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

Optimal. Leaf size=1221 \[ \text{result too large to display} \]

[Out]

8*f^3*p^2*x - (1408*d^3*f*g^2*p^2*x)/(245*e^3) - (3*d*f^2*g*p^2*x^2)/e + (d^4*g^3*p^2*x^2)/e^4 + (568*d^2*f*g^
2*p^2*x^3)/(735*e^2) - (288*d*f*g^2*p^2*x^5)/(1225*e) + (24*f*g^2*p^2*x^7)/343 + (3*f^2*g*p^2*(d + e*x^2)^2)/(
8*e^2) - (d^3*g^3*p^2*(d + e*x^2)^2)/(2*e^5) + (2*d^2*g^3*p^2*(d + e*x^2)^3)/(9*e^5) - (d*g^3*p^2*(d + e*x^2)^
4)/(16*e^5) + (g^3*p^2*(d + e*x^2)^5)/(125*e^5) - (8*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (1
408*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(245*e^(7/2)) + ((4*I)*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/S
qrt[d]]^2)/Sqrt[e] - (((12*I)/7)*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(7/2) + (8*Sqrt[d]*f^3*p^2
*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (24*d^(7/2)*f*g^2*p^2*ArcTan[
(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2)) - (d^5*g^3*p^2*Log[d + e*x^2]^2)/(1
0*e^5) - 4*f^3*p*x*Log[c*(d + e*x^2)^p] + (12*d^3*f*g^2*p*x*Log[c*(d + e*x^2)^p])/(7*e^3) - (4*d^2*f*g^2*p*x^3
*Log[c*(d + e*x^2)^p])/(7*e^2) + (12*d*f*g^2*p*x^5*Log[c*(d + e*x^2)^p])/(35*e) - (12*f*g^2*p*x^7*Log[c*(d + e
*x^2)^p])/49 + (3*d*f^2*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 - (d^4*g^3*p*(d + e*x^2)*Log[c*(d + e*x^2)^p
])/e^5 - (3*f^2*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) + (d^3*g^3*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p
])/e^5 - (2*d^2*g^3*p*(d + e*x^2)^3*Log[c*(d + e*x^2)^p])/(3*e^5) + (d*g^3*p*(d + e*x^2)^4*Log[c*(d + e*x^2)^p
])/(4*e^5) - (g^3*p*(d + e*x^2)^5*Log[c*(d + e*x^2)^p])/(25*e^5) + (4*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]
]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (12*d^(7/2)*f*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(7*e^(
7/2)) + (d^5*g^3*p*Log[d + e*x^2]*Log[c*(d + e*x^2)^p])/(5*e^5) + f^3*x*Log[c*(d + e*x^2)^p]^2 + (3*f*g^2*x^7*
Log[c*(d + e*x^2)^p]^2)/7 + (g^3*x^10*Log[c*(d + e*x^2)^p]^2)/10 - (3*d*f^2*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]
^2)/(2*e^2) + (3*f^2*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + ((4*I)*Sqrt[d]*f^3*p^2*PolyLog[2, 1 - (
2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (((12*I)/7)*d^(7/2)*f*g^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[
d] + I*Sqrt[e]*x)])/e^(7/2)

________________________________________________________________________________________

Rubi [A]  time = 1.64153, antiderivative size = 1139, normalized size of antiderivative = 0.93, number of steps used = 55, number of rules used = 29, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.208, Rules used = {2471, 2450, 2476, 2448, 321, 205, 2470, 12, 4920, 4854, 2402, 2315, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2457, 2455, 302, 2398, 2411, 43, 2334, 14, 2301} \[ \frac{1}{10} g^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^{10}+\frac{24}{343} f g^2 p^2 x^7+\frac{3}{7} f g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac{12}{49} f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7-\frac{288 d f g^2 p^2 x^5}{1225 e}+\frac{12 d f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}+\frac{568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac{4 d^2 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{7 e^2}+\frac{d^4 g^3 p^2 x^2}{e^4}-\frac{3 d f^2 g p^2 x^2}{e}+8 f^3 p^2 x-\frac{1408 d^3 f g^2 p^2 x}{245 e^3}+f^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x-4 f^3 p \log \left (c \left (e x^2+d\right )^p\right ) x+\frac{12 d^3 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+\frac{g^3 p^2 \left (e x^2+d\right )^5}{125 e^5}-\frac{d g^3 p^2 \left (e x^2+d\right )^4}{16 e^5}+\frac{2 d^2 g^3 p^2 \left (e x^2+d\right )^3}{9 e^5}-\frac{d^3 g^3 p^2 \left (e x^2+d\right )^2}{2 e^5}+\frac{3 f^2 g p^2 \left (e x^2+d\right )^2}{8 e^2}+\frac{4 i \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{7 e^{7/2}}-\frac{d^5 g^3 p^2 \log ^2\left (e x^2+d\right )}{10 e^5}+\frac{3 f^2 g \left (e x^2+d\right )^2 \log ^2\left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac{3 d f^2 g \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac{8 \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{1408 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{245 e^{7/2}}+\frac{8 \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}\right )}{\sqrt{e}}-\frac{24 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}\right )}{7 e^{7/2}}-\frac{3 f^2 g p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^2}+\frac{3 d f^2 g p \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{e^2}+\frac{4 \sqrt{d} f^3 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\sqrt{e}}-\frac{12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{7 e^{7/2}}-\frac{1}{300} g^3 p \left (-\frac{60 \log \left (e x^2+d\right ) d^5}{e^5}+\frac{300 \left (e x^2+d\right ) d^4}{e^5}-\frac{300 \left (e x^2+d\right )^2 d^3}{e^5}+\frac{200 \left (e x^2+d\right )^3 d^2}{e^5}-\frac{75 \left (e x^2+d\right )^4 d}{e^5}+\frac{12 \left (e x^2+d\right )^5}{e^5}\right ) \log \left (c \left (e x^2+d\right )^p\right )+\frac{4 i \sqrt{d} f^3 p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}\right )}{\sqrt{e}}-\frac{12 i d^{7/2} f g^2 p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}\right )}{7 e^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

8*f^3*p^2*x - (1408*d^3*f*g^2*p^2*x)/(245*e^3) - (3*d*f^2*g*p^2*x^2)/e + (d^4*g^3*p^2*x^2)/e^4 + (568*d^2*f*g^
2*p^2*x^3)/(735*e^2) - (288*d*f*g^2*p^2*x^5)/(1225*e) + (24*f*g^2*p^2*x^7)/343 + (3*f^2*g*p^2*(d + e*x^2)^2)/(
8*e^2) - (d^3*g^3*p^2*(d + e*x^2)^2)/(2*e^5) + (2*d^2*g^3*p^2*(d + e*x^2)^3)/(9*e^5) - (d*g^3*p^2*(d + e*x^2)^
4)/(16*e^5) + (g^3*p^2*(d + e*x^2)^5)/(125*e^5) - (8*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (1
408*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(245*e^(7/2)) + ((4*I)*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/S
qrt[d]]^2)/Sqrt[e] - (((12*I)/7)*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(7/2) + (8*Sqrt[d]*f^3*p^2
*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (24*d^(7/2)*f*g^2*p^2*ArcTan[
(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2)) - (d^5*g^3*p^2*Log[d + e*x^2]^2)/(1
0*e^5) - 4*f^3*p*x*Log[c*(d + e*x^2)^p] + (12*d^3*f*g^2*p*x*Log[c*(d + e*x^2)^p])/(7*e^3) - (4*d^2*f*g^2*p*x^3
*Log[c*(d + e*x^2)^p])/(7*e^2) + (12*d*f*g^2*p*x^5*Log[c*(d + e*x^2)^p])/(35*e) - (12*f*g^2*p*x^7*Log[c*(d + e
*x^2)^p])/49 + (3*d*f^2*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 - (3*f^2*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)
^p])/(4*e^2) + (4*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (12*d^(7/2)*f*g^2*
p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(7*e^(7/2)) - (g^3*p*((300*d^4*(d + e*x^2))/e^5 - (300*d^3
*(d + e*x^2)^2)/e^5 + (200*d^2*(d + e*x^2)^3)/e^5 - (75*d*(d + e*x^2)^4)/e^5 + (12*(d + e*x^2)^5)/e^5 - (60*d^
5*Log[d + e*x^2])/e^5)*Log[c*(d + e*x^2)^p])/300 + f^3*x*Log[c*(d + e*x^2)^p]^2 + (3*f*g^2*x^7*Log[c*(d + e*x^
2)^p]^2)/7 + (g^3*x^10*Log[c*(d + e*x^2)^p]^2)/10 - (3*d*f^2*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/(2*e^2) + (
3*f^2*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + ((4*I)*Sqrt[d]*f^3*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqr
t[d] + I*Sqrt[e]*x)])/Sqrt[e] - (((12*I)/7)*d^(7/2)*f*g^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*
x)])/e^(7/2)

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
]

Rule 2457

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

Rule 2455

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+3 f^2 g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+3 f g^2 x^6 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^3 x^9 \log ^2\left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^3 \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f^2 g\right ) \int x^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f g^2\right ) \int x^6 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+g^3 \int x^9 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} \left (3 f^2 g\right ) \operatorname{Subst}\left (\int x \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac{1}{2} g^3 \operatorname{Subst}\left (\int x^4 \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (4 e f^3 p\right ) \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{1}{7} \left (12 e f g^2 p\right ) \int \frac{x^8 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{2} \left (3 f^2 g\right ) \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 )-\left (4 e f^3 p\right ) \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-\frac{1}{7} \left (12 e f g^2 p\right ) \int \left (-\frac{d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^4}+\frac{d^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac{d x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^6 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{d^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx-\frac{1}{5} \left (e g^3 p\right ) \operatorname{Subst}\left (\int \frac{x^5 \log \left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^2\right )\\ &=f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{\left (3 f^2 g\right ) \operatorname{Subst}\left (\int (d+e x) \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 e}-\frac{\left (3 d f^2 g\right ) \operatorname{Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 e}-\left (4 f^3 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (4 d f^3 p\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{1}{7} \left (12 f g^2 p\right ) \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\frac{\left (12 d^3 f g^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac{\left (12 d^4 f g^2 p\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}-\frac{\left (12 d^2 f g^2 p\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}+\frac{\left (12 d f g^2 p\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e}-\frac{1}{5} \left (g^3 p\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^5 \log \left (c x^p\right )}{x} \, dx,x,d+e x^2\right )\\ &=-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac{4 \sqrt{d} f^3 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{12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac{1}{300} g^3 p \left (\frac{300 d^4 \left (d+e x^2\right )}{e^5}-\frac{300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac{200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac{75 d \left (d+e x^2\right )^4}{e^5}+\frac{12 \left (d+e x^2\right )^5}{e^5}-\frac{60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{\left (3 f^2 g\right ) \operatorname{Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}-\frac{\left (3 d f^2 g\right ) \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^3 p^2\right ) \int \frac{x^2}{d+e x^2} \, dx-\left (8 d e f^3 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-\frac{1}{35} \left (24 d f g^2 p^2\right ) \int \frac{x^6}{d+e x^2} \, dx-\frac{\left (24 d^3 f g^2 p^2\right ) \int \frac{x^2}{d+e x^2} \, dx}{7 e^2}+\frac{\left (24 d^4 f g^2 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}{7 e^2}+\frac{\left (8 d^2 f g^2 p^2\right ) \int \frac{x^4}{d+e x^2} \, dx}{7 e}+\frac{1}{49} \left (24 e f g^2 p^2\right ) \int \frac{x^8}{d+e x^2} \, dx+\frac{1}{5} \left (g^3 p^2\right ) \operatorname{Subst}\left (\int \frac{300 d^4 x-300 d^3 x^2+200 d^2 x^3-75 d x^4+12 x^5-60 d^5 \log (x)}{60 e^5 x} \, dx,x,d+e x^2\right )\\ &=8 f^3 p^2 x-\frac{24 d^3 f g^2 p^2 x}{7 e^3}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac{4 \sqrt{d} f^3 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{12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac{1}{300} g^3 p \left (\frac{300 d^4 \left (d+e x^2\right )}{e^5}-\frac{300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac{200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac{75 d \left (d+e x^2\right )^4}{e^5}+\frac{12 \left (d+e x^2\right )^5}{e^5}-\frac{60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac{3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{3 f^2 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 (3 f^2 g p\right ) \operatorname{Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\frac{\left (3 d f^2 g p\right ) \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\left (8 d f^3 p^2\right ) \int \frac{1}{d+e x^2} \, dx-\left (8 \sqrt{d} \sqrt{e} f^3 p^2\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx-\frac{1}{35} \left (24 d f g^2 p^2\right ) \int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx+\frac{\left (24 d^4 f g^2 p^2\right ) \int \frac{1}{d+e x^2} \, dx}{7 e^3}+\frac{\left (24 d^{7/2} f g^2 p^2\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{7 e^{5/2}}+\frac{\left (8 d^2 f g^2 p^2\right ) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx}{7 e}+\frac{1}{49} \left (24 e f g^2 p^2\right ) \int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx+\frac{\left (g^3 p^2\right ) \operatorname{Subst}\left (\int \frac{300 d^4 x-300 d^3 x^2+200 d^2 x^3-75 d x^4+12 x^5-60 d^5 \log (x)}{x} \, dx,x,d+e x^2\right )}{300 e^5}\\ &=8 f^3 p^2 x-\frac{1408 d^3 f g^2 p^2 x}{245 e^3}-\frac{3 d f^2 g p^2 x^2}{e}+\frac{568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac{288 d f g^2 p^2 x^5}{1225 e}+\frac{24}{343} f g^2 p^2 x^7+\frac{3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac{8 \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{24 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{7 e^{7/2}}+\frac{4 i \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{7 e^{7/2}}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac{3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f^2 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^3 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{12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac{1}{300} g^3 p \left (\frac{300 d^4 \left (d+e x^2\right )}{e^5}-\frac{300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac{200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac{75 d \left (d+e x^2\right )^4}{e^5}+\frac{12 \left (d+e x^2\right )^5}{e^5}-\frac{60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac{3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{3 f^2 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^3 p^2\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx-\frac{\left (24 d^3 f g^2 p^2\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{7 e^3}+\frac{\left (24 d^4 f g^2 p^2\right ) \int \frac{1}{d+e x^2} \, dx}{49 e^3}+\frac{\left (24 d^4 f g^2 p^2\right ) \int \frac{1}{d+e x^2} \, dx}{35 e^3}+\frac{\left (8 d^4 f g^2 p^2\right ) \int \frac{1}{d+e x^2} \, dx}{7 e^3}+\frac{\left (g^3 p^2\right ) \operatorname{Subst}\left (\int \left (300 d^4-300 d^3 x+200 d^2 x^2-75 d x^3+12 x^4-\frac{60 d^5 \log (x)}{x}\right ) \, dx,x,d+e x^2\right )}{300 e^5}\\ &=8 f^3 p^2 x-\frac{1408 d^3 f g^2 p^2 x}{245 e^3}-\frac{3 d f^2 g p^2 x^2}{e}+\frac{d^4 g^3 p^2 x^2}{e^4}+\frac{568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac{288 d f g^2 p^2 x^5}{1225 e}+\frac{24}{343} f g^2 p^2 x^7+\frac{3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac{d^3 g^3 p^2 \left (d+e x^2\right )^2}{2 e^5}+\frac{2 d^2 g^3 p^2 \left (d+e x^2\right )^3}{9 e^5}-\frac{d g^3 p^2 \left (d+e x^2\right )^4}{16 e^5}+\frac{g^3 p^2 \left (d+e x^2\right )^5}{125 e^5}-\frac{8 \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{1408 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{245 e^{7/2}}+\frac{4 i \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{7 e^{7/2}}+\frac{8 \sqrt{d} f^3 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}}-\frac{24 d^{7/2} f g^2 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 )}{7 e^{7/2}}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac{3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f^2 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^3 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{12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac{1}{300} g^3 p \left (\frac{300 d^4 \left (d+e x^2\right )}{e^5}-\frac{300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac{200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac{75 d \left (d+e x^2\right )^4}{e^5}+\frac{12 \left (d+e x^2\right )^5}{e^5}-\frac{60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac{3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{3 f^2 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^3 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+\frac{\left (24 d^3 f g^2 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}{7 e^3}-\frac{\left (d^5 g^3 p^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e x^2\right )}{5 e^5}\\ &=8 f^3 p^2 x-\frac{1408 d^3 f g^2 p^2 x}{245 e^3}-\frac{3 d f^2 g p^2 x^2}{e}+\frac{d^4 g^3 p^2 x^2}{e^4}+\frac{568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac{288 d f g^2 p^2 x^5}{1225 e}+\frac{24}{343} f g^2 p^2 x^7+\frac{3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac{d^3 g^3 p^2 \left (d+e x^2\right )^2}{2 e^5}+\frac{2 d^2 g^3 p^2 \left (d+e x^2\right )^3}{9 e^5}-\frac{d g^3 p^2 \left (d+e x^2\right )^4}{16 e^5}+\frac{g^3 p^2 \left (d+e x^2\right )^5}{125 e^5}-\frac{8 \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{1408 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{245 e^{7/2}}+\frac{4 i \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{7 e^{7/2}}+\frac{8 \sqrt{d} f^3 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}}-\frac{24 d^{7/2} f g^2 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 )}{7 e^{7/2}}-\frac{d^5 g^3 p^2 \log ^2\left (d+e x^2\right )}{10 e^5}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac{3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f^2 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^3 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{12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac{1}{300} g^3 p \left (\frac{300 d^4 \left (d+e x^2\right )}{e^5}-\frac{300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac{200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac{75 d \left (d+e x^2\right )^4}{e^5}+\frac{12 \left (d+e x^2\right )^5}{e^5}-\frac{60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac{3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{3 f^2 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^3 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}}-\frac{\left (24 i d^{7/2} f g^2 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 )}{7 e^{7/2}}\\ &=8 f^3 p^2 x-\frac{1408 d^3 f g^2 p^2 x}{245 e^3}-\frac{3 d f^2 g p^2 x^2}{e}+\frac{d^4 g^3 p^2 x^2}{e^4}+\frac{568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac{288 d f g^2 p^2 x^5}{1225 e}+\frac{24}{343} f g^2 p^2 x^7+\frac{3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac{d^3 g^3 p^2 \left (d+e x^2\right )^2}{2 e^5}+\frac{2 d^2 g^3 p^2 \left (d+e x^2\right )^3}{9 e^5}-\frac{d g^3 p^2 \left (d+e x^2\right )^4}{16 e^5}+\frac{g^3 p^2 \left (d+e x^2\right )^5}{125 e^5}-\frac{8 \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{1408 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{245 e^{7/2}}+\frac{4 i \sqrt{d} f^3 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}-\frac{12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{7 e^{7/2}}+\frac{8 \sqrt{d} f^3 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}}-\frac{24 d^{7/2} f g^2 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 )}{7 e^{7/2}}-\frac{d^5 g^3 p^2 \log ^2\left (d+e x^2\right )}{10 e^5}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac{12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac{4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac{12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac{12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac{3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac{3 f^2 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^3 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{12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac{1}{300} g^3 p \left (\frac{300 d^4 \left (d+e x^2\right )}{e^5}-\frac{300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac{200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac{75 d \left (d+e x^2\right )^4}{e^5}+\frac{12 \left (d+e x^2\right )^5}{e^5}-\frac{60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac{3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac{3 f^2 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^3 p^2 \text{Li}_2\left (1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}-\frac{12 i d^{7/2} f g^2 p^2 \text{Li}_2\left (1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{7 e^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.966562, size = 1020, normalized size = 0.84 \[ \frac{1}{125} g^3 p^2 x^{10}+\frac{1}{10} g^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^{10}-\frac{1}{25} g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^{10}-\frac{9 d g^3 p^2 x^8}{400 e}+\frac{d g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^8}{20 e}+\frac{24}{343} f g^2 p^2 x^7+\frac{3}{7} f g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac{12}{49} f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7+\frac{47 d^2 g^3 p^2 x^6}{900 e^2}-\frac{d^2 g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^6}{15 e^2}-\frac{288 d f g^2 p^2 x^5}{1225 e}+\frac{12 d f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}-\frac{77 d^3 g^3 p^2 x^4}{600 e^3}+\frac{3}{8} f^2 g p^2 x^4+\frac{3}{4} f^2 g \log ^2\left (c \left (e x^2+d\right )^p\right ) x^4+\frac{d^3 g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^4}{10 e^3}-\frac{3}{4} f^2 g p \log \left (c \left (e x^2+d\right )^p\right ) x^4+\frac{568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac{4 d^2 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{7 e^2}+\frac{137 d^4 g^3 p^2 x^2}{300 e^4}-\frac{9 d f^2 g p^2 x^2}{4 e}-\frac{d^4 g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^2}{5 e^4}+\frac{3 d f^2 g p \log \left (c \left (e x^2+d\right )^p\right ) x^2}{2 e}+8 f^3 p^2 x-\frac{1408 d^3 f g^2 p^2 x}{245 e^3}+f^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x-4 f^3 p \log \left (c \left (e x^2+d\right )^p\right ) x+\frac{12 d^3 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}-\frac{4 i \sqrt{d} f \left (3 d^3 g^2-7 e^3 f^2\right ) p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{7 e^{7/2}}+\frac{d^5 g^3 \log ^2\left (c \left (e x^2+d\right )^p\right )}{10 e^5}-\frac{3 d^2 f^2 g \log ^2\left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac{77 d^5 g^3 p^2 \log \left (e x^2+d\right )}{300 e^5}+\frac{3 d^2 f^2 g p^2 \log \left (e x^2+d\right )}{4 e^2}-\frac{d^5 g^3 p \log \left (c \left (e x^2+d\right )^p\right )}{5 e^5}+\frac{3 d^2 f^2 g p \log \left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac{4 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-352 g^2 p d^3+490 e^3 f^2 p-70 \left (7 e^3 f^2-3 d^3 g^2\right ) p \log \left (\frac{2 \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}\right )-35 \left (7 e^3 f^2-3 d^3 g^2\right ) \log \left (c \left (e x^2+d\right )^p\right )\right )}{245 e^{7/2}}-\frac{4 i \sqrt{d} f \left (3 d^3 g^2-7 e^3 f^2\right ) p^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x+i \sqrt{d}}{\sqrt{e} x-i \sqrt{d}}\right )}{7 e^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

8*f^3*p^2*x - (1408*d^3*f*g^2*p^2*x)/(245*e^3) - (9*d*f^2*g*p^2*x^2)/(4*e) + (137*d^4*g^3*p^2*x^2)/(300*e^4) +
(568*d^2*f*g^2*p^2*x^3)/(735*e^2) + (3*f^2*g*p^2*x^4)/8 - (77*d^3*g^3*p^2*x^4)/(600*e^3) - (288*d*f*g^2*p^2*x
^5)/(1225*e) + (47*d^2*g^3*p^2*x^6)/(900*e^2) + (24*f*g^2*p^2*x^7)/343 - (9*d*g^3*p^2*x^8)/(400*e) + (g^3*p^2*
x^10)/125 - (((4*I)/7)*Sqrt[d]*f*(-7*e^3*f^2 + 3*d^3*g^2)*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(7/2) + (3*d^2*
f^2*g*p^2*Log[d + e*x^2])/(4*e^2) - (77*d^5*g^3*p^2*Log[d + e*x^2])/(300*e^5) + (3*d^2*f^2*g*p*Log[c*(d + e*x^
2)^p])/(2*e^2) - (d^5*g^3*p*Log[c*(d + e*x^2)^p])/(5*e^5) - 4*f^3*p*x*Log[c*(d + e*x^2)^p] + (12*d^3*f*g^2*p*x
*Log[c*(d + e*x^2)^p])/(7*e^3) + (3*d*f^2*g*p*x^2*Log[c*(d + e*x^2)^p])/(2*e) - (d^4*g^3*p*x^2*Log[c*(d + e*x^
2)^p])/(5*e^4) - (4*d^2*f*g^2*p*x^3*Log[c*(d + e*x^2)^p])/(7*e^2) - (3*f^2*g*p*x^4*Log[c*(d + e*x^2)^p])/4 + (
d^3*g^3*p*x^4*Log[c*(d + e*x^2)^p])/(10*e^3) + (12*d*f*g^2*p*x^5*Log[c*(d + e*x^2)^p])/(35*e) - (d^2*g^3*p*x^6
*Log[c*(d + e*x^2)^p])/(15*e^2) - (12*f*g^2*p*x^7*Log[c*(d + e*x^2)^p])/49 + (d*g^3*p*x^8*Log[c*(d + e*x^2)^p]
)/(20*e) - (g^3*p*x^10*Log[c*(d + e*x^2)^p])/25 - (3*d^2*f^2*g*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + (d^5*g^3*Log[
c*(d + e*x^2)^p]^2)/(10*e^5) + f^3*x*Log[c*(d + e*x^2)^p]^2 + (3*f^2*g*x^4*Log[c*(d + e*x^2)^p]^2)/4 + (3*f*g^
2*x^7*Log[c*(d + e*x^2)^p]^2)/7 + (g^3*x^10*Log[c*(d + e*x^2)^p]^2)/10 - (4*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqr
t[d]]*(490*e^3*f^2*p - 352*d^3*g^2*p - 70*(7*e^3*f^2 - 3*d^3*g^2)*p*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] -
35*(7*e^3*f^2 - 3*d^3*g^2)*Log[c*(d + e*x^2)^p]))/(245*e^(7/2)) - (((4*I)/7)*Sqrt[d]*f*(-7*e^3*f^2 + 3*d^3*g^
2)*p^2*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)])/e^(7/2)

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Maple [F]  time = 1.704, size = 0, normalized size = 0. \begin{align*} \int \left ( g{x}^{3}+f \right ) ^{3} \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)^3*ln(c*(e*x^2+d)^p)^2,x)

[Out]

int((g*x^3+f)^3*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)^3*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^{3} x^{9} + 3 \, f g^{2} x^{6} + 3 \, f^{2} g x^{3} + f^{3}\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)^3*log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")

[Out]

integral((g^3*x^9 + 3*f*g^2*x^6 + 3*f^2*g*x^3 + f^3)*log((e*x^2 + d)^p*c)^2, x)

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

[Out]

Timed out

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

[Out]

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