\(\int (f+g x^3)^2 \log ^2(c (d+e x^2)^p) \, dx\) [294]

Optimal result

Integrand size = 24, antiderivative size = 835 \[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\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^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\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 {8 d^{7/2} 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 )}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 i d^{7/2} g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{7 e^{7/2}} \]

[Out]

-2*d*f*g*p^2*x^2/e+4/7*d^3*g^2*p*x*ln(c*(e*x^2+d)^p)/e^3-4/21*d^2*g^2*p*x^3*ln(c*(e*x^2+d)^p)/e^2+4/35*d*g^2*p
*x^5*ln(c*(e*x^2+d)^p)/e-1/2*f*g*p*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)/e^2-4/7*d^(7/2)*g^2*p*arctan(x*e^(1/2)/d^(1/2
))*ln(c*(e*x^2+d)^p)/e^(7/2)-8/7*d^(7/2)*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))
/e^(7/2)-4/7*I*d^(7/2)*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))^2/e^(7/2)-4/7*I*d^(7/2)*g^2*p^2*polylog(2,1-2*d^(1/2)
/(d^(1/2)+I*x*e^(1/2)))/e^(7/2)-d*f*g*(e*x^2+d)*ln(c*(e*x^2+d)^p)^2/e^2-8*f^2*p^2*arctan(x*e^(1/2)/d^(1/2))*d^
(1/2)/e^(1/2)+8*f^2*p^2*x+8/343*g^2*p^2*x^7+1/7*g^2*x^7*ln(c*(e*x^2+d)^p)^2-4*f^2*p*x*ln(c*(e*x^2+d)^p)-4/49*g
^2*p*x^7*ln(c*(e*x^2+d)^p)+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*arcta
n(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)+f^2*x*ln(c*(e*x^2+d)^p)^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)+2*d*f*g*p*(e*x^2+d)*ln(c*(e*x^2+d)^p)/e^2+1408/735*d^(7/2)*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))/e^(7/2)+1
/2*f*g*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^2/e^2-1408/735*d^3*g^2*p^2*x/e^3+568/2205*d^2*g^2*p^2*x^3/e^2-96/1225*d*g
^2*p^2*x^5/e+1/4*f*g*p^2*(e*x^2+d)^2/e^2

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.958, Rules used = {2521, 2500, 2526, 2498, 327, 211, 2520, 12, 5040, 4964, 2449, 2352, 2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341, 2507, 2505, 308} \[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {8}{343} g^2 p^2 x^7+\frac {1}{7} g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac {4}{49} g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {4 d g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {4 d^2 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{21 e^2}-\frac {2 d f g p^2 x^2}{e}+8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}+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^3 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+\frac {f g p^2 \left (e x^2+d\right )^2}{4 e^2}+\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^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {f g \left (e x^2+d\right )^2 \log ^2\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {d f g \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{e^2}-\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\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^{7/2} 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 )}{7 e^{7/2}}-\frac {f g p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{2 e^2}+\frac {2 d f 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^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^{7/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{7 e^{7/2}}+\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^{7/2} g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{7 e^{7/2}} \]

[In]

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

[Out]

8*f^2*p^2*x - (1408*d^3*g^2*p^2*x)/(735*e^3) - (2*d*f*g*p^2*x^2)/e + (568*d^2*g^2*p^2*x^3)/(2205*e^2) - (96*d*
g^2*p^2*x^5)/(1225*e) + (8*g^2*p^2*x^7)/343 + (f*g*p^2*(d + e*x^2)^2)/(4*e^2) - (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqr
t[e]*x)/Sqrt[d]])/Sqrt[e] + (1408*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(735*e^(7/2)) + ((4*I)*Sqrt[d]*
f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (((4*I)/7)*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(
7/2) + (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (8*d
^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2)) - 4*f^2*p*x*L
og[c*(d + e*x^2)^p] + (4*d^3*g^2*p*x*Log[c*(d + e*x^2)^p])/(7*e^3) - (4*d^2*g^2*p*x^3*Log[c*(d + e*x^2)^p])/(2
1*e^2) + (4*d*g^2*p*x^5*Log[c*(d + e*x^2)^p])/(35*e) - (4*g^2*p*x^7*Log[c*(d + e*x^2)^p])/49 + (2*d*f*g*p*(d +
e*x^2)*Log[c*(d + e*x^2)^p])/e^2 - (f*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(2*e^2) + (4*Sqrt[d]*f^2*p*ArcT
an[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (4*d^(7/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d
+ e*x^2)^p])/(7*e^(7/2)) + f^2*x*Log[c*(d + e*x^2)^p]^2 + (g^2*x^7*Log[c*(d + e*x^2)^p]^2)/7 - (d*f*g*(d + e*x
^2)*Log[c*(d + e*x^2)^p]^2)/e^2 + (f*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(2*e^2) + ((4*I)*Sqrt[d]*f^2*p^2*
PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (((4*I)/7)*d^(7/2)*g^2*p^2*PolyLog[2, 1 - (2*Sq
rt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/e^(7/2)

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 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 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^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \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^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \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 {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+(f g) \text {Subst}\left (\int x \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\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}{7} \left (4 e 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^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+(f 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 )-\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}{7} \left (4 e 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 \\ & = f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {(f g) \text {Subst}\left (\int (d+e x) \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{e}-\frac {(d f g) \text {Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{e}-\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}{7} \left (4 g^2 p\right ) \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {\left (4 d^3 g^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac {\left (4 d^4 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 (4 d^2 g^2 p\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}+\frac {\left (4 d g^2 p\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e} \\ & = -4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {(f g) \text {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\frac {(d f g) \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}+\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}{35} \left (8 d g^2 p^2\right ) \int \frac {x^6}{d+e x^2} \, dx-\frac {\left (8 d^3 g^2 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx}{7 e^2}+\frac {\left (8 d^4 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 g^2 p^2\right ) \int \frac {x^4}{d+e x^2} \, dx}{21 e}+\frac {1}{49} \left (8 e g^2 p^2\right ) \int \frac {x^8}{d+e x^2} \, dx \\ & = 8 f^2 p^2 x-\frac {8 d^3 g^2 p^2 x}{7 e^3}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac {(f g p) \text {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}+\frac {(2 d f 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^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 {1}{35} \left (8 d 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 (8 d^4 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}+\frac {\left (8 d^{7/2} 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 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}{21 e}+\frac {1}{49} \left (8 e 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 \\ & = 8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {8 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 (8 d^3 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 (8 d^4 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{49 e^3}+\frac {\left (8 d^4 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{35 e^3}+\frac {\left (8 d^4 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{21 e^3} \\ & = 8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\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 {8 d^{7/2} 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 )}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\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 (8 d^3 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} \\ & = 8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\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 {8 d^{7/2} 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 )}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 (8 i d^{7/2} 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 )}{7 e^{7/2}} \\ & = 8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\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 {8 d^{7/2} 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 )}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 i d^{7/2} 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] (verified)

Time = 0.34 (sec) , antiderivative size = 475, normalized size of antiderivative = 0.57 \[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {-176400 i \sqrt {d} \left (-7 e^3 f^2+d^3 g^2\right ) p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2-1680 \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (2 \left (735 e^3 f^2-176 d^3 g^2\right ) p-210 \left (7 e^3 f^2-d^3 g^2\right ) p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )-105 \left (7 e^3 f^2-d^3 g^2\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {e} \left (p^2 x \left (-591360 d^3 g^2+79520 d^2 e g^2 x^2-378 d e^2 g x \left (1225 f+64 g x^3\right )+225 e^3 \left (10976 f^2+343 f g x^3+32 g^2 x^6\right )\right )+154350 d^2 e f g p^2 \log \left (d+e x^2\right )-210 p \left (-840 d^3 g^2 x+70 d^2 e g \left (-21 f+4 g x^3\right )-42 d e^2 g x^2 \left (35 f+4 g x^3\right )+15 e^3 x \left (392 f^2+49 f g x^3+8 g^2 x^6\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+22050 \left (-7 d^2 e f g+e^3 x \left (14 f^2+7 f g x^3+2 g^2 x^6\right )\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )\right )-176400 i \sqrt {d} \left (-7 e^3 f^2+d^3 g^2\right ) p^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )}{308700 e^{7/2}} \]

[In]

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

[Out]

((-176400*I)*Sqrt[d]*(-7*e^3*f^2 + d^3*g^2)*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 - 1680*Sqrt[d]*p*ArcTan[(Sqrt[e]
*x)/Sqrt[d]]*(2*(735*e^3*f^2 - 176*d^3*g^2)*p - 210*(7*e^3*f^2 - d^3*g^2)*p*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[
e]*x)] - 105*(7*e^3*f^2 - d^3*g^2)*Log[c*(d + e*x^2)^p]) + Sqrt[e]*(p^2*x*(-591360*d^3*g^2 + 79520*d^2*e*g^2*x
^2 - 378*d*e^2*g*x*(1225*f + 64*g*x^3) + 225*e^3*(10976*f^2 + 343*f*g*x^3 + 32*g^2*x^6)) + 154350*d^2*e*f*g*p^
2*Log[d + e*x^2] - 210*p*(-840*d^3*g^2*x + 70*d^2*e*g*(-21*f + 4*g*x^3) - 42*d*e^2*g*x^2*(35*f + 4*g*x^3) + 15
*e^3*x*(392*f^2 + 49*f*g*x^3 + 8*g^2*x^6))*Log[c*(d + e*x^2)^p] + 22050*(-7*d^2*e*f*g + e^3*x*(14*f^2 + 7*f*g*
x^3 + 2*g^2*x^6))*Log[c*(d + e*x^2)^p]^2) - (176400*I)*Sqrt[d]*(-7*e^3*f^2 + d^3*g^2)*p^2*PolyLog[2, (I*Sqrt[d
] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)])/(308700*e^(7/2))

Maple [C] (warning: unable to verify)

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

Time = 3.75 (sec) , antiderivative size = 1127, normalized size of antiderivative = 1.35

method	result	size
risch	\(\text {Expression too large to display}\)	\(1127\)

[In]

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

[Out]

-4/49*p*g^2*x^7*ln((e*x^2+d)^p)-4/21*p/e^2*d^2*g^2*x^3*ln((e*x^2+d)^p)+4/7*p/e^3*x*d^3*g^2*ln((e*x^2+d)^p)+p^2
/e^2*d^2*f*g*ln(e*x^2+d)^2+3/2*p^2/e^2*d^2*f*g*ln(e*x^2+d)+1408/735*p^2/e^3*g^2*d^4/(d*e)^(1/2)*arctan(x*e/(d*
e)^(1/2))-4*p*f^2*x*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*(1/7*g^2*x^7+1/2*f*g*x^4+f^2*x)+1/4*p^2*f*g*x^4-1/2*p*f*g*x^4*ln((e*x^2+d)^p)-4*p^2*d/(d*e)^(1/2)*arctan
(x*e/(d*e)^(1/2))*f^2*ln(e*x^2+d)+4*p*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^2*ln((e*x^2+d)^p)+4/35*p/e*d*g^2
*x^5*ln((e*x^2+d)^p)-8*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^2+ln((e*x^2+d)^p)^2*x*f^2+1/2*ln((e*x^2+d)^
p)^2*f*g*x^4+1/7*ln((e*x^2+d)^p)^2*g^2*x^7-2/7*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*(7
*_alpha*d*e^2*f*g+2*d^3*g^2-14*e^3*f^2)/e^5/_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)*csgn(I*c)-I*Pi*csgn(I*c*(e*x^2+d)^p)^3+I*Pi*csg
n(I*c*(e*x^2+d)^p)^2*csgn(I*c)+2*ln(c))*(1/7*ln((e*x^2+d)^p)*g^2*x^7+1/2*ln((e*x^2+d)^p)*f*g*x^4+ln((e*x^2+d)^
p)*x*f^2-1/7*p*e*(1/e^4*(2/7*e^3*g^2*x^7-2/5*d*e^2*g^2*x^5+7/4*e^3*f*g*x^4+2/3*d^2*e*g^2*x^3-7/2*d*f*g*x^2*e^2
-2*x*d^3*g^2+14*x*e^3*f^2)+d/e^4*(7/2*d*e*f*g*ln(e*x^2+d)+(2*d^3*g^2-14*e^3*f^2)/(d*e)^(1/2)*arctan(x*e/(d*e)^
(1/2)))))+p/e*d*f*g*x^2*ln((e*x^2+d)^p)-p/e^2*d^2*f*g*ln(e*x^2+d)*ln((e*x^2+d)^p)-4/7*p/e^3*d^4/(d*e)^(1/2)*ar
ctan(x*e/(d*e)^(1/2))*g^2*ln((e*x^2+d)^p)+4/7*p^2/e^3*d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*g^2*ln(e*x^2+d)-
3/2*d*f*g*p^2*x^2/e+8*f^2*p^2*x+8/343*g^2*p^2*x^7-1408/735*d^3*g^2*p^2*x/e^3+568/2205*d^2*g^2*p^2*x^3/e^2-96/1
225*d*g^2*p^2*x^5/e

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((g*x^3+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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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