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

Optimal. Leaf size=250 \[ \frac{(d+e x)^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{3 e}-\frac{b \left (3 c d^2+e^2\right ) \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2} c^{3/2}}+\frac{b \left (3 c d^2+e^2\right ) \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2} c^{3/2}}+\frac{b \left (3 c d^2-e^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{3 \sqrt{2} c^{3/2}}-\frac{b \left (3 c d^2-e^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{3 \sqrt{2} c^{3/2}}-\frac{b d e \log \left (c^2 x^4+1\right )}{2 c}-\frac{b d^3 \tan ^{-1}\left (c x^2\right )}{3 e}-\frac{2 b e^2 x}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.303477, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {5205, 12, 1831, 1248, 635, 203, 260, 1280, 1168, 1162, 617, 204, 1165, 628} \[ \frac{(d+e x)^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{3 e}-\frac{b \left (3 c d^2+e^2\right ) \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2} c^{3/2}}+\frac{b \left (3 c d^2+e^2\right ) \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2} c^{3/2}}+\frac{b \left (3 c d^2-e^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{3 \sqrt{2} c^{3/2}}-\frac{b \left (3 c d^2-e^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{3 \sqrt{2} c^{3/2}}-\frac{b d e \log \left (c^2 x^4+1\right )}{2 c}-\frac{b d^3 \tan ^{-1}\left (c x^2\right )}{3 e}-\frac{2 b e^2 x}{3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5205

Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 + u^2), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1831

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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 635

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 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]

Rule 1280

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(e*f*(f*x)^(m - 1)*
(a + c*x^4)^(p + 1))/(c*(m + 4*p + 3)), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*
(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] &
& IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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

Mathematica [A]  time = 3.24062, size = 252, normalized size = 1.01 \[ \frac{1}{12} \left (12 a d^2 x+12 a d e x^2+4 a e^2 x^3-\frac{\sqrt{2} b \left (3 c d^2+e^2\right ) \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{c^{3/2}}+\frac{\sqrt{2} b \left (3 c d^2+e^2\right ) \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{c^{3/2}}+\frac{2 \sqrt{2} b \left (3 c d^2-e^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{c^{3/2}}-\frac{2 \sqrt{2} b \left (3 c d^2-e^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{c^{3/2}}-\frac{6 b d e \log \left (c^2 x^4+1\right )}{c}+4 b x \tan ^{-1}\left (c x^2\right ) \left (3 d^2+3 d e x+e^2 x^2\right )-\frac{8 b e^2 x}{c}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*a*d^2*x - (8*b*e^2*x)/c + 12*a*d*e*x^2 + 4*a*e^2*x^3 + 4*b*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTan[c*x^2] + (
2*Sqrt[2]*b*(3*c*d^2 - e^2)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/c^(3/2) - (2*Sqrt[2]*b*(3*c*d^2 - e^2)*ArcTan[1 + S
qrt[2]*Sqrt[c]*x])/c^(3/2) - (Sqrt[2]*b*(3*c*d^2 + e^2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/c^(3/2) + (Sqrt[2]
*b*(3*c*d^2 + e^2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/c^(3/2) - (6*b*d*e*Log[1 + c^2*x^4])/c)/12

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Maple [A]  time = 0.033, size = 381, normalized size = 1.5 \begin{align*}{\frac{a{e}^{2}{x}^{3}}{3}}+ae{x}^{2}d+ax{d}^{2}+{\frac{a{d}^{3}}{3\,e}}+{\frac{b{e}^{2}\arctan \left ( c{x}^{2} \right ){x}^{3}}{3}}+be\arctan \left ( c{x}^{2} \right ){x}^{2}d+b\arctan \left ( c{x}^{2} \right ) x{d}^{2}+{\frac{b{d}^{3}\arctan \left ( c{x}^{2} \right ) }{3\,e}}-{\frac{2\,b{e}^{2}x}{3\,c}}+{\frac{b{e}^{2}\sqrt{2}}{6\,c}\sqrt [4]{{c}^{-2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{c}^{-2}}}}}+1 \right ) }+{\frac{b{e}^{2}\sqrt{2}}{6\,c}\sqrt [4]{{c}^{-2}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{c}^{-2}}}}}-1 \right ) }+{\frac{b{e}^{2}\sqrt{2}}{12\,c}\sqrt [4]{{c}^{-2}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) \left ({x}^{2}-\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) ^{-1}} \right ) }-{\frac{bc{d}^{3}}{3\,e}\arctan \left ({x}^{2}\sqrt{{c}^{2}} \right ){\frac{1}{\sqrt{{c}^{2}}}}}-{\frac{b{d}^{2}\sqrt{2}}{4\,c}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) \left ({x}^{2}+\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{b{d}^{2}\sqrt{2}}{2\,c}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{c}^{-2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{b{d}^{2}\sqrt{2}}{2\,c}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{c}^{-2}}}}}-1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{bed\ln \left ({c}^{2}{x}^{4}+1 \right ) }{2\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a+b*arctan(c*x^2)),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^2)*x^3+b*e*arctan(c*x^2)*x^2*d+b*arctan(c*x^2
)*x*d^2+1/3*b*d^3*arctan(c*x^2)/e-2/3*b*e^2*x/c+1/6*b*e^2/c*(1/c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)
*x+1)+1/6*b*e^2/c*(1/c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)*x-1)+1/12*b*e^2/c*(1/c^2)^(1/4)*2^(1/2)*l
n((x^2+(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2))/(x^2-(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2)))-1/3*b/e*c*d^3/(c^2)
^(1/2)*arctan(x^2*(c^2)^(1/2))-1/4*b/c*d^2/(1/c^2)^(1/4)*2^(1/2)*ln((x^2-(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2)
)/(x^2+(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2)))-1/2*b/c*d^2/(1/c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)*
x+1)-1/2*b/c*d^2/(1/c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)*x-1)-1/2*b*d*e*ln(c^2*x^4+1)/c

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Maxima [B]  time = 1.5398, size = 771, normalized size = 3.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*e^2*x^3 + a*d*e*x^2 + 1/4*(c*(sqrt(2)*log(sqrt(c^2)*x^2 + sqrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(3/4) - sqrt(
2)*log(sqrt(c^2)*x^2 - sqrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(3/4) - sqrt(2)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqr
t(c^2)) + sqrt(2)*(c^2)^(1/4))/(2*sqrt(c^2)*x + sqrt(2)*sqrt(-sqrt(c^2)) + sqrt(2)*(c^2)^(1/4)))/(sqrt(c^2)*sq
rt(-sqrt(c^2))) - sqrt(2)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqrt(c^2)) - sqrt(2)*(c^2)^(1/4))/(2*sqrt(c^2)*x
+ sqrt(2)*sqrt(-sqrt(c^2)) - sqrt(2)*(c^2)^(1/4)))/(sqrt(c^2)*sqrt(-sqrt(c^2)))) + 4*x*arctan(c*x^2))*b*d^2 +
1/12*(4*x^3*arctan(c*x^2) + c*((sqrt(2)*log(sqrt(c^2)*x^2 + sqrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(1/4) - sqrt(2)*l
og(sqrt(c^2)*x^2 - sqrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(1/4) + sqrt(2)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqrt(c^
2)) + sqrt(2)*(c^2)^(1/4))/(2*sqrt(c^2)*x + sqrt(2)*sqrt(-sqrt(c^2)) + sqrt(2)*(c^2)^(1/4)))/sqrt(-sqrt(c^2))
+ sqrt(2)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqrt(c^2)) - sqrt(2)*(c^2)^(1/4))/(2*sqrt(c^2)*x + sqrt(2)*sqrt(-
sqrt(c^2)) - sqrt(2)*(c^2)^(1/4)))/sqrt(-sqrt(c^2)))/c^2 - 8*x/c^2))*b*e^2 + a*d^2*x + 1/2*(2*c*x^2*arctan(c*x
^2) - log(c^2*x^4 + 1))*b*d*e/c

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Fricas [B]  time = 4.64603, size = 9956, normalized size = 39.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*sqrt(2)*c^7*sqrt((81*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8
+ 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b
^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(3/4)*sqrt((81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)*arctan((sqrt(2)
*(c^11*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)*sqrt((81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4
+ b^4*e^8)/c^6) + 3*(9*b^2*c^11*d^6 + b^2*c^9*d^2*e^4)*sqrt((81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c
^6))*sqrt((6561*b^6*c^8*d^16 - 162*b^6*c^4*d^8*e^8 + b^6*e^16)*x^2 + sqrt(2)*(3*(81*b^3*c^9*d^10 - 18*b^3*c^7*
d^6*e^4 + b^3*c^5*d^2*e^8)*x*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6) + (729*b^5*c^7*d^12*e^2
- 81*b^5*c^5*d^8*e^6 - 9*b^5*c^3*d^4*e^10 + b^5*c*e^14)*x)*sqrt((81*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^
8 - 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*
e^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(1/4) + (729*b^4*c^8*d^12 - 81*b^4*c^6*d
^8*e^4 - 9*b^4*c^4*d^4*e^8 + b^4*c^2*e^12)*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))*sqrt((81
*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^
8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6
)^(3/4) + sqrt(2)*((81*b^3*c^15*d^8*e^2 - b^3*c^11*e^10)*x*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8
)/c^6)*sqrt((81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6) + 3*(729*b^5*c^15*d^14 + 81*b^5*c^13*d^10*e^4
- 9*b^5*c^11*d^6*e^8 - b^5*c^9*d^2*e^12)*x*sqrt((81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))*sqrt((8
1*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e
^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^
6)^(3/4) + (6561*b^6*c^14*d^16 + 1458*b^6*c^12*d^12*e^4 - 18*b^6*c^8*d^4*e^12 - b^6*c^6*e^16)*sqrt((81*b^4*c^4
*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)*sqrt((81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(531441*b
^10*c^12*d^24 + 118098*b^10*c^10*d^20*e^4 - 6561*b^10*c^8*d^16*e^8 - 2916*b^10*c^6*d^12*e^12 - 81*b^10*c^4*d^8
*e^16 + 18*b^10*c^2*d^4*e^20 + b^10*e^24)) + 4*sqrt(2)*c^7*sqrt((81*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8
- 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e
^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(3/4)*sqrt((81*b^4*c^4*d^8 - 18*b^4*c^2*d
^4*e^4 + b^4*e^8)/c^6)*arctan((sqrt(2)*(c^11*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)*sqr
t((81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6) + 3*(9*b^2*c^11*d^6 + b^2*c^9*d^2*e^4)*sqrt((81*b^4*c^4
*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))*sqrt((6561*b^6*c^8*d^16 - 162*b^6*c^4*d^8*e^8 + b^6*e^16)*x^2 - sqr
t(2)*(3*(81*b^3*c^9*d^10 - 18*b^3*c^7*d^6*e^4 + b^3*c^5*d^2*e^8)*x*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 +
b^4*e^8)/c^6) + (729*b^5*c^7*d^12*e^2 - 81*b^5*c^5*d^8*e^6 - 9*b^5*c^3*d^4*e^10 + b^5*c*e^14)*x)*sqrt((81*b^2
*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c
^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(1
/4) + (729*b^4*c^8*d^12 - 81*b^4*c^6*d^8*e^4 - 9*b^4*c^4*d^4*e^8 + b^4*c^2*e^12)*sqrt((81*b^4*c^4*d^8 + 18*b^4
*c^2*d^4*e^4 + b^4*e^8)/c^6))*sqrt((81*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2*sqrt((81*b^4
*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*((81*b^4*c^4*d
^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(3/4) + sqrt(2)*((81*b^3*c^15*d^8*e^2 - b^3*c^11*e^10)*x*sqrt((81*b^4*
c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)*sqrt((81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6) + 3*(72
9*b^5*c^15*d^14 + 81*b^5*c^13*d^10*e^4 - 9*b^5*c^11*d^6*e^8 - b^5*c^9*d^2*e^12)*x*sqrt((81*b^4*c^4*d^8 - 18*b^
4*c^2*d^4*e^4 + b^4*e^8)/c^6))*sqrt((81*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2*sqrt((81*b^
4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*((81*b^4*c^4*
d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(3/4) - (6561*b^6*c^14*d^16 + 1458*b^6*c^12*d^12*e^4 - 18*b^6*c^8*d^4
*e^12 - b^6*c^6*e^16)*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)*sqrt((81*b^4*c^4*d^8 - 18*b^4*
c^2*d^4*e^4 + b^4*e^8)/c^6))/(531441*b^10*c^12*d^24 + 118098*b^10*c^10*d^20*e^4 - 6561*b^10*c^8*d^16*e^8 - 291
6*b^10*c^6*d^12*e^12 - 81*b^10*c^4*d^8*e^16 + 18*b^10*c^2*d^4*e^20 + b^10*e^24)) - 4*(81*a*b^4*c^5*d^8*e^2 + 1
8*a*b^4*c^3*d^4*e^6 + a*b^4*c*e^10)*x^3 - 12*(81*a*b^4*c^5*d^9*e + 18*a*b^4*c^3*d^5*e^5 + a*b^4*c*d*e^9)*x^2 -
4*(243*a*b^4*c^5*d^10 - 162*b^5*c^4*d^8*e^2 + 54*a*b^4*c^3*d^6*e^4 - 36*b^5*c^2*d^4*e^6 + 3*a*b^4*c*d^2*e^8 -
2*b^5*e^10)*x - 4*((81*b^5*c^5*d^8*e^2 + 18*b^5*c^3*d^4*e^6 + b^5*c*e^10)*x^3 + 3*(81*b^5*c^5*d^9*e + 18*b^5*
c^3*d^5*e^5 + b^5*c*d*e^9)*x^2 + 3*(81*b^5*c^5*d^10 + 18*b^5*c^3*d^6*e^4 + b^5*c*d^2*e^8)*x)*arctan(c*x^2) + (
486*b^5*c^4*d^9*e + 108*b^5*c^2*d^5*e^5 + 6*b^5*d*e^9 - sqrt(2)*(81*b^4*c^5*d^8 + 18*b^4*c^3*d^4*e^4 + b^4*c*e
^8 + 6*b^2*c^5*d^2*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))*sqrt((81*b^2*c^4*d^8 + 18*b^
2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4
*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(1/4))*log((6561*b
^6*c^8*d^16 - 162*b^6*c^4*d^8*e^8 + b^6*e^16)*x^2 + sqrt(2)*(3*(81*b^3*c^9*d^10 - 18*b^3*c^7*d^6*e^4 + b^3*c^5
*d^2*e^8)*x*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6) + (729*b^5*c^7*d^12*e^2 - 81*b^5*c^5*d^8
*e^6 - 9*b^5*c^3*d^4*e^10 + b^5*c*e^14)*x)*sqrt((81*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2
*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*(
(81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(1/4) + (729*b^4*c^8*d^12 - 81*b^4*c^6*d^8*e^4 - 9*b^4*c^
4*d^4*e^8 + b^4*c^2*e^12)*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)) + (486*b^5*c^4*d^9*e + 10
8*b^5*c^2*d^5*e^5 + 6*b^5*d*e^9 + sqrt(2)*(81*b^4*c^5*d^8 + 18*b^4*c^3*d^4*e^4 + b^4*c*e^8 + 6*b^2*c^5*d^2*e^2
*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))*sqrt((81*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^
8 - 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*
e^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(1/4))*log((6561*b^6*c^8*d^16 - 162*b^6*
c^4*d^8*e^8 + b^6*e^16)*x^2 - sqrt(2)*(3*(81*b^3*c^9*d^10 - 18*b^3*c^7*d^6*e^4 + b^3*c^5*d^2*e^8)*x*sqrt((81*b
^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6) + (729*b^5*c^7*d^12*e^2 - 81*b^5*c^5*d^8*e^6 - 9*b^5*c^3*d^4*e
^10 + b^5*c*e^14)*x)*sqrt((81*b^2*c^4*d^8 + 18*b^2*c^2*d^4*e^4 + b^2*e^8 - 6*c^4*d^2*e^2*sqrt((81*b^4*c^4*d^8
+ 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/(81*b^2*c^4*d^8 - 18*b^2*c^2*d^4*e^4 + b^2*e^8))*((81*b^4*c^4*d^8 + 18*b
^4*c^2*d^4*e^4 + b^4*e^8)/c^6)^(1/4) + (729*b^4*c^8*d^12 - 81*b^4*c^6*d^8*e^4 - 9*b^4*c^4*d^4*e^8 + b^4*c^2*e^
12)*sqrt((81*b^4*c^4*d^8 + 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6)))/(81*b^4*c^5*d^8 + 18*b^4*c^3*d^4*e^4 + b^4*c*e
^8)

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Sympy [A]  time = 49.9778, size = 3135, normalized size = 12.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*(d**2*x + d*e*x**2 + e**2*x**3/3), Eq(c, 0)), ((a - oo*I*b)*(d**2*x + d*e*x**2 + e**2*x**3/3), Eq
(c, -I/x**2)), ((a + oo*I*b)*(d**2*x + d*e*x**2 + e**2*x**3/3), Eq(c, I/x**2)), (11*(-1)**(1/4)*a*c**10*e**2*x
**4*(c**(-2))**(17/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(
-2))**(3/2)) + 11*(-1)**(1/4)*a*c**8*e**2*(c**(-2))**(17/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(-24*I*c**6*x
**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 11*(-1)**(3/4)*a*c**7*e**2*(c**(-2))**(15/4)*atan(c*x**2)
/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*I*a*c**6*d**2*x**5*(c**(-2))**(3/2)/(-24
*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*I*a*c**6*d*e*x**6*(c**(-2))**(3/2)/(-24*I*c**
6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 8*I*a*c**6*e**2*x**7*(c**(-2))**(3/2)/(-24*I*c**6*x**4
*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 11*(-1)**(1/4)*a*c**6*e**2*x**4*(c**(-2))**(9/4)*atan((-1)**
(3/4)*x/(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*I*a*c**4*d**2*x
*(c**(-2))**(3/2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*I*a*c**4*d*e*x**2*(c**(
-2))**(3/2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 8*I*a*c**4*e**2*x**3*(c**(-2))**
(3/2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 11*(-1)**(1/4)*a*c**4*e**2*(c**(-2))**
(9/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 1
1*(-1)**(3/4)*a*c**3*e**2*(c**(-2))**(7/4)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2)
)**(3/2)) + 66*(-1)**(1/4)*b*c**11*d**2*x**4*(c**(-2))**(17/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(-24*I*c**
6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 66*(-1)**(1/4)*b*c**9*d**2*(c**(-2))**(17/4)*atan((-1)
**(3/4)*x/(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 114*(-1)**(3/4)*
b*c**8*d**2*(c**(-2))**(15/4)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 9
0*(-1)**(1/4)*b*c**7*d**2*x**4*(c**(-2))**(9/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2)
)**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*I*b*c**6*d**2*x**5*(c**(-2))**(3/2)*atan(c*x**2)/(-24*I*c**6*x**4*
(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*I*b*c**6*d*e*x**6*(c**(-2))**(3/2)*atan(c*x**2)/(-24*I*c**
6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 8*I*b*c**6*e**2*x**7*(c**(-2))**(3/2)*atan(c*x**2)/(-2
4*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 24*(-1)**(1/4)*b*c**5*d**2*x**4*(c**(-2))**(5/4
)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 12*(
-1)**(1/4)*b*c**5*d**2*x**4*(c**(-2))**(5/4)*log(x**2 + I*sqrt(c**(-2)))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 2
4*I*c**4*(c**(-2))**(3/2)) - 90*(-1)**(1/4)*b*c**5*d**2*(c**(-2))**(9/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/
(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 24*I*b*c**5*d*e*x**4*(c**(-2))**(3/2)*log(x*
*2 + I*sqrt(c**(-2)))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 16*I*b*c**5*e**2*x**5*
(c**(-2))**(3/2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 8*(-1)**(3/4)*b*c**5*e**2*x
**4*(c**(-2))**(7/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(
-2))**(3/2)) - 4*(-1)**(3/4)*b*c**5*e**2*x**4*(c**(-2))**(7/4)*log(x**2 + I*sqrt(c**(-2)))/(-24*I*c**6*x**4*(c
**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 8*(-1)**(3/4)*b*c**5*e**2*x**4*(c**(-2))**(7/4)*atan((-1)**(3/4
)*x/(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*(-1)**(3/4)*b*c**4*
d**2*x**4*(c**(-2))**(3/4)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*I
*b*c**4*d**2*x*(c**(-2))**(3/2)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) +
90*(-1)**(3/4)*b*c**4*d**2*(c**(-2))**(7/4)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-
2))**(3/2)) - 24*I*b*c**4*d*e*x**2*(c**(-2))**(3/2)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4
*(c**(-2))**(3/2)) - 8*I*b*c**4*e**2*x**3*(c**(-2))**(3/2)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24
*I*c**4*(c**(-2))**(3/2)) + 24*(-1)**(1/4)*b*c**3*d**2*(c**(-2))**(5/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/
(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 12*(-1)**(1/4)*b*c**3*d**2*(c**(-2))**(5/4)*
log(x**2 + I*sqrt(c**(-2)))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 24*I*b*c**3*d*e*
(c**(-2))**(3/2)*log(x**2 + I*sqrt(c**(-2)))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) +
16*I*b*c**3*e**2*x*(c**(-2))**(3/2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 8*(-1)*
*(3/4)*b*c**3*e**2*(c**(-2))**(7/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) -
24*I*c**4*(c**(-2))**(3/2)) - 4*(-1)**(3/4)*b*c**3*e**2*(c**(-2))**(7/4)*log(x**2 + I*sqrt(c**(-2)))/(-24*I*c*
*6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 8*(-1)**(3/4)*b*c**3*e**2*(c**(-2))**(7/4)*atan((-1)*
*(3/4)*x/(c**(-2))**(1/4))/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*b*c**2*d*e*x**
4*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 8*(-1)**(1/4)*b*c**2*e**2*x**
4*(c**(-2))**(1/4)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) - 24*b*d*e*ata
n(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)) + 8*(-1)**(1/4)*b*e**2*(c**(-2))**(1
/4)*atan(c*x**2)/(-24*I*c**6*x**4*(c**(-2))**(3/2) - 24*I*c**4*(c**(-2))**(3/2)), True))

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Giac [A]  time = 1.8809, size = 510, normalized size = 2.04 \begin{align*} \frac{1}{12} \, b c^{5}{\left (\frac{2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{6} \sqrt{{\left | c \right |}}} + \frac{2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{6} \sqrt{{\left | c \right |}}} + \frac{\sqrt{2} \log \left (x^{2} + \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{6} \sqrt{{\left | c \right |}}} - \frac{\sqrt{2} \log \left (x^{2} - \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{6} \sqrt{{\left | c \right |}}}\right )} e^{2} - \frac{1}{4} \, b c^{3} d^{2}{\left (\frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{4}} + \frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{4}} - \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (x^{2} + \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{4}} + \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (x^{2} - \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{4}}\right )} + \frac{2 \, b c x^{3} \arctan \left (c x^{2}\right ) e^{2} + 6 \, b c d x^{2} \arctan \left (c x^{2}\right ) e + 6 \, b c d^{2} x \arctan \left (c x^{2}\right ) + 2 \, a c x^{3} e^{2} + 6 \, a c d x^{2} e + 6 \, a c d^{2} x - 3 \, b d e \log \left (c^{2} x^{4} + 1\right ) - 4 \, b x 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^2)),x, algorithm="giac")

[Out]

1/12*b*c^5*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/(c^6*sqrt(abs(c))) + 2*sqr
t(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/(c^6*sqrt(abs(c))) + sqrt(2)*log(x^2 + sqrt
(2)*x/sqrt(abs(c)) + 1/abs(c))/(c^6*sqrt(abs(c))) - sqrt(2)*log(x^2 - sqrt(2)*x/sqrt(abs(c)) + 1/abs(c))/(c^6*
sqrt(abs(c))))*e^2 - 1/4*b*c^3*d^2*(2*sqrt(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(c)))*sqr
t(abs(c)))/c^4 + 2*sqrt(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/c^4 - sq
rt(2)*sqrt(abs(c))*log(x^2 + sqrt(2)*x/sqrt(abs(c)) + 1/abs(c))/c^4 + sqrt(2)*sqrt(abs(c))*log(x^2 - sqrt(2)*x
/sqrt(abs(c)) + 1/abs(c))/c^4) + 1/6*(2*b*c*x^3*arctan(c*x^2)*e^2 + 6*b*c*d*x^2*arctan(c*x^2)*e + 6*b*c*d^2*x*
arctan(c*x^2) + 2*a*c*x^3*e^2 + 6*a*c*d*x^2*e + 6*a*c*d^2*x - 3*b*d*e*log(c^2*x^4 + 1) - 4*b*x*e^2)/c