3.295 \(\int (f+g x^3) \log ^2(c (d+e x^2)^p) \, dx\)

Optimal. Leaf size=395 \[ \frac{4 i \sqrt{d} f p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{d g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac{d g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{4 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{g p^2 \left (d+e x^2\right )^2}{8 e^2}+\frac{4 i \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{8 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{8 \sqrt{d} f p^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{d g p^2 x^2}{e}+8 f p^2 x \]

[Out]

8*f*p^2*x - (d*g*p^2*x^2)/e + (g*p^2*(d + e*x^2)^2)/(8*e^2) - (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sq
rt[e] + ((4*I)*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] + (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt
[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - 4*f*p*x*Log[c*(d + e*x^2)^p] + (d*g*p*(d + e*x^2)*Log
[c*(d + e*x^2)^p])/e^2 - (g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) + (4*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/
Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] + f*x*Log[c*(d + e*x^2)^p]^2 - (d*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)
/(2*e^2) + (g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + ((4*I)*Sqrt[d]*f*p^2*PolyLog[2, 1 - (2*Sqrt[d])/
(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e]

________________________________________________________________________________________

Rubi [A]  time = 0.506887, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 20, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.909, Rules used = {2471, 2450, 2476, 2448, 321, 205, 2470, 12, 4920, 4854, 2402, 2315, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{4 i \sqrt{d} f p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac{d g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac{g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac{d g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{4 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{g p^2 \left (d+e x^2\right )^2}{8 e^2}+\frac{4 i \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{8 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{8 \sqrt{d} f p^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{d g p^2 x^2}{e}+8 f p^2 x \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x^3)*Log[c*(d + e*x^2)^p]^2,x]

[Out]

8*f*p^2*x - (d*g*p^2*x^2)/e + (g*p^2*(d + e*x^2)^2)/(8*e^2) - (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sq
rt[e] + ((4*I)*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] + (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt
[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - 4*f*p*x*Log[c*(d + e*x^2)^p] + (d*g*p*(d + e*x^2)*Log
[c*(d + e*x^2)^p])/e^2 - (g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) + (4*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/
Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] + f*x*Log[c*(d + e*x^2)^p]^2 - (d*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)
/(2*e^2) + (g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + ((4*I)*Sqrt[d]*f*p^2*PolyLog[2, 1 - (2*Sqrt[d])/
(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e]

Rule 2471

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

Rule 2450

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x^
n)^p])^q, x] - Dist[b*e*n*p*q, Int[(x^n*(a + b*Log[c*(d + e*x^n)^p])^(q - 1))/(d + e*x^n), x], x] /; FreeQ[{a,
b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])

Rule 2476

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

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

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

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2401

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

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

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

Rule 2390

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

Rule 2305

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

Rule 2304

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

Rubi steps

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

Mathematica [A]  time = 0.157656, size = 415, normalized size = 1.05 \[ \frac{4 i \sqrt{d} f p^2 \text{PolyLog}\left (2,-\frac{\sqrt{e} x+i \sqrt{d}}{-\sqrt{e} x+i \sqrt{d}}\right )}{\sqrt{e}}-e g p \left (\frac{d^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^3 p}+\frac{d \left (p x^2-\frac{\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}\right )}{2 e^2}+\frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 e}+\frac{1}{8} p \left (-\frac{2 d^2 \log \left (d+e x^2\right )}{e^3}+\frac{2 d x^2}{e^2}-\frac{x^4}{e}\right )\right )+f x \log ^2\left (c \left (d+e x^2\right )^p\right )-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{4 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{1}{4} g x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{4 i \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{8 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{8 \sqrt{d} f p^2 \log \left (\frac{2 i \sqrt{d}}{-\sqrt{e} x+i \sqrt{d}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+8 f p^2 x \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x^3)*Log[c*(d + e*x^2)^p]^2,x]

[Out]

8*f*p^2*x - (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + ((4*I)*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sq
rt[d]]^2)/Sqrt[e] + (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[((2*I)*Sqrt[d])/(I*Sqrt[d] - Sqrt[e]*x)])
/Sqrt[e] - 4*f*p*x*Log[c*(d + e*x^2)^p] + (4*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqr
t[e] + f*x*Log[c*(d + e*x^2)^p]^2 + (g*x^4*Log[c*(d + e*x^2)^p]^2)/4 - e*g*p*((p*((2*d*x^2)/e^2 - x^4/e - (2*d
^2*Log[d + e*x^2])/e^3))/8 + (x^4*Log[c*(d + e*x^2)^p])/(4*e) + (d^2*Log[c*(d + e*x^2)^p]^2)/(4*e^3*p) + (d*(p
*x^2 - ((d + e*x^2)*Log[c*(d + e*x^2)^p])/e))/(2*e^2)) + ((4*I)*Sqrt[d]*f*p^2*PolyLog[2, -((I*Sqrt[d] + Sqrt[e
]*x)/(I*Sqrt[d] - Sqrt[e]*x))])/Sqrt[e]

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Maple [F]  time = 1.532, size = 0, normalized size = 0. \begin{align*} \int \left ( g{x}^{3}+f \right ) \left ( \ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^3+f)*ln(c*(e*x^2+d)^p)^2,x)

[Out]

int((g*x^3+f)*ln(c*(e*x^2+d)^p)^2,x)

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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((g*x^3+f)*log(c*(e*x^2+d)^p)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)*log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")

[Out]

integral((g*x^3 + f)*log((e*x^2 + d)^p*c)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f + g x^{3}\right ) \log{\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**3+f)*ln(c*(e*x**2+d)**p)**2,x)

[Out]

Integral((f + g*x**3)*log(c*(d + e*x**2)**p)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^3+f)*log(c*(e*x^2+d)^p)^2,x, algorithm="giac")

[Out]

integrate((g*x^3 + f)*log((e*x^2 + d)^p*c)^2, x)