\(\int \frac {x^2 (a+b \arctan (c x))}{(d+e x^2)^3} \, dx\) [1170]

Optimal result

Integrand size = 21, antiderivative size = 966 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac {x (a+b \arctan (c x))}{4 e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{8 d e \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+\frac {i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d \left (c^2 d-e\right )^2 e}-\frac {b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right ) e}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d \left (c^2 d-e\right )^2 e}+\frac {b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right ) e}+\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{3/2} e^{3/2}} \]

[Out]

1/8*b*c/(c^2*d-e)/e/(e*x^2+d)-1/4*x*(a+b*arctan(c*x))/e/(e*x^2+d)^2+1/8*x*(a+b*arctan(c*x))/d/e/(e*x^2+d)+1/8*
(a+b*arctan(c*x))*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)+1/16*b*c*(5*c^2*d-3*e)*ln(c^2*x^2+1)/d/(c^2*d-e)^2
/e-1/4*b*c*ln(c^2*x^2+1)/d/(c^2*d-e)/e-1/16*b*c*(5*c^2*d-3*e)*ln(e*x^2+d)/d/(c^2*d-e)^2/e+1/4*b*c*ln(e*x^2+d)/
d/(c^2*d-e)/e-1/32*I*b*c*ln(-(1-x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)-e^(1/2)))*ln(1+I*x*e^(1/2)/d^(
1/2))/d^(3/2)/e^(3/2)/(-c^2)^(1/2)-1/32*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)+I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/
2)+I*e^(1/2)))/d^(3/2)/e^(3/2)/(-c^2)^(1/2)+1/32*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)-I*x*e^(1/2))/((-c^2)^(1
/2)*d^(1/2)-I*e^(1/2)))/d^(3/2)/e^(3/2)/(-c^2)^(1/2)-1/32*I*b*c*ln(-(1+x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)
*d^(1/2)-e^(1/2)))*ln(1-I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)/(-c^2)^(1/2)-1/32*I*b*c*polylog(2,(-c^2)^(1/2)*(d
^(1/2)-I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)+I*e^(1/2)))/d^(3/2)/e^(3/2)/(-c^2)^(1/2)+1/32*I*b*c*polylog(2,(-c^2)
^(1/2)*(d^(1/2)+I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)-I*e^(1/2)))/d^(3/2)/e^(3/2)/(-c^2)^(1/2)+1/32*I*b*c*ln((1-x
*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)+e^(1/2)))*ln(1-I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(3/2)/(-c^2)^(1/2
)+1/32*I*b*c*ln((1+x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)+e^(1/2)))*ln(1+I*x*e^(1/2)/d^(1/2))/d^(3/2)
/e^(3/2)/(-c^2)^(1/2)

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 966, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5100, 205, 211, 5032, 6857, 585, 78, 5028, 2456, 2441, 2440, 2438, 455, 36, 31} \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {i b \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \log \left (-\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b \log \left (\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {b \log \left (c^2 x^2+1\right ) c}{4 d \left (c^2 d-e\right ) e}+\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac {b \log \left (e x^2+d\right ) c}{4 d \left (c^2 d-e\right ) e}-\frac {b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {b c}{8 \left (c^2 d-e\right ) e \left (e x^2+d\right )}+\frac {x (a+b \arctan (c x))}{8 d e \left (e x^2+d\right )}-\frac {x (a+b \arctan (c x))}{4 e \left (e x^2+d\right )^2}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}} \]

[In]

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

[Out]

(b*c)/(8*(c^2*d - e)*e*(d + e*x^2)) - (x*(a + b*ArcTan[c*x]))/(4*e*(d + e*x^2)^2) + (x*(a + b*ArcTan[c*x]))/(8
*d*e*(d + e*x^2)) + ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + ((I/32)*b*c*Log[(S
qrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/
2)*e^(3/2)) - ((I/32)*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqr
t[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*
Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + ((I/32)*b*c*Log[(Sqrt[e]*(
1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/
2)) + (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(16*d*(c^2*d - e)^2*e) - (b*c*Log[1 + c^2*x^2])/(4*d*(c^2*d - e)*
e) - (b*c*(5*c^2*d - 3*e)*Log[d + e*x^2])/(16*d*(c^2*d - e)^2*e) + (b*c*Log[d + e*x^2])/(4*d*(c^2*d - e)*e) +
((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(
3/2)*e^(3/2)) - ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])
/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d
] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(S
qrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2))

Rule 31

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

Rule 36

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

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

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 585

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 5028

Int[ArcTan[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[I/2, Int[Log[1 - I*c*x]/(d + e*x^2), x], x] -
Dist[I/2, Int[Log[1 + I*c*x]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]

Rule 5032

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x]
&& (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 5100

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With
[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b,
c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d (a+b \arctan (c x))}{e \left (d+e x^2\right )^3}+\frac {a+b \arctan (c x)}{e \left (d+e x^2\right )^2}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx}{e}-\frac {d \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^3} \, dx}{e} \\ & = -\frac {x (a+b \arctan (c x))}{4 e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{8 d e \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}-\frac {(b c) \int \frac {\frac {x}{2 d \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}}{1+c^2 x^2} \, dx}{e}+\frac {(b c d) \int \frac {\frac {x}{4 d \left (d+e x^2\right )^2}+\frac {3 x}{8 d^2 \left (d+e x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}}{1+c^2 x^2} \, dx}{e} \\ & = -\frac {x (a+b \arctan (c x))}{4 e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{8 d e \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}-\frac {(b c) \int \left (\frac {x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx}{e}+\frac {(b c d) \int \left (\frac {x \left (5 d+3 e x^2\right )}{8 d^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx}{e} \\ & = -\frac {x (a+b \arctan (c x))}{4 e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{8 d e \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+\frac {(3 b c) \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{8 d^{3/2} e^{3/2}}-\frac {(b c) \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{2 d^{3/2} e^{3/2}}+\frac {(b c) \int \frac {x \left (5 d+3 e x^2\right )}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{8 d e}-\frac {(b c) \int \frac {x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d e} \\ & = -\frac {x (a+b \arctan (c x))}{4 e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{8 d e \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{16 d^{3/2} e^{3/2}}-\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{16 d^{3/2} e^{3/2}}-\frac {(i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{4 d^{3/2} e^{3/2}}+\frac {(i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{4 d^{3/2} e^{3/2}}+\frac {(b c) \text {Subst}\left (\int \frac {5 d+3 e x}{\left (1+c^2 x\right ) (d+e x)^2} \, dx,x,x^2\right )}{16 d e}-\frac {(b c) \text {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 d e} \\ & = -\frac {x (a+b \arctan (c x))}{4 e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{8 d e \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+\frac {(b c) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{4 d \left (c^2 d-e\right )}+\frac {(3 i b c) \int \left (\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{16 d^{3/2} e^{3/2}}-\frac {(3 i b c) \int \left (\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{16 d^{3/2} e^{3/2}}-\frac {(i b c) \int \left (\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{4 d^{3/2} e^{3/2}}+\frac {(i b c) \int \left (\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{4 d^{3/2} e^{3/2}}+\frac {(b c) \text {Subst}\left (\int \left (\frac {5 c^4 d-3 c^2 e}{\left (c^2 d-e\right )^2 \left (1+c^2 x\right )}-\frac {2 d e}{\left (c^2 d-e\right ) (d+e x)^2}+\frac {e \left (-5 c^2 d+3 e\right )}{\left (-c^2 d+e\right )^2 (d+e x)}\right ) \, dx,x,x^2\right )}{16 d e}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{4 d \left (c^2 d-e\right ) e} \\ & = \frac {b c}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac {x (a+b \arctan (c x))}{4 e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))}{8 d e \left (d+e x^2\right )}+\frac {(a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d \left (c^2 d-e\right )^2 e}-\frac {b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right ) e}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d \left (c^2 d-e\right )^2 e}+\frac {b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right ) e}+\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{32 d^{3/2} e^{3/2}}+\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{32 d^{3/2} e^{3/2}}-\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{32 d^{3/2} e^{3/2}}-\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{32 d^{3/2} e^{3/2}}-\frac {(i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{8 d^{3/2} e^{3/2}}-\frac {(i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{8 d^{3/2} e^{3/2}}+\frac {(i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{8 d^{3/2} e^{3/2}}+\frac {(i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{8 d^{3/2} e^{3/2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 12.54 (sec) , antiderivative size = 1744, normalized size of antiderivative = 1.81 \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {a x}{4 e \left (d+e x^2\right )^2}+\frac {a x}{8 d e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+\frac {b c^3 \left (-\frac {2 \log \left (1+\frac {\left (c^2 d-e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )}{c^2 d}-\frac {2 \log \left (1+\frac {\left (c^2 d-e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )}{e}+\frac {\left (c^2 d-e\right ) e \left (-4 \arctan (c x) \text {arctanh}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )+2 \arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right ) \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (-i e+\sqrt {-c^2 d e}\right ) (-i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (i e+\sqrt {-c^2 d e}\right ) (i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )\right )\right )}{\left (-c^2 d e\right )^{3/2}}+\frac {\left (c^2 d-e\right ) \left (-4 \arctan (c x) \text {arctanh}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )+2 \arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right ) \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (-i e+\sqrt {-c^2 d e}\right ) (-i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 c^2 d \left (i e+\sqrt {-c^2 d e}\right ) (i+c x)}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+\left (\arccos \left (-\frac {c^2 d+e}{c^2 d-e}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (c^2 d+c \sqrt {-c^2 d e} x\right )}\right )\right )\right )}{e \sqrt {-c^2 d e}}+\frac {16 \left (c^2 d-e\right ) \arctan (c x) \sin (2 \arctan (c x))}{\left (c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))\right )^2}+\frac {-8 c^2 d e-4 \left (c^4 d^2-e^2\right ) \arctan (c x) \sin (2 \arctan (c x))}{c^2 d e \left (c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))\right )}\right )}{32 \left (-c^2 d+e\right )^2} \]

[In]

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

[Out]

-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2))
+ (b*c^3*((-2*Log[1 + ((c^2*d - e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(c^2*d) - (2*Log[1 + ((c^2*d - e)*Cos[2*
ArcTan[c*x]])/(c^2*d + e)])/e + ((c^2*d - e)*e*(-4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-(
(c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcT
anh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sq
rt[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^
2*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/
(c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*
Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (Arc
Cos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)
]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcT
an[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)
*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e
)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))])))/(-(c^2*d*e))^(3/2) + ((c^2*d - e)*(-4*ArcTan[c*x]*ArcT
anh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-((c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcC
os[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d*
e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*A
rcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sqr
t[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTan
h[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e
+ (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*
d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d -
e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(
c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*
Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))])))/(e*Sqrt[-(c
^2*d*e)]) + (16*(c^2*d - e)*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])^2 + (
-8*c^2*d*e - 4*(c^4*d^2 - e^2)*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d*e*(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[
c*x]]))))/(32*(-(c^2*d) + e)^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3773 vs. \(2 (750 ) = 1500\).

Time = 2.10 (sec) , antiderivative size = 3774, normalized size of antiderivative = 3.91

method	result	size
parts	\(\text {Expression too large to display}\)	\(3774\)
derivativedivides	\(\text {Expression too large to display}\)	\(3815\)
default	\(\text {Expression too large to display}\)	\(3815\)
risch	\(\text {Expression too large to display}\)	\(6982\)

[In]

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

[Out]

a*((1/8/d*x^3-1/8/e*x)/(e*x^2+d)^2+1/8/e/d/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2)))-1/16*I*b*c^7*d^2*ln(1-(c^2*d-e
)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))*arctan(c*x)/(c^4*d^2-2*c^2*d*e+e^2)^2/e^2*(c^2*d*e)^(1
/2)+1/4*I*b*c^5*d*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))*arctan(c*x)/(c^4*d^2-2*
c^2*d*e+e^2)^2/e*(c^2*d*e)^(1/2)-1/8*I*b*c^5/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*e^2*arctan(c*x)*x^4
-1/8*I*b*c*(c^2*d*e)^(1/2)/d/e/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^
2*d-2*(c^2*d*e)^(1/2)-e))+1/4*I*b*c*e*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))*arc
tan(c*x)/(c^4*d^2-2*c^2*d*e+e^2)^2/d*(c^2*d*e)^(1/2)-1/16*I*b/c*e^2*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c
^2*d+2*(c^2*d*e)^(1/2)-e))*arctan(c*x)/(c^4*d^2-2*c^2*d*e+e^2)^2/d^2*(c^2*d*e)^(1/2)-3/16*b*c^3*polylog(2,(c^2
*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*d*e)^(1/2)-1/8*b*c^
5/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d-1/16*b*(e*d)^(1/2)/d^2*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c
^2*x^2+1)+2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)+1/32*b/c*(c^2*d*e)^(1/2)/d^2/(c^4*d^2-2*c^2*d*e+
e^2)*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))+1/16*b/c*(c^2*d*e)^(1/2)/d^2/(c
^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^2+1/32*b*c^3*(c^2*d*e)^(1/2)/e^2/(c^4*d^2-2*c^2*d*e+e^2)*polylog(2,(c^2*d-e)
*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))+1/16*b*c^3*(c^2*d*e)^(1/2)/e^2/(c^4*d^2-2*c^2*d*e+e^2)*
arctan(c*x)^2-1/8*b*c^7/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*x^4-1/8*b*c^5/(c^4*d^2-2*c^2*d*e+e^2)/(c
^2*e*x^2+c^2*d)^2*e*x^2-1/16*b*c^4*(e*d)^(1/2)/e^2*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*
e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)-1/8*b*c^7/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*x^2-3/8*b*c^
3*arctan(c*x)^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*d*e)^(1/2)+1/8*b*c^5*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1
)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))/e/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*d*e)^(1/2)*d+1/8*b*c*polylog(2,(c^2*d-e)*(1+I
*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))/d/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*d*e)^(1/2)*e+1/4*b*c^5*arct
an(c*x)^2/e/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*d*e)^(1/2)*d-1/16*b*c*(c^2*d*e)^(1/2)/d/e/(c^4*d^2-2*c^2*d*e+e^2)*p
olylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))+1/4*b*c*arctan(c*x)^2/d/(c^4*d^2-2*c^
2*d*e+e^2)^2*(c^2*d*e)^(1/2)*e+1/8*b*c^8/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*arctan(c*x)*x^3-1/4*b*c
^6/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)*x^3+1/4*b*c^6/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+
c^2*d)^2*d*arctan(c*x)*x-1/4*b*c/d*e/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))-1/16*b/
c*e^2*arctan(c*x)^2/(c^4*d^2-2*c^2*d*e+e^2)^2/d^2*(c^2*d*e)^(1/2)-1/32*b*c^7*d^2*polylog(2,(c^2*d-e)*(1+I*c*x)
^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))/(c^4*d^2-2*c^2*d*e+e^2)^2/e^2*(c^2*d*e)^(1/2)-1/16*b*c^2*(e*d)^(1
/2)/d*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)/(c^
2*d-e)-1/32*b/c*e^2*polylog(2,(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))/(c^4*d^2-2*c^2*d
*e+e^2)^2/d^2*(c^2*d*e)^(1/2)-1/16*b*c^7*d^2*arctan(c*x)^2/(c^4*d^2-2*c^2*d*e+e^2)^2/e^2*(c^2*d*e)^(1/2)+1/16*
b*c/d*e/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)*ln(c^2*d*(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I*c*x)^2/(c^2*x^2+1)-e
*(1+I*c*x)^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)+1/4*b*c^5*d/e/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e
)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))+1/8*b*c^4/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/d*e^2*arctan(c*x)*x^3+
1/16*b*c^6*(e*d)^(1/2)/e^2*d*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4
*d^2-2*c^2*d*e+e^2)/(c^2*d-e)-1/8*b*c^8/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d^2/e*arctan(c*x)*x-1/8*I*
b*c^7/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d^2/e*arctan(c*x)-1/4*I*b*c^7/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e
*x^2+c^2*d)^2*d*arctan(c*x)*x^2-1/4*I*b*c^5/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)*x^2-1/8*
I*b*c^7/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)*x^4+1/16*I*b*c^3*(c^2*d*e)^(1/2)/e^2/(c^4*d^
2-2*c^2*d*e+e^2)*arctan(c*x)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d*e)^(1/2)-e))+1/16*I*b/c*(
c^2*d*e)^(1/2)/d^2/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d-2*(c^2*d
*e)^(1/2)-e))-1/16*b*c^5*d/e/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)*ln(c^2*d*(1+I*c*x)^4/(c^2*x^2+1)^2+2*c^2*d*(1+I
*c*x)^2/(c^2*x^2+1)-e*(1+I*c*x)^4/(c^2*x^2+1)^2+c^2*d+2*e*(1+I*c*x)^2/(c^2*x^2+1)-e)-1/8*b*c*(c^2*d*e)^(1/2)/d
/e/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^2-1/8*b*c^2*(e*d)^(1/2)/e/d*arctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x
^2+1)+2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)+1/16*b*c^4*(e*d)^(1/2)/e*arctanh(1/4*(2*(c^2*d-e)*(1
+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)-1/16*b*(e*d)^(1/2)*e/d^2*a
rctanh(1/4*(2*(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)+2*c^2*d+2*e)/c/(e*d)^(1/2))/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)-
1/8*b*c^4/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)*x-1/8*I*b*c^5/(c^4*d^2-2*c^2*d*e+e^2)/(c^2
*e*x^2+c^2*d)^2*d*arctan(c*x)-3/8*I*b*c^3*ln(1-(c^2*d-e)*(1+I*c*x)^2/(c^2*x^2+1)/(-c^2*d+2*(c^2*d*e)^(1/2)-e))
*arctan(c*x)/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*d*e)^(1/2)

Fricas [F]

\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

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

[Out]

integral((b*x^2*arctan(c*x) + a*x^2)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

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

[Out]

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