\(\int (f+g x^2)^2 \log ^2(c (d+e x^2)^p) \, dx\) [273]

Optimal result

Integrand size = 24, antiderivative size = 945 \[ \int \left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {64 d^{3/2} f g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {184 d^{5/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}+\frac {8 \sqrt {d} f^2 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 {16 d^{3/2} f g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+\frac {8 d^{5/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {8 d^{3/2} f g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+\frac {4 d^{5/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {4 i \sqrt {d} f^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}} \]

[Out]

-8/9*f*g*p*x^3*ln(c*(e*x^2+d)^p)-8*f^2*p^2*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(1/2)-184/75*d^(5/2)*g^2*p^2*ar
ctan(x*e^(1/2)/d^(1/2))/e^(5/2)+8*f^2*p^2*x+8/125*g^2*p^2*x^5+1/5*g^2*x^5*ln(c*(e*x^2+d)^p)^2+184/75*d^2*g^2*p
^2*x/e^2-64/225*d*g^2*p^2*x^3/e+8/3*d*f*g*p*x*ln(c*(e*x^2+d)^p)/e+16/27*f*g*p^2*x^3-4*f^2*p*x*ln(c*(e*x^2+d)^p
)-4/25*g^2*p*x^5*ln(c*(e*x^2+d)^p)+2/3*f*g*x^3*ln(c*(e*x^2+d)^p)^2-64/9*d*f*g*p^2*x/e+8/5*d^(5/2)*g^2*p^2*arct
an(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))/e^(5/2)+4*f^2*p*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^
2+d)^p)*d^(1/2)/e^(1/2)+8*f^2*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
)+64/9*d^(3/2)*f*g*p^2*arctan(x*e^(1/2)/d^(1/2))/e^(3/2)-4/5*d^2*g^2*p*x*ln(c*(e*x^2+d)^p)/e^2+4/15*d*g^2*p*x^
3*ln(c*(e*x^2+d)^p)/e+4/5*d^(5/2)*g^2*p*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(5/2)+f^2*x*ln(c*(e*x^2+
d)^p)^2+4/5*I*d^(5/2)*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))^2/e^(5/2)+4/5*I*d^(5/2)*g^2*p^2*polylog(2,1-2*d^(1/2)/
(d^(1/2)+I*x*e^(1/2)))/e^(5/2)+4*I*f^2*p^2*arctan(x*e^(1/2)/d^(1/2))^2*d^(1/2)/e^(1/2)+4*I*f^2*p^2*polylog(2,1
-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)-8/3*d^(3/2)*f*g*p*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p
)/e^(3/2)-16/3*d^(3/2)*f*g*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))/e^(3/2)-8/3*I*d^(
3/2)*f*g*p^2*arctan(x*e^(1/2)/d^(1/2))^2/e^(3/2)-8/3*I*d^(3/2)*f*g*p^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1
/2)))/e^(3/2)

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 945, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {2521, 2500, 2526, 2498, 327, 211, 2520, 12, 5040, 4964, 2449, 2352, 2507, 2505, 308} \[ \int \left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {8}{125} g^2 p^2 x^5+\frac {1}{5} g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^5-\frac {4}{25} g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {16}{27} f g p^2 x^3+\frac {2}{3} f g \log ^2\left (c \left (e x^2+d\right )^p\right ) x^3+\frac {4 d g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{15 e}-\frac {8}{9} f g p \log \left (c \left (e x^2+d\right )^p\right ) x^3+8 f^2 p^2 x+\frac {184 d^2 g^2 p^2 x}{75 e^2}-\frac {64 d f g p^2 x}{9 e}+f^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x-4 f^2 p \log \left (c \left (e x^2+d\right )^p\right ) x-\frac {4 d^2 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{5 e^2}+\frac {8 d f g p \log \left (c \left (e x^2+d\right )^p\right ) x}{3 e}+\frac {4 i \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {4 i d^{5/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}-\frac {8 i d^{3/2} f g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {184 d^{5/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {64 d^{3/2} f g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}+\frac {8 d^{5/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{5 e^{5/2}}-\frac {16 d^{3/2} f g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{3 e^{3/2}}+\frac {4 \sqrt {d} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\sqrt {e}}+\frac {4 d^{5/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{5 e^{5/2}}-\frac {8 d^{3/2} f g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{3 e^{3/2}}+\frac {4 i \sqrt {d} f^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}+\frac {4 i d^{5/2} g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{5 e^{5/2}}-\frac {8 i d^{3/2} f g p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{3 e^{3/2}} \]

[In]

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

[Out]

8*f^2*p^2*x - (64*d*f*g*p^2*x)/(9*e) + (184*d^2*g^2*p^2*x)/(75*e^2) + (16*f*g*p^2*x^3)/27 - (64*d*g^2*p^2*x^3)
/(225*e) + (8*g^2*p^2*x^5)/125 - (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (64*d^(3/2)*f*g*p^2
*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*e^(3/2)) - (184*d^(5/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(75*e^(5/2)) + (
(4*I)*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (((8*I)/3)*d^(3/2)*f*g*p^2*ArcTan[(Sqrt[e]*x)/S
qrt[d]]^2)/e^(3/2) + (((4*I)/5)*d^(5/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(5/2) + (8*Sqrt[d]*f^2*p^2*Ar
cTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (16*d^(3/2)*f*g*p^2*ArcTan[(Sqrt
[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) + (8*d^(5/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/
Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(5*e^(5/2)) - 4*f^2*p*x*Log[c*(d + e*x^2)^p] + (8*d*f*g*p*x
*Log[c*(d + e*x^2)^p])/(3*e) - (4*d^2*g^2*p*x*Log[c*(d + e*x^2)^p])/(5*e^2) - (8*f*g*p*x^3*Log[c*(d + e*x^2)^p
])/9 + (4*d*g^2*p*x^3*Log[c*(d + e*x^2)^p])/(15*e) - (4*g^2*p*x^5*Log[c*(d + e*x^2)^p])/25 + (4*Sqrt[d]*f^2*p*
ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (8*d^(3/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c
*(d + e*x^2)^p])/(3*e^(3/2)) + (4*d^(5/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(5*e^(5/2))
+ f^2*x*Log[c*(d + e*x^2)^p]^2 + (2*f*g*x^3*Log[c*(d + e*x^2)^p]^2)/3 + (g^2*x^5*Log[c*(d + e*x^2)^p]^2)/5 + (
(4*I)*Sqrt[d]*f^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (((8*I)/3)*d^(3/2)*f*g*p^
2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/e^(3/2) + (((4*I)/5)*d^(5/2)*g^2*p^2*PolyLog[2, 1 - (2*
Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/e^(5/2)

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 308

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 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 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 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 2505

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 2507

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 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^2 \log ^2\left (c \left (d+e x^2\right )^p\right )+2 f g x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^2 x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )\right ) \, dx \\ & = f^2 \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^4 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx \\ & = f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (4 e f^2 p\right ) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{3} (8 e f g p) \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{5} \left (4 e g^2 p\right ) \int \frac {x^6 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx \\ & = f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (4 e f^2 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}{3} (8 e f g p) \int \left (-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx-\frac {1}{5} \left (4 e g^2 p\right ) \int \left (\frac {d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^3 \left (d+e x^2\right )}\right ) \, dx \\ & = f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (4 f^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (4 d f^2 p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{3} (8 f g p) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {(8 d f g p) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{3 e}-\frac {\left (8 d^2 f g p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{3 e}-\frac {1}{5} \left (4 g^2 p\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\frac {\left (4 d^2 g^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{5 e^2}+\frac {\left (4 d^3 g^2 p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{5 e^2}+\frac {\left (4 d g^2 p\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{5 e} \\ & = -4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 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 {8 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\left (8 e f^2 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx-\left (8 d e f^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-\frac {1}{3} \left (16 d f g p^2\right ) \int \frac {x^2}{d+e x^2} \, dx+\frac {1}{3} \left (16 d^2 f g 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}{9} \left (16 e f g p^2\right ) \int \frac {x^4}{d+e x^2} \, dx-\frac {1}{15} \left (8 d g^2 p^2\right ) \int \frac {x^4}{d+e x^2} \, dx+\frac {\left (8 d^2 g^2 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx}{5 e}-\frac {\left (8 d^3 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}{5 e}+\frac {1}{25} \left (8 e g^2 p^2\right ) \int \frac {x^6}{d+e x^2} \, dx \\ & = 8 f^2 p^2 x-\frac {16 d f g p^2 x}{3 e}+\frac {8 d^2 g^2 p^2 x}{5 e^2}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 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 {8 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (8 d f^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx-\left (8 \sqrt {d} \sqrt {e} f^2 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx+\frac {\left (16 d^2 f g p^2\right ) \int \frac {1}{d+e x^2} \, dx}{3 e}+\frac {\left (16 d^{3/2} f g p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{3 \sqrt {e}}+\frac {1}{9} \left (16 e f g 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-\frac {1}{15} \left (8 d 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-\frac {\left (8 d^3 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{5 e^2}-\frac {\left (8 d^{5/2} g^2 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{5 e^{3/2}}+\frac {1}{25} \left (8 e 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 \\ & = 8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {16 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {8 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 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 {8 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\left (8 f^2 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 (16 d f g p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx}{3 e}+\frac {\left (16 d^2 f g p^2\right ) \int \frac {1}{d+e x^2} \, dx}{9 e}+\frac {\left (8 d^2 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}{5 e^2}-\frac {\left (8 d^3 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{25 e^2}-\frac {\left (8 d^3 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{15 e^2} \\ & = 8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {64 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {184 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}+\frac {8 \sqrt {d} f^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 )}{\sqrt {e}}-\frac {16 d^{3/2} f g 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 )}{3 e^{3/2}}+\frac {8 d^{5/2} 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 )}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 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 {8 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (8 f^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+\frac {\left (16 d f g 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}{3 e}-\frac {\left (8 d^2 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}{5 e^2} \\ & = 8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {64 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {184 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}+\frac {8 \sqrt {d} f^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 )}{\sqrt {e}}-\frac {16 d^{3/2} f g 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 )}{3 e^{3/2}}+\frac {8 d^{5/2} 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 )}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 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 {8 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {\left (8 i \sqrt {d} f^2 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}}-\frac {\left (16 i d^{3/2} f g 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 )}{3 e^{3/2}}+\frac {\left (8 i d^{5/2} g^2 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 )}{5 e^{5/2}} \\ & = 8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {64 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {184 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}+\frac {8 \sqrt {d} f^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 )}{\sqrt {e}}-\frac {16 d^{3/2} f g 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 )}{3 e^{3/2}}+\frac {8 d^{5/2} 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 )}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 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 {8 d^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {4 i \sqrt {d} f^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.46 \[ \int \left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {900 i \sqrt {d} \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right ) p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2+60 \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-2 \left (225 e^2 f^2-200 d e f g+69 d^2 g^2\right ) p+30 \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right ) p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+15 \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {e} x \left (8 p^2 \left (1035 d^2 g^2-120 d e g \left (25 f+g x^2\right )+e^2 \left (3375 f^2+250 f g x^2+27 g^2 x^4\right )\right )-60 p \left (45 d^2 g^2-15 d e g \left (10 f+g x^2\right )+e^2 \left (225 f^2+50 f g x^2+9 g^2 x^4\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+225 e^2 \left (15 f^2+10 f g x^2+3 g^2 x^4\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )\right )+900 i \sqrt {d} \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right ) p^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )}{3375 e^{5/2}} \]

[In]

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

[Out]

((900*I)*Sqrt[d]*(15*e^2*f^2 - 10*d*e*f*g + 3*d^2*g^2)*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + 60*Sqrt[d]*p*ArcTan
[(Sqrt[e]*x)/Sqrt[d]]*(-2*(225*e^2*f^2 - 200*d*e*f*g + 69*d^2*g^2)*p + 30*(15*e^2*f^2 - 10*d*e*f*g + 3*d^2*g^2
)*p*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + 15*(15*e^2*f^2 - 10*d*e*f*g + 3*d^2*g^2)*Log[c*(d + e*x^2)^p])
+ Sqrt[e]*x*(8*p^2*(1035*d^2*g^2 - 120*d*e*g*(25*f + g*x^2) + e^2*(3375*f^2 + 250*f*g*x^2 + 27*g^2*x^4)) - 60*
p*(45*d^2*g^2 - 15*d*e*g*(10*f + g*x^2) + e^2*(225*f^2 + 50*f*g*x^2 + 9*g^2*x^4))*Log[c*(d + e*x^2)^p] + 225*e
^2*(15*f^2 + 10*f*g*x^2 + 3*g^2*x^4)*Log[c*(d + e*x^2)^p]^2) + (900*I)*Sqrt[d]*(15*e^2*f^2 - 10*d*e*f*g + 3*d^
2*g^2)*p^2*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)])/(3375*e^(5/2))

Maple [C] (warning: unable to verify)

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

Time = 2.22 (sec) , antiderivative size = 1077, normalized size of antiderivative = 1.14

method	result	size
risch	\(\text {Expression too large to display}\)	\(1077\)

[In]

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

[Out]

-4/25*p*g^2*x^5*ln((e*x^2+d)^p)-4*p*f^2*x*ln((e*x^2+d)^p)-184/75*p^2/e^2*g^2*d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^
(1/2))-4*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^2*ln(e*x^2+d)+4/15*p/e*g^2*d*x^3*ln((e*x^2+d)^p)-4/5*p/e^
2*g^2*d^2*x*ln((e*x^2+d)^p)+4*p*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^2*ln((e*x^2+d)^p)+2/3*ln((e*x^2+d)^p)^
2*f*g*x^3+8/3*p/e*d*f*g*x*ln((e*x^2+d)^p)-8*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^2-8/9*p*f*g*x^3*ln((e*
x^2+d)^p)-8/3*p/e*d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f*g*ln((e*x^2+d)^p)+8/3*p^2/e*d^2/(d*e)^(1/2)*arctan
(x*e/(d*e)^(1/2))*f*g*ln(e*x^2+d)+1/5*ln((e*x^2+d)^p)^2*g^2*x^5+ln((e*x^2+d)^p)^2*x*f^2-4/15*p^2*e*Sum(-1/2*(l
n(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*(3*d^2*g^2-10*d*e*f*g+15*e^2*f^2)/e^4/_alpha,_alpha=RootOf(_Z^2*e
+d))+64/9*p^2/e*d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f*g-4/5*p^2/e^2*g^2*d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(
1/2))*ln(e*x^2+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)*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))*(1/5*ln((e*x^2+d)^
p)*g^2*x^5+2/3*ln((e*x^2+d)^p)*f*g*x^3+ln((e*x^2+d)^p)*x*f^2-2/15*p*e*(1/e^3*(3/5*e^2*g^2*x^5-d*e*g^2*x^3+10/3
*e^2*f*g*x^3+3*d^2*g^2*x-10*d*e*f*g*x+15*e^2*f^2*x)-d*(3*d^2*g^2-10*d*e*f*g+15*e^2*f^2)/e^3/(d*e)^(1/2)*arctan
(x*e/(d*e)^(1/2))))+4/5*p/e^2*g^2*d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*ln((e*x^2+d)^p)+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*(1/5*g^2*x^5+2/3*f*g*x^3+f^2*x)+8*f^2*p^2*x+8/125*
g^2*p^2*x^5-64/9*d*f*g*p^2*x/e+184/75*d^2*g^2*p^2*x/e^2-64/225*d*g^2*p^2*x^3/e+16/27*f*g*p^2*x^3

Fricas [F]

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

[In]

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

[Out]

integral((g^2*x^4 + 2*f*g*x^2 + f^2)*log((e*x^2 + d)^p*c)^2, x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((g*x^2+f)^2*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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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