d+e x)^3}{3 e}+\frac{b d^2 \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac{b d^2 \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac{\sqrt{3} b d^2 \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}-\frac{\sqrt{3} b d e \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}+\frac{\sqrt{3} b d e \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{4 c^{2/3}}-\frac{b d e \tan ^{-1}\left (\sqrt [3]{c} x\right )}{c^{2/3}}+\frac{b d e \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d e \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{2 c^{2/3}}-\frac{b e^2 \log \left (c^2 x^6+1\right )}{6 c}+b d^2 x \tan ^{-1}\left (c x^3\right )+b d e x^2 \tan ^{-1}\left (c x^3\right )+\frac{1}{3} b e^2 x^3 \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(d + e*x)^3)/(3*e) - (b*d*e*ArcTan[c^(1/3)*x])/c^(2/3) + b*d^2*x*ArcTan[c*x^3] + b*d*e*x^2*ArcTan[c*x^3] +
(b*e^2*x^3*ArcTan[c*x^3])/3 + (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)) + (Sqrt[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)) - (Sqrt[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*L
og[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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 5027

Int[ArcTan[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTan[c*x^n], x] - Dist[c*n, Int[x^n/(1 + c^2*x^(2*n)), x],
x] /; FreeQ[{c, n}, x]

Rule 275

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 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]

Rule 5033

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

Rule 295

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)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
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 618

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 203

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

Rule 260

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]

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )-\frac{(b d e) \operatorname{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) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 143.84, size = 297, normalized size = 0.94 \[ \frac{12 a c d^2 x+12 a c d e x^2+4 a c e^2 x^3+6 b c^{2/3} d^2 \log \left (c^{2/3} x^2+1\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d+\sqrt{3} e\right ) \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )-3 b \sqrt [3]{c} d \left (\sqrt [3]{c} d-\sqrt{3} e\right ) \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )-2 b e^2 \log \left (c^2 x^6+1\right )+4 b c x \tan ^{-1}\left (c x^3\right ) \left (3 d^2+3 d e x+e^2 x^2\right )-12 b \sqrt [3]{c} d e \tan ^{-1}\left (\sqrt [3]{c} x\right )+6 b \sqrt [3]{c} d \left (\sqrt{3} \sqrt [3]{c} d+e\right ) \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )+6 b \sqrt [3]{c} d \left (\sqrt{3} \sqrt [3]{c} d-e\right ) \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{12 c} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.095, size = 536, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*e^2*x^3+a*e*x^2*d+a*x*d^2+1/3*a/e*d^3+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)*arct
an(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)*ar
ctan(x/(1/c^2)^(1/6))*d

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Maxima [A]  time = 1.45915, size = 448, normalized size = 1.42 \begin{align*} \frac{1}{3} \, a e^{2} x^{3} + a d e x^{2} - \frac{1}{4} \,{\left (c{\left (\frac{2 \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}}{\left (2 \, x^{2} - \frac{1}{c^{2}}^{\frac{1}{3}}\right )}\right )}{c^{2}} + \frac{{\left (c^{2}\right )}^{\frac{1}{3}} \log \left (x^{4} - \frac{1}{c^{2}}^{\frac{1}{3}} x^{2} + \frac{1}{c^{2}}^{\frac{2}{3}}\right )}{c^{2}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} \log \left (x^{2} + \frac{1}{c^{2}}^{\frac{1}{3}}\right )}{c^{2}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b d^{2} + \frac{1}{4} \,{\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c{\left (\frac{\sqrt{3} \log \left ({\left (c^{2}\right )}^{\frac{1}{3}} x^{2} + \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{5}{6}}} - \frac{\sqrt{3} \log \left ({\left (c^{2}\right )}^{\frac{1}{3}} x^{2} - \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{5}{6}}} - \frac{4 \, \arctan \left ({\left (c^{2}\right )}^{\frac{1}{6}} x\right )}{{\left (c^{2}\right )}^{\frac{5}{6}}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{1}{6}} \arctan \left (\frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} x + \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}}}{{\left (c^{2}\right )}^{\frac{1}{6}}}\right )}{c^{2}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{1}{6}} \arctan \left (\frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} x - \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}}}{{\left (c^{2}\right )}^{\frac{1}{6}}}\right )}{c^{2}}\right )}\right )} b d e + a d^{2} x + \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{align*}

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*e^2*x^3 + a*d*e*x^2 - 1/4*(c*(2*sqrt(3)*(c^2)^(1/3)*arctan(1/3*sqrt(3)*(c^2)^(1/3)*(2*x^2 - (c^(-2))^(1/
3)))/c^2 + (c^2)^(1/3)*log(x^4 - (c^(-2))^(1/3)*x^2 + (c^(-2))^(2/3))/c^2 - 2*(c^2)^(1/3)*log(x^2 + (c^(-2))^(
1/3))/c^2) - 4*x*arctan(c*x^3))*b*d^2 + 1/4*(4*x^2*arctan(c*x^3) + c*(sqrt(3)*log((c^2)^(1/3)*x^2 + sqrt(3)*(c
^2)^(1/6)*x + 1)/(c^2)^(5/6) - sqrt(3)*log((c^2)^(1/3)*x^2 - sqrt(3)*(c^2)^(1/6)*x + 1)/(c^2)^(5/6) - 4*arctan
((c^2)^(1/6)*x)/(c^2)^(5/6) - 2*(c^2)^(1/6)*arctan((2*(c^2)^(1/3)*x + sqrt(3)*(c^2)^(1/6))/(c^2)^(1/6))/c^2 -
2*(c^2)^(1/6)*arctan((2*(c^2)^(1/3)*x - sqrt(3)*(c^2)^(1/6))/(c^2)^(1/6))/c^2))*b*d*e + a*d^2*x + 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)]  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((e*x+d)^2*(a+b*arctan(c*x^3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 113.241, size = 151, normalized size = 0.48 \begin{align*} 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{align*}

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 = 4.89821, size = 436, normalized size = 1.38 \begin{align*} \frac{1}{4} \, b c^{5} d{\left (\frac{\sqrt{3}{\left | c \right |}^{\frac{1}{3}} \log \left (x^{2} + \frac{\sqrt{3} x}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{6}} - \frac{\sqrt{3}{\left | c \right |}^{\frac{1}{3}} \log \left (x^{2} - \frac{\sqrt{3} x}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{6}} - \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left ({\left (2 \, x + \frac{\sqrt{3}}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}} - \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left ({\left (2 \, x - \frac{\sqrt{3}}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}} - \frac{4 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left (x{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}}\right )} e - \frac{1}{4} \, b c^{3} d^{2}{\left (\frac{2 \, \sqrt{3}{\left | c \right |}^{\frac{2}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{\left | c \right |}^{\frac{2}{3}}\right )}{c^{4}} + \frac{{\left | c \right |}^{\frac{2}{3}} \log \left (x^{4} - \frac{x^{2}}{{\left | c \right |}^{\frac{2}{3}}} + \frac{1}{{\left | c \right |}^{\frac{4}{3}}}\right )}{c^{4}} - \frac{2 \, \log \left (x^{2} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{2}{\left | c \right |}^{\frac{4}{3}}}\right )} + \frac{2 \, b c x^{3} \arctan \left (c x^{3}\right ) e^{2} + 6 \, b c d x^{2} \arctan \left (c x^{3}\right ) e + 6 \, b c d^{2} x \arctan \left (c x^{3}\right ) + 2 \, a c x^{3} e^{2} + 6 \, a c d x^{2} e + 6 \, a c d^{2} x - b e^{2} \log \left (c^{2} x^{6} + 1\right )}{6 \, c} \end{align*}

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]

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