\(\int (d+e x)^2 (a+b \arctan (c x^2)) \, dx\) [21]

Optimal result

Integrand size = 18, antiderivative size = 250 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {2 b e^2 x}{3 c}-\frac {b d^3 \arctan \left (c x^2\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \left (c x^2\right )\right )}{3 e}+\frac {b \left (3 c d^2-e^2\right ) \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}-\frac {b \left (3 c d^2-e^2\right ) \arctan \left (1+\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}-\frac {b \left (3 c d^2+e^2\right ) \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \left (3 c d^2+e^2\right ) \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{6 \sqrt {2} c^{3/2}}-\frac {b d e \log \left (1+c^2 x^4\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {4980, 1845, 1262, 649, 209, 266, 1294, 1182, 1176, 631, 210, 1179, 642} \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\frac {(d+e x)^3 \left (a+b \arctan \left (c x^2\right )\right )}{3 e}+\frac {b \arctan \left (1-\sqrt {2} \sqrt {c} x\right ) \left (3 c d^2-e^2\right )}{3 \sqrt {2} c^{3/2}}-\frac {b \arctan \left (\sqrt {2} \sqrt {c} x+1\right ) \left (3 c d^2-e^2\right )}{3 \sqrt {2} c^{3/2}}-\frac {b d^3 \arctan \left (c x^2\right )}{3 e}-\frac {b \left (3 c d^2+e^2\right ) \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{6 \sqrt {2} c^{3/2}}+\frac {b \left (3 c d^2+e^2\right ) \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{6 \sqrt {2} c^{3/2}}-\frac {b d e \log \left (c^2 x^4+1\right )}{2 c}-\frac {2 b e^2 x}{3 c} \]

[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 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 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 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 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 1176

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 1179

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 1182

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 1262

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 1294

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 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*} \text {integral}& = \frac {(d+e x)^3 \left (a+b \arctan \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 \arctan \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 \arctan \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 \arctan \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) \text {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 \arctan \left (c x^2\right )\right )}{3 e}-\frac {\left (b c d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^2\right )}{3 e}-(b c d e) \text {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 \arctan \left (c x^2\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \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 \arctan \left (c x^2\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \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 ) \text {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 ) \text {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 \arctan \left (c x^2\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \arctan \left (c x^2\right )\right )}{3 e}+\frac {b \left (3 c d^2-e^2\right ) \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{3 \sqrt {2} c^{3/2}}-\frac {b \left (3 c d^2-e^2\right ) \arctan \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] (verified)

Time = 3.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\frac {1}{12} \left (12 a d^2 x-\frac {8 b e^2 x}{c}+12 a d e x^2+4 a e^2 x^3+4 b x \left (3 d^2+3 d e x+e^2 x^2\right ) \arctan \left (c x^2\right )+\frac {2 \sqrt {2} b \left (3 c d^2-e^2\right ) \arctan \left (1-\sqrt {2} \sqrt {c} x\right )}{c^{3/2}}-\frac {2 \sqrt {2} b \left (3 c d^2-e^2\right ) \arctan \left (1+\sqrt {2} \sqrt {c} x\right )}{c^{3/2}}-\frac {\sqrt {2} b \left (3 c d^2+e^2\right ) \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{c^{3/2}}+\frac {\sqrt {2} b \left (3 c d^2+e^2\right ) \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{c^{3/2}}-\frac {6 b d e \log \left (1+c^2 x^4\right )}{c}\right ) \]

[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

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.21

method	result	size
default	\(\frac {a \left (e x +d \right )^{3}}{3 e}+b \left (\frac {e^{2} \arctan \left (c \,x^{2}\right ) x^{3}}{3}+e \arctan \left (c \,x^{2}\right ) d \,x^{2}+\arctan \left (c \,x^{2}\right ) d^{2} x +\frac {\arctan \left (c \,x^{2}\right ) d^{3}}{3 e}-\frac {2 c \left (\frac {e^{3} x}{c^{2}}+\frac {-\frac {e^{3} \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {c^{2} d^{3} \arctan \left (x^{2} \sqrt {c^{2}}\right )}{2 \sqrt {c^{2}}}+\frac {3 d^{2} e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+\frac {3 d \,e^{2} \ln \left (c^{2} x^{4}+1\right )}{4}}{c^{2}}\right )}{3 e}\right )\)	\(303\)
parts	\(\frac {a \left (e x +d \right )^{3}}{3 e}+b \left (\frac {e^{2} \arctan \left (c \,x^{2}\right ) x^{3}}{3}+e \arctan \left (c \,x^{2}\right ) d \,x^{2}+\arctan \left (c \,x^{2}\right ) d^{2} x +\frac {\arctan \left (c \,x^{2}\right ) d^{3}}{3 e}-\frac {2 c \left (\frac {e^{3} x}{c^{2}}+\frac {-\frac {e^{3} \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {c^{2} d^{3} \arctan \left (x^{2} \sqrt {c^{2}}\right )}{2 \sqrt {c^{2}}}+\frac {3 d^{2} e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+\frac {3 d \,e^{2} \ln \left (c^{2} x^{4}+1\right )}{4}}{c^{2}}\right )}{3 e}\right )\)	\(303\)

[In]

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

[Out]

1/3*a*(e*x+d)^3/e+b*(1/3*e^2*arctan(c*x^2)*x^3+e*arctan(c*x^2)*d*x^2+arctan(c*x^2)*d^2*x+1/3/e*arctan(c*x^2)*d
^3-2/3*c/e*(1/c^2*e^3*x+1/c^2*(-1/8*e^3*(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)))+2*arctan(2^(1/2)/(1/c^2)^(1/4)*x+1)+2*arctan(2^(1/2)/(1/c^2)^(1/4
)*x-1))+1/2*c^2*d^3/(c^2)^(1/2)*arctan(x^2*(c^2)^(1/2))+3/8*d^2*e/(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)))+2*arctan(2^(1/2)/(1/c^2)^(1/4)*x+1)+2*a
rctan(2^(1/2)/(1/c^2)^(1/4)*x-1))+3/4*d*e^2*ln(c^2*x^4+1))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1013 vs. \(2 (201) = 402\).

Time = 0.28 (sec) , antiderivative size = 1013, normalized size of antiderivative = 4.05 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\frac {2 \, a c e^{2} x^{3} + 6 \, a c d e x^{2} + 2 \, {\left (3 \, a c d^{2} - 2 \, b e^{2}\right )} x + 2 \, {\left (b c e^{2} x^{3} + 3 \, b c d e x^{2} + 3 \, b c d^{2} x\right )} \arctan \left (c x^{2}\right ) - {\left (3 \, b d e + c \sqrt {\frac {6 \, b^{2} d^{2} e^{2} + c^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}}{c^{2}}}\right )} \log \left (-{\left (81 \, b^{3} c^{4} d^{8} - b^{3} e^{8}\right )} x + {\left (9 \, b^{2} c^{3} d^{4} e^{2} - b^{2} c e^{6} - 3 \, c^{5} d^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}\right )} \sqrt {\frac {6 \, b^{2} d^{2} e^{2} + c^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}}{c^{2}}}\right ) - {\left (3 \, b d e - c \sqrt {\frac {6 \, b^{2} d^{2} e^{2} + c^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}}{c^{2}}}\right )} \log \left (-{\left (81 \, b^{3} c^{4} d^{8} - b^{3} e^{8}\right )} x - {\left (9 \, b^{2} c^{3} d^{4} e^{2} - b^{2} c e^{6} - 3 \, c^{5} d^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}\right )} \sqrt {\frac {6 \, b^{2} d^{2} e^{2} + c^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}}{c^{2}}}\right ) - {\left (3 \, b d e + c \sqrt {\frac {6 \, b^{2} d^{2} e^{2} - c^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}}{c^{2}}}\right )} \log \left (-{\left (81 \, b^{3} c^{4} d^{8} - b^{3} e^{8}\right )} x + {\left (9 \, b^{2} c^{3} d^{4} e^{2} - b^{2} c e^{6} + 3 \, c^{5} d^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}\right )} \sqrt {\frac {6 \, b^{2} d^{2} e^{2} - c^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}}{c^{2}}}\right ) - {\left (3 \, b d e - c \sqrt {\frac {6 \, b^{2} d^{2} e^{2} - c^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}}{c^{2}}}\right )} \log \left (-{\left (81 \, b^{3} c^{4} d^{8} - b^{3} e^{8}\right )} x - {\left (9 \, b^{2} c^{3} d^{4} e^{2} - b^{2} c e^{6} + 3 \, c^{5} d^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}\right )} \sqrt {\frac {6 \, b^{2} d^{2} e^{2} - c^{2} \sqrt {-\frac {81 \, b^{4} c^{4} d^{8} - 18 \, b^{4} c^{2} d^{4} e^{4} + b^{4} e^{8}}{c^{6}}}}{c^{2}}}\right )}{6 \, c} \]

[In]

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

[Out]

1/6*(2*a*c*e^2*x^3 + 6*a*c*d*e*x^2 + 2*(3*a*c*d^2 - 2*b*e^2)*x + 2*(b*c*e^2*x^3 + 3*b*c*d*e*x^2 + 3*b*c*d^2*x)
*arctan(c*x^2) - (3*b*d*e + c*sqrt((6*b^2*d^2*e^2 + c^2*sqrt(-(81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/
c^6))/c^2))*log(-(81*b^3*c^4*d^8 - b^3*e^8)*x + (9*b^2*c^3*d^4*e^2 - b^2*c*e^6 - 3*c^5*d^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((6*b^2*d^2*e^2 + c^2*sqrt(-(81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4
+ b^4*e^8)/c^6))/c^2)) - (3*b*d*e - c*sqrt((6*b^2*d^2*e^2 + c^2*sqrt(-(81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b
^4*e^8)/c^6))/c^2))*log(-(81*b^3*c^4*d^8 - b^3*e^8)*x - (9*b^2*c^3*d^4*e^2 - b^2*c*e^6 - 3*c^5*d^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((6*b^2*d^2*e^2 + c^2*sqrt(-(81*b^4*c^4*d^8 - 18*b^4*c^2*
d^4*e^4 + b^4*e^8)/c^6))/c^2)) - (3*b*d*e + c*sqrt((6*b^2*d^2*e^2 - c^2*sqrt(-(81*b^4*c^4*d^8 - 18*b^4*c^2*d^4
*e^4 + b^4*e^8)/c^6))/c^2))*log(-(81*b^3*c^4*d^8 - b^3*e^8)*x + (9*b^2*c^3*d^4*e^2 - b^2*c*e^6 + 3*c^5*d^2*sqr
t(-(81*b^4*c^4*d^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))*sqrt((6*b^2*d^2*e^2 - c^2*sqrt(-(81*b^4*c^4*d^8 - 18*
b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/c^2)) - (3*b*d*e - c*sqrt((6*b^2*d^2*e^2 - c^2*sqrt(-(81*b^4*c^4*d^8 - 18*b^4
*c^2*d^4*e^4 + b^4*e^8)/c^6))/c^2))*log(-(81*b^3*c^4*d^8 - b^3*e^8)*x - (9*b^2*c^3*d^4*e^2 - b^2*c*e^6 + 3*c^5
*d^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((6*b^2*d^2*e^2 - c^2*sqrt(-(81*b^4*c^4*d
^8 - 18*b^4*c^2*d^4*e^4 + b^4*e^8)/c^6))/c^2)))/c

Sympy [A] (verification not implemented)

Time = 10.18 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.61 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {atan}{\left (c x^{2} \right )} + b d e x^{2} \operatorname {atan}{\left (c x^{2} \right )} + \frac {b e^{2} x^{3} \operatorname {atan}{\left (c x^{2} \right )}}{3} - \frac {b d^{2} \log {\left (x - \sqrt [4]{- \frac {1}{c^{2}}} \right )}}{c \sqrt [4]{- \frac {1}{c^{2}}}} + \frac {b d^{2} \log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{2 c \sqrt [4]{- \frac {1}{c^{2}}}} - \frac {b d^{2} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{c^{2}}}} \right )}}{c \sqrt [4]{- \frac {1}{c^{2}}}} - \frac {b d e \log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{c} - \frac {2 b e^{2} x}{3 c} - \frac {b d^{2} \operatorname {atan}{\left (c x^{2} \right )}}{c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{4}}} + \frac {b d e \operatorname {atan}{\left (c x^{2} \right )}}{c^{2} \sqrt {- \frac {1}{c^{2}}}} - \frac {b e^{2} \operatorname {atan}{\left (c x^{2} \right )}}{3 c^{2} \sqrt [4]{- \frac {1}{c^{2}}}} + \frac {b e^{2} \log {\left (x - \sqrt [4]{- \frac {1}{c^{2}}} \right )}}{3 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{4}}} - \frac {b e^{2} \log {\left (x^{2} + \sqrt {- \frac {1}{c^{2}}} \right )}}{6 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{4}}} - \frac {b e^{2} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{c^{2}}}} \right )}}{3 c^{3} \left (- \frac {1}{c^{2}}\right )^{\frac {3}{4}}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 + b*d**2*x*atan(c*x**2) + b*d*e*x**2*atan(c*x**2) + b*e**2*x*
*3*atan(c*x**2)/3 - b*d**2*log(x - (-1/c**2)**(1/4))/(c*(-1/c**2)**(1/4)) + b*d**2*log(x**2 + sqrt(-1/c**2))/(
2*c*(-1/c**2)**(1/4)) - b*d**2*atan(x/(-1/c**2)**(1/4))/(c*(-1/c**2)**(1/4)) - b*d*e*log(x**2 + sqrt(-1/c**2))
/c - 2*b*e**2*x/(3*c) - b*d**2*atan(c*x**2)/(c**2*(-1/c**2)**(3/4)) + b*d*e*atan(c*x**2)/(c**2*sqrt(-1/c**2))
- b*e**2*atan(c*x**2)/(3*c**2*(-1/c**2)**(1/4)) + b*e**2*log(x - (-1/c**2)**(1/4))/(3*c**3*(-1/c**2)**(3/4)) -
b*e**2*log(x**2 + sqrt(-1/c**2))/(6*c**3*(-1/c**2)**(3/4)) - b*e**2*atan(x/(-1/c**2)**(1/4))/(3*c**3*(-1/c**2
)**(3/4)), Ne(c, 0)), (a*(d**2*x + d*e*x**2 + e**2*x**3/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.29 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} - \frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b d^{2} + \frac {1}{12} \, {\left (4 \, x^{3} \arctan \left (c x^{2}\right ) - c {\left (\frac {8 \, x}{c^{2}} - \frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{\sqrt {c}} - \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{\sqrt {c}}}{c^{2}}\right )}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b d e}{2 \, c} \]

[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*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/c^(3/2) +
2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/c^(3/2) - sqrt(2)*log(c*x^2 + sqrt(2)*sqrt(c)*
x + 1)/c^(3/2) + sqrt(2)*log(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/c^(3/2)) - 4*x*arctan(c*x^2))*b*d^2 + 1/12*(4*x^3*
arctan(c*x^2) - c*(8*x/c^2 - (2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/sqrt(c) + 2*sqrt
(2)*arctan(1/2*sqrt(2)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/sqrt(c) + sqrt(2)*log(c*x^2 + sqrt(2)*sqrt(c)*x + 1)
/sqrt(c) - sqrt(2)*log(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/sqrt(c))/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

Giac [A] (verification not implemented)

none

Time = 1.05 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.22 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^2\right )\right ) \, dx=-\frac {b d e \log \left (c^{2} x^{4} + 1\right )}{2 \, c} + \frac {b c e^{2} x^{3} \arctan \left (c x^{2}\right ) + a c e^{2} x^{3} + 3 \, b c d e x^{2} \arctan \left (c x^{2}\right ) + 3 \, a c d e x^{2} + 3 \, b c d^{2} x \arctan \left (c x^{2}\right ) + 3 \, a c d^{2} x - 2 \, b e^{2} x}{3 \, c} - \frac {\sqrt {2} {\left (3 \, b c^{2} d^{2} - b e^{2} {\left | c \right |}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{6 \, c {\left | c \right |}^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (3 \, b c^{2} d^{2} - b e^{2} {\left | c \right |}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{6 \, c {\left | c \right |}^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (3 \, b c^{2} d^{2} + b e^{2} {\left | c \right |}\right )} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{12 \, c {\left | c \right |}^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (3 \, b c^{2} d^{2} \sqrt {{\left | c \right |}} + b e^{2} {\left | c \right |}^{\frac {3}{2}}\right )} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{12 \, c^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.59 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.68 \[ \int (d+e x)^2 \left (a+b \arctan \left (c x^2\right )\right ) \, dx=\frac {a\,e^2\,x^3}{3}+a\,d^2\,x+\frac {b\,e^2\,x^3\,\mathrm {atan}\left (c\,x^2\right )}{3}+a\,d\,e\,x^2-\frac {3\,b\,d^2\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}-1\right )\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}}{2}+\frac {3\,b\,d^2\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}+1\right )\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}}{2}-\frac {b\,d^2\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}+1{}\mathrm {i}\right )\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}\,3{}\mathrm {i}}{2}+\frac {b\,d^2\,\ln \left (1+c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}\,3{}\mathrm {i}\right )\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}\,3{}\mathrm {i}}{2}-\frac {2\,b\,e^2\,x}{3\,c}+b\,d^2\,x\,\mathrm {atan}\left (c\,x^2\right )+\frac {b\,e^2\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}-1\right )\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}\,1{}\mathrm {i}}{2\,c}-\frac {b\,e^2\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}+1\right )\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}\,1{}\mathrm {i}}{2\,c}+\frac {b\,e^2\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}+1{}\mathrm {i}\right )\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}}{2\,c}-\frac {b\,e^2\,\ln \left (1+c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}\,3{}\mathrm {i}\right )\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}}{2\,c}+b\,d\,e\,x^2\,\mathrm {atan}\left (c\,x^2\right )-\frac {b\,d\,e\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}-1\right )}{2\,c}-\frac {b\,d\,e\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}+1\right )}{2\,c}-\frac {b\,d\,e\,\ln \left (3\,c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}+1{}\mathrm {i}\right )}{2\,c}-\frac {b\,d\,e\,\ln \left (1+c\,x\,\sqrt {\frac {1{}\mathrm {i}}{9\,c}}\,3{}\mathrm {i}\right )}{2\,c} \]

[In]

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

[Out]

(a*e^2*x^3)/3 + a*d^2*x + (b*e^2*x^3*atan(c*x^2))/3 + a*d*e*x^2 - (3*b*d^2*log(3*c*x*(1i/(9*c))^(1/2) - 1)*(1i
/(9*c))^(1/2))/2 + (3*b*d^2*log(3*c*x*(1i/(9*c))^(1/2) + 1)*(1i/(9*c))^(1/2))/2 - (b*d^2*log(3*c*x*(1i/(9*c))^
(1/2) + 1i)*(1i/(9*c))^(1/2)*3i)/2 + (b*d^2*log(c*x*(1i/(9*c))^(1/2)*3i + 1)*(1i/(9*c))^(1/2)*3i)/2 - (2*b*e^2
*x)/(3*c) + b*d^2*x*atan(c*x^2) + (b*e^2*log(3*c*x*(1i/(9*c))^(1/2) - 1)*(1i/(9*c))^(1/2)*1i)/(2*c) - (b*e^2*l
og(3*c*x*(1i/(9*c))^(1/2) + 1)*(1i/(9*c))^(1/2)*1i)/(2*c) + (b*e^2*log(3*c*x*(1i/(9*c))^(1/2) + 1i)*(1i/(9*c))
^(1/2))/(2*c) - (b*e^2*log(c*x*(1i/(9*c))^(1/2)*3i + 1)*(1i/(9*c))^(1/2))/(2*c) + b*d*e*x^2*atan(c*x^2) - (b*d
*e*log(3*c*x*(1i/(9*c))^(1/2) - 1))/(2*c) - (b*d*e*log(3*c*x*(1i/(9*c))^(1/2) + 1))/(2*c) - (b*d*e*log(3*c*x*(
1i/(9*c))^(1/2) + 1i))/(2*c) - (b*d*e*log(c*x*(1i/(9*c))^(1/2)*3i + 1))/(2*c)