\(\int (f+g x^3) \log ^2(c (d+e x^2)^p) \, dx\) [295]

Optimal result

Integrand size = 22, antiderivative size = 395 \[ \int \left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=8 f p^2 x-\frac {d g p^2 x^2}{e}+\frac {g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac {8 \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {4 i \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {8 \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {d g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 \sqrt {d} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 i \sqrt {d} f p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}} \]

[Out]

8*f*p^2*x-d*g*p^2*x^2/e+1/8*g*p^2*(e*x^2+d)^2/e^2-4*f*p*x*ln(c*(e*x^2+d)^p)+d*g*p*(e*x^2+d)*ln(c*(e*x^2+d)^p)/
e^2-1/4*g*p*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)/e^2+f*x*ln(c*(e*x^2+d)^p)^2-1/2*d*g*(e*x^2+d)*ln(c*(e*x^2+d)^p)^2/e^
2+1/4*g*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^2/e^2-8*f*p^2*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(1/2)+4*I*f*p^2*arctan
(x*e^(1/2)/d^(1/2))^2*d^(1/2)/e^(1/2)+4*f*p*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)*d^(1/2)/e^(1/2)+8*f*p^
2*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)+4*I*f*p^2*polylog(2,1-2*d^(1/2
)/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {2521, 2500, 2526, 2498, 327, 211, 2520, 12, 5040, 4964, 2449, 2352, 2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int \left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {4 \sqrt {d} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {4 i \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {8 \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac {d g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac {g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {d g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {g p^2 \left (d+e x^2\right )^2}{8 e^2}+\frac {4 i \sqrt {d} f p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {d g p^2 x^2}{e}+8 f p^2 x \]

[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 12

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

Rule 211

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

Rule 327

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 2332

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

Rule 2333

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 2341

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 2342

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 2352

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

Rule 2436

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 2437

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 2448

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 2449

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 2498

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 2500

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 2504

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 2520

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 2521

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 2526

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 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[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 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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]

Rubi steps \begin{align*} \text {integral}& = \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 \text {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 \text {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 \text {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) \text {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 \text {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}-\frac {(d g) \text {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) \text {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\frac {(d g p) \text {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 ) \text {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] (verified)

Time = 0.08 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.01 \[ \int \left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} g x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {1}{2} g p \left (\frac {3 d p x^2}{2 e}-\frac {p x^4}{4}-\frac {d^2 p \log \left (d+e x^2\right )}{2 e^2}-\frac {d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {d x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{2} x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {d^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2 p}\right )-4 e f p \left (-\frac {2 p x}{e}+\frac {2 \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2}}+\frac {x \log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^{3/2}}-\frac {\sqrt {d} p \left (\frac {i \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{e}+\frac {2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 i \sqrt {d}}{i \sqrt {d}-\sqrt {e} x}\right )}{e}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {d}+\sqrt {e} x}{i \sqrt {d}-\sqrt {e} x}\right )}{e}\right )}{\sqrt {e}}\right ) \]

[In]

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

[Out]

f*x*Log[c*(d + e*x^2)^p]^2 + (g*x^4*Log[c*(d + e*x^2)^p]^2)/4 - (g*p*((3*d*p*x^2)/(2*e) - (p*x^4)/4 - (d^2*p*L
og[d + e*x^2])/(2*e^2) - (d^2*Log[c*(d + e*x^2)^p])/e^2 - (d*x^2*Log[c*(d + e*x^2)^p])/e + (x^4*Log[c*(d + e*x
^2)^p])/2 + (d^2*Log[c*(d + e*x^2)^p]^2)/(2*e^2*p)))/2 - 4*e*f*p*((-2*p*x)/e + (2*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)
/Sqrt[d]])/e^(3/2) + (x*Log[c*(d + e*x^2)^p])/e - (Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/e
^(3/2) - (Sqrt[d]*p*((I*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e + (2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[((2*I)*Sqrt[d])/
(I*Sqrt[d] - Sqrt[e]*x)])/e + (I*PolyLog[2, -((I*Sqrt[d] + Sqrt[e]*x)/(I*Sqrt[d] - Sqrt[e]*x))])/e))/Sqrt[e])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.02 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(724\)

[In]

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

[Out]

1/4*ln((e*x^2+d)^p)^2*g*x^4+ln((e*x^2+d)^p)^2*x*f+1/2*p^2/e^2*d^2*g*ln(e*x^2+d)^2-1/2*p/e^2*d^2*g*ln(e*x^2+d)*
ln((e*x^2+d)^p)-4*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f*ln(e*x^2+d)+4*p*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(
1/2))*f*ln((e*x^2+d)^p)-1/4*p*g*x^4*ln((e*x^2+d)^p)+1/2*p/e*d*g*x^2*ln((e*x^2+d)^p)-4*p*f*x*ln((e*x^2+d)^p)+1/
8*p^2*x^4*g-3/4*d*g*p^2*x^2/e+3/4*p^2/e^2*d^2*g*ln(e*x^2+d)+8*f*p^2*x-8*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/
2))*f-p^2*e*Sum(1/2*(ln(x-_alpha)*ln(e*x^2+d)-2*e*(1/4/_alpha/e*ln(x-_alpha)^2+1/2*_alpha/d*ln(x-_alpha)*ln(1/
2*(x+_alpha)/_alpha)+1/2*_alpha/d*dilog(1/2*(x+_alpha)/_alpha)))*d*(_alpha*d*g-4*e*f)/e^3/_alpha,_alpha=RootOf
(_Z^2*e+d))+(I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*c
sgn(I*c)-I*Pi*csgn(I*c*(e*x^2+d)^p)^3+I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+2*ln(c))*(1/4*ln((e*x^2+d)^p)*g*x
^4+ln((e*x^2+d)^p)*x*f-1/2*p*e*(d/e^2*(1/2*d*g/e*ln(e*x^2+d)-4*e*f/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)))+1/e^2*
(1/4*e*g*x^4-1/2*d*g*x^2+4*e*f*x)))+1/4*(I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-I*Pi*csgn(I*(e*x^2+d
)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-I*Pi*csgn(I*c*(e*x^2+d)^p)^3+I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+2*ln(
c))^2*(1/4*g*x^4+f*x)

Fricas [F]

\[ \int \left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]

[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)

Sympy [F]

\[ \int \left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{3}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,\left (g\,x^3+f\right ) \,d x \]

[In]

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

[Out]

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