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

Optimal. Leaf size=945 \[ \text{result too large to display} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.25121, antiderivative size = 945, normalized size of antiderivative = 1., number of steps used = 50, number of rules used = 15, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {2471, 2450, 2476, 2448, 321, 205, 2470, 12, 4920, 4854, 2402, 2315, 2457, 2455, 302} \[ \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 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\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 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\frac{8 \sqrt{d} f^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{184 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{75 e^{5/2}}+\frac{64 d^{3/2} f g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}+\frac{8 \sqrt{d} f^2 p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}\right )}{\sqrt{e}}+\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}}{i \sqrt{e} x+\sqrt{d}}\right )}{5 e^{5/2}}-\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}}{i \sqrt{e} x+\sqrt{d}}\right )}{3 e^{3/2}}+\frac{4 \sqrt{d} f^2 p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\sqrt{e}}+\frac{4 d^{5/2} g^2 p \tan ^{-1}\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 \tan ^{-1}\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 \text{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 \text{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 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}\right )}{3 e^{3/2}} \]

Antiderivative was successfully verified.

[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 2471

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

Rule 2450

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

Rule 2476

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

Rule 2448

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

Rule 321

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

Rule 205

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

Rule 2470

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

Rule 12

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

Rule 4920

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

Rule 4854

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

Rule 2402

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

Rule 2315

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

Rule 2457

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

Rule 2455

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

Rule 302

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

Rubi steps

\begin{align*} \int \left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx &=\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 ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{\sqrt{e}}-\frac{\left (16 i d^{3/2} f g p^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{3 e^{3/2}}+\frac{\left (8 i d^{5/2} g^2 p^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{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]  time = 0.509005, size = 435, normalized size = 0.46 \[ \frac{900 i \sqrt{d} p^2 \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} x+i \sqrt{d}}{\sqrt{e} x-i \sqrt{d}}\right )+\sqrt{e} x \left (-60 p \left (45 d^2 g^2-15 d e g \left (10 f+g x^2\right )+e^2 \left (225 f^2+50 f g x^2+9 g^2 x^4\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+225 e^2 \left (15 f^2+10 f g x^2+3 g^2 x^4\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )+8 p^2 \left (1035 d^2 g^2-120 d e g \left (25 f+g x^2\right )+e^2 \left (3375 f^2+250 f g x^2+27 g^2 x^4\right )\right )\right )+60 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (15 \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right ) \log \left (c \left (d+e x^2\right )^p\right )+30 p \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )-2 p \left (69 d^2 g^2-200 d e f g+225 e^2 f^2\right )\right )+900 i \sqrt{d} p^2 \left (3 d^2 g^2-10 d e f g+15 e^2 f^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3375 e^{5/2}} \]

Antiderivative was successfully verified.

[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))

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Maple [F]  time = 0.945, size = 0, normalized size = 0. \begin{align*} \int \left ( g{x}^{2}+f \right ) ^{2} \left ( \ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (g^{2} x^{4} + 2 \, f g x^{2} + f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^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)