3.1162 \(\int \frac{x^2 (a+b \tan ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=1335 \[ \text{result too large to display} \]

[Out]

-(x*(a + b*ArcTan[c*x]))/(2*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(3/2)) - ((a + b*ArcTa
n[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^(3/2)) - ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*
x))/(c*Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*
Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d
] - I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*S
qrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/8)*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*L
og[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*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]*Sqrt[d]*e^(3/2)) + ((I/8)*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]
*Sqrt[d]*e^(3/2)) - ((I/8)*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*S
qrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + (b*c*Log[1 + c^2*x^2])/(4*(c^2*d - e)*e) - (b*c*Log[d + e*x
^2])/(4*(c^2*d - e)*e) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2))
- ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)*b*PolyLog[
2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I + c*x
))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/
(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] -
I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) - ((I/8)*b*c*PolyLog[2, (Sqrt[-
c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*Pol
yLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 1.96414, antiderivative size = 1335, normalized size of antiderivative = 1., number of steps used = 45, number of rules used = 14, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4980, 199, 205, 4912, 6725, 444, 36, 31, 4908, 2409, 2394, 2393, 2391, 4910} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 \sqrt{d} e^{3/2}}-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (e x^2+d\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{i b \log (i c x+1) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} c+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{e} x+\sqrt{-d}\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (i c x+1) \log \left (\frac{c \left (\sqrt{e} x+\sqrt{-d}\right )}{\sqrt{-d} c+i \sqrt{e}}\right )}{4 \sqrt{-d} 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 )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \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 )}{8 \sqrt{-c^2} \sqrt{d} 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 (\frac{i \sqrt{e} x}{\sqrt{d}}+1\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}-\frac{i b c \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 )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{b c \log \left (c^2 x^2+1\right )}{4 \left (c^2 d-e\right ) e}-\frac{b c \log \left (e x^2+d\right )}{4 \left (c^2 d-e\right ) e}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{PolyLog}\left (2,\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}-\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \text{PolyLog}\left (2,\frac{\sqrt{-c^2} \left (i \sqrt{e} x+\sqrt{d}\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(a + b*ArcTan[c*x]))/(2*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(3/2)) - ((a + b*ArcTa
n[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^(3/2)) - ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*
x))/(c*Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*
Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d
] - I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*S
qrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/8)*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*L
og[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*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]*Sqrt[d]*e^(3/2)) + ((I/8)*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]
*Sqrt[d]*e^(3/2)) - ((I/8)*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*S
qrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + (b*c*Log[1 + c^2*x^2])/(4*(c^2*d - e)*e) - (b*c*Log[d + e*x
^2])/(4*(c^2*d - e)*e) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2))
- ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)*b*PolyLog[
2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I + c*x
))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/
(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] -
I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) - ((I/8)*b*c*PolyLog[2, (Sqrt[-
c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*Pol
yLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2))

Rule 4980

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 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

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

Rule 4912

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 6725

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]

Rule 444

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 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 31

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

Rule 4908

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 2409

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 2394

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 2393

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 2391

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 4910

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

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{e \left (d+e x^2\right )^2}+\frac{a+b \tan ^{-1}(c x)}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \tan ^{-1}(c x)}{d+e x^2} \, dx}{e}-\frac{d \int \frac{a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}+\frac{a \int \frac{1}{d+e x^2} \, dx}{e}+\frac{b \int \frac{\tan ^{-1}(c x)}{d+e x^2} \, dx}{e}+\frac{(b c d) \int \frac{\frac{x}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}}{1+c^2 x^2} \, dx}{e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}+\frac{(i b) \int \frac{\log (1-i c x)}{d+e x^2} \, dx}{2 e}-\frac{(i b) \int \frac{\log (1+i c x)}{d+e x^2} \, dx}{2 e}+\frac{(b c d) \int \left (\frac{x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac{\tan ^{-1}\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{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}+\frac{(b c) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{2 \sqrt{d} e^{3/2}}+\frac{(i b) \int \left (\frac{\sqrt{-d} \log (1-i c x)}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \log (1-i c x)}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 e}-\frac{(i b) \int \left (\frac{\sqrt{-d} \log (1+i c x)}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \log (1+i c x)}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 e}+\frac{(b c) \int \frac{x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} 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 \sqrt{d} 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 \sqrt{d} e^{3/2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 e}-\frac{(i b) \int \frac{\log (1-i c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 \sqrt{-d} e}-\frac{(i b) \int \frac{\log (1-i c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 \sqrt{-d} e}+\frac{(i b) \int \frac{\log (1+i c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 \sqrt{-d} e}+\frac{(i b) \int \frac{\log (1+i c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 \sqrt{-d} e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^2\right )}{4 \left (c^2 d-e\right )}-\frac{(b c) \int \frac{\log \left (-\frac{i c \left (\sqrt{-d}-\sqrt{e} x\right )}{-i c \sqrt{-d}+\sqrt{e}}\right )}{1-i c x} \, dx}{4 \sqrt{-d} e^{3/2}}-\frac{(b c) \int \frac{\log \left (\frac{i c \left (\sqrt{-d}-\sqrt{e} x\right )}{i c \sqrt{-d}+\sqrt{e}}\right )}{1+i c x} \, dx}{4 \sqrt{-d} e^{3/2}}+\frac{(b c) \int \frac{\log \left (-\frac{i c \left (\sqrt{-d}+\sqrt{e} x\right )}{-i c \sqrt{-d}-\sqrt{e}}\right )}{1-i c x} \, dx}{4 \sqrt{-d} e^{3/2}}+\frac{(b c) \int \frac{\log \left (\frac{i c \left (\sqrt{-d}+\sqrt{e} x\right )}{i c \sqrt{-d}-\sqrt{e}}\right )}{1+i c x} \, dx}{4 \sqrt{-d} 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 \sqrt{d} 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 \sqrt{d} e^{3/2}}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )}{4 \left (c^2 d-e\right ) e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac{b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{-i c \sqrt{-d}-\sqrt{e}}\right )}{x} \, dx,x,1-i c x\right )}{4 \sqrt{-d} e^{3/2}}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{i c \sqrt{-d}-\sqrt{e}}\right )}{x} \, dx,x,1+i c x\right )}{4 \sqrt{-d} e^{3/2}}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{-i c \sqrt{-d}+\sqrt{e}}\right )}{x} \, dx,x,1-i c x\right )}{4 \sqrt{-d} e^{3/2}}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{i c \sqrt{-d}+\sqrt{e}}\right )}{x} \, dx,x,1+i c x\right )}{4 \sqrt{-d} 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 \sqrt{d} 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 \sqrt{d} 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 \sqrt{d} 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 \sqrt{d} e^{3/2}}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac{b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (i-c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (1+i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (i+c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{(b c) \int \frac{\log \left (-\frac{i \sqrt{e} \left (1-\sqrt{-c^2} x\right )}{\sqrt{d} \left (\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1-\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d e}+\frac{(b c) \int \frac{\log \left (\frac{i \sqrt{e} \left (1-\sqrt{-c^2} x\right )}{\sqrt{d} \left (\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1+\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d e}-\frac{(b c) \int \frac{\log \left (-\frac{i \sqrt{e} \left (1+\sqrt{-c^2} x\right )}{\sqrt{d} \left (-\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1-\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d e}-\frac{(b c) \int \frac{\log \left (\frac{i \sqrt{e} \left (1+\sqrt{-c^2} x\right )}{\sqrt{d} \left (-\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}\right )}\right )}{1+\frac{i \sqrt{e} x}{\sqrt{d}}} \, dx}{8 \sqrt{-c^2} d e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac{b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (i-c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (1+i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (i+c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{(i b c) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c^2} x}{-\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{(i b c) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c^2} x}{\sqrt{-c^2}-\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{(i b c) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-c^2} x}{-\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}-\frac{(i b c) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-c^2} x}{\sqrt{-c^2}+\frac{i \sqrt{e}}{\sqrt{d}}}\right )}{x} \, dx,x,1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (d+e x^2\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{3/2}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}-\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} 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 )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{b c \log \left (1+c^2 x^2\right )}{4 \left (c^2 d-e\right ) e}-\frac{b c \log \left (d+e x^2\right )}{4 \left (c^2 d-e\right ) e}+\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (i-c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (1+i c x)}{i c \sqrt{-d}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left (\frac{\sqrt{e} (i+c x)}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}-i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}-\frac{i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}+i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \text{Li}_2\left (\frac{\sqrt{-c^2} \left (\sqrt{d}+i \sqrt{e} x\right )}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right )}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}\\ \end{align*}

Mathematica [A]  time = 10.2903, size = 881, normalized size = 0.66 \[ -\frac{a x}{2 e \left (e x^2+d\right )}+\frac{a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e^{3/2}}+\frac{b c \left (-\frac{2 \log \left (\frac{\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}{d c^2+e}+1\right )}{c^2 d-e}+\frac{-4 \tan ^{-1}(c x) \tanh ^{-1}\left (\frac{\sqrt{-c^2 d e}}{c e x}\right )+2 \cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right ) \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )+\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )-2 i \tanh ^{-1}\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 ) (c x-i)}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt{-c^2 d e} x\right )}\right )+\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )+2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right ) \log \left (\frac{2 i c^2 d \left (e+i \sqrt{-c^2 d e}\right ) (c x+i)}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt{-c^2 d e} x\right )}\right )-\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )-2 i \tanh ^{-1}\left (\frac{\sqrt{-c^2 d e}}{c e x}\right )+2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{e-c^2 d} \sqrt{-d c^2-e+\left (e-c^2 d\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )-\left (\cos ^{-1}\left (-\frac{d c^2+e}{c^2 d-e}\right )+2 i \tanh ^{-1}\left (\frac{\sqrt{-c^2 d e}}{c e x}\right )-2 i \tanh ^{-1}\left (\frac{c e x}{\sqrt{-c^2 d e}}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{e-c^2 d} \sqrt{-d c^2-e+\left (e-c^2 d\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (d c^2+e-2 i \sqrt{-c^2 d e}\right ) \left (d c^2+\sqrt{-c^2 d e} x c\right )}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt{-c^2 d e} x\right )}\right )-\text{PolyLog}\left (2,\frac{\left (d c^2+e+2 i \sqrt{-c^2 d e}\right ) \left (d c^2+\sqrt{-c^2 d e} x c\right )}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt{-c^2 d e} x\right )}\right )\right )}{\sqrt{-c^2 d e}}-\frac{4 \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )}{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}\right )}{8 e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.815, size = 2315, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{2} \arctan \left (c x\right ) + a x^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctan(c*x) + a)*x^2/(e*x^2 + d)^2, x)