3.310 \(\int \frac{x^5 (a+b \log (c (d+e x)^n))^2}{f+g x^2} \, dx\)

Optimal. Leaf size=831 \[ \frac{b^2 n^2 \log ^2(d+e x) d^4}{4 e^4 g}-\frac{b n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^4}{2 e^4 g}-\frac{2 b^2 n^2 x d^3}{e^3 g}+\frac{2 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{e^4 g}+\frac{3 b^2 n^2 (d+e x)^2 d^2}{4 e^4 g}-\frac{3 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{2 e^4 g}-\frac{2 b^2 n^2 (d+e x)^3 d}{9 e^4 g}+\frac{f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 d}{e^2 g^2}+\frac{2 b^2 f n^2 x d}{e g^2}-\frac{2 a b f n x d}{e g^2}-\frac{2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right ) d}{e^2 g^2}+\frac{2 b n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{3 e^4 g}+\frac{b^2 n^2 (d+e x)^4}{32 e^4 g}-\frac{b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac{f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac{b n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 e^4 g}+\frac{b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{\sqrt{g} d+e \sqrt{-f}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{g} x+\sqrt{-f}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}+\frac{b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}+\frac{b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right )}{g^3}-\frac{b^2 f^2 n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}-\frac{b^2 f^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+e \sqrt{-f}}\right )}{g^3} \]

[Out]

(-2*a*b*d*f*n*x)/(e*g^2) + (2*b^2*d*f*n^2*x)/(e*g^2) - (2*b^2*d^3*n^2*x)/(e^3*g) - (b^2*f*n^2*(d + e*x)^2)/(4*
e^2*g^2) + (3*b^2*d^2*n^2*(d + e*x)^2)/(4*e^4*g) - (2*b^2*d*n^2*(d + e*x)^3)/(9*e^4*g) + (b^2*n^2*(d + e*x)^4)
/(32*e^4*g) + (b^2*d^4*n^2*Log[d + e*x]^2)/(4*e^4*g) - (2*b^2*d*f*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g^2) +
(2*b*d^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(e^4*g) + (b*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e
^2*g^2) - (3*b*d^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^4*g) + (2*b*d*n*(d + e*x)^3*(a + b*Log[c*(d
+ e*x)^n]))/(3*e^4*g) - (b*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(8*e^4*g) - (b*d^4*n*Log[d + e*x]*(a + b*
Log[c*(d + e*x)^n]))/(2*e^4*g) + (x^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g) + (d*f*(d + e*x)*(a + b*Log[c*(d + e
*x)^n])^2)/(e^2*g^2) - (f*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*g^2) + (f^2*(a + b*Log[c*(d + e*x)^
n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3) + (f^2*(a + b*Log[c*(d + e*x)^n])^2*Lo
g[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^3) + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[
2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqr
t[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3 - (b^2*f^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d
*Sqrt[g]))])/g^3 - (b^2*f^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3

________________________________________________________________________________________

Rubi [A]  time = 1.08527, antiderivative size = 752, normalized size of antiderivative = 0.9, number of steps used = 28, number of rules used = 19, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.655, Rules used = {2416, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2398, 2411, 43, 2334, 12, 14, 2301, 2396, 2433, 2374, 6589} \[ \frac{b f^2 n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac{b f^2 n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}-\frac{b^2 f^2 n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}-\frac{b^2 f^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{g^3}+\frac{b n \left (\frac{48 d^3 (d+e x)}{e^4}-\frac{36 d^2 (d+e x)^2}{e^4}-\frac{12 d^4 \log (d+e x)}{e^4}+\frac{16 d (d+e x)^3}{e^4}-\frac{3 (d+e x)^4}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac{f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac{b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{f^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3}+\frac{f^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^3}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac{2 a b d f n x}{e g^2}-\frac{2 b^2 d f n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}-\frac{2 b^2 d^3 n^2 x}{e^3 g}+\frac{3 b^2 d^2 n^2 (d+e x)^2}{4 e^4 g}+\frac{b^2 d^4 n^2 \log ^2(d+e x)}{4 e^4 g}-\frac{b^2 f n^2 (d+e x)^2}{4 e^2 g^2}-\frac{2 b^2 d n^2 (d+e x)^3}{9 e^4 g}+\frac{b^2 n^2 (d+e x)^4}{32 e^4 g}+\frac{2 b^2 d f n^2 x}{e g^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2),x]

[Out]

(-2*a*b*d*f*n*x)/(e*g^2) + (2*b^2*d*f*n^2*x)/(e*g^2) - (2*b^2*d^3*n^2*x)/(e^3*g) - (b^2*f*n^2*(d + e*x)^2)/(4*
e^2*g^2) + (3*b^2*d^2*n^2*(d + e*x)^2)/(4*e^4*g) - (2*b^2*d*n^2*(d + e*x)^3)/(9*e^4*g) + (b^2*n^2*(d + e*x)^4)
/(32*e^4*g) + (b^2*d^4*n^2*Log[d + e*x]^2)/(4*e^4*g) - (2*b^2*d*f*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g^2) +
(b*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2*g^2) + (b*n*((48*d^3*(d + e*x))/e^4 - (36*d^2*(d + e*x)^
2)/e^4 + (16*d*(d + e*x)^3)/e^4 - (3*(d + e*x)^4)/e^4 - (12*d^4*Log[d + e*x])/e^4)*(a + b*Log[c*(d + e*x)^n]))
/(24*g) + (x^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g) + (d*f*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e^2*g^2) -
(f*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*g^2) + (f^2*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f]
- Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3) + (f^2*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g
]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^3) + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x)
)/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt
[-f] + d*Sqrt[g])])/g^3 - (b^2*f^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 - (b^2
*f^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3

Rule 2416

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

Rule 2401

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

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

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

Rule 2390

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

Rule 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2398

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

Rule 2411

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

Rule 43

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

Rule 2334

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

Rule 12

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

Rule 14

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

Rule 2301

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

Rule 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx &=\int \left (-\frac{f x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac{f^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac{f \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g^2}+\frac{f^2 \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{g^2}+\frac{\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g}\\ &=\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac{f \int \left (-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\right ) \, dx}{g^2}+\frac{f^2 \int \left (-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{g^2}-\frac{(b e n) \int \frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{2 g}\\ &=\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac{f^2 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 g^{5/2}}+\frac{f^2 \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 g^{5/2}}-\frac{f \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{e g^2}+\frac{(d f) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{e g^2}-\frac{(b n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^4 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{2 g}\\ &=\frac{b n \left (\frac{48 d^3 (d+e x)}{e^4}-\frac{36 d^2 (d+e x)^2}{e^4}+\frac{16 d (d+e x)^3}{e^4}-\frac{3 (d+e x)^4}{e^4}-\frac{12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}-\frac{f \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g^2}+\frac{(d f) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g^2}-\frac{\left (b e f^2 n\right ) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{d+e x} \, dx}{g^3}-\frac{\left (b e f^2 n\right ) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{d+e x} \, dx}{g^3}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3\right )+12 d^4 \log (x)}{12 e^4 x} \, dx,x,d+e x\right )}{2 g}\\ &=\frac{b n \left (\frac{48 d^3 (d+e x)}{e^4}-\frac{36 d^2 (d+e x)^2}{e^4}+\frac{16 d (d+e x)^3}{e^4}-\frac{3 (d+e x)^4}{e^4}-\frac{12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac{f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}-\frac{\left (b f^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e \sqrt{-f}+d \sqrt{g}}{e}-\frac{\sqrt{g} x}{e}\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{g^3}-\frac{\left (b f^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e \sqrt{-f}-d \sqrt{g}}{e}+\frac{\sqrt{g} x}{e}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{g^3}+\frac{(b f n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g^2}-\frac{(2 b d f n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g^2}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3\right )+12 d^4 \log (x)}{x} \, dx,x,d+e x\right )}{24 e^4 g}\\ &=-\frac{2 a b d f n x}{e g^2}-\frac{b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac{b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{b n \left (\frac{48 d^3 (d+e x)}{e^4}-\frac{36 d^2 (d+e x)^2}{e^4}+\frac{16 d (d+e x)^3}{e^4}-\frac{3 (d+e x)^4}{e^4}-\frac{12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac{f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}+\frac{b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}+\frac{b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^3}-\frac{\left (2 b^2 d f n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2 g^2}-\frac{\left (b^2 f^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{\sqrt{g} x}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{g^3}-\frac{\left (b^2 f^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{g} x}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{g^3}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3+\frac{12 d^4 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{24 e^4 g}\\ &=-\frac{2 a b d f n x}{e g^2}+\frac{2 b^2 d f n^2 x}{e g^2}-\frac{2 b^2 d^3 n^2 x}{e^3 g}-\frac{b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac{3 b^2 d^2 n^2 (d+e x)^2}{4 e^4 g}-\frac{2 b^2 d n^2 (d+e x)^3}{9 e^4 g}+\frac{b^2 n^2 (d+e x)^4}{32 e^4 g}-\frac{2 b^2 d f n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}+\frac{b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{b n \left (\frac{48 d^3 (d+e x)}{e^4}-\frac{36 d^2 (d+e x)^2}{e^4}+\frac{16 d (d+e x)^3}{e^4}-\frac{3 (d+e x)^4}{e^4}-\frac{12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac{f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}+\frac{b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}+\frac{b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^3}-\frac{b^2 f^2 n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}-\frac{b^2 f^2 n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^3}+\frac{\left (b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e x\right )}{2 e^4 g}\\ &=-\frac{2 a b d f n x}{e g^2}+\frac{2 b^2 d f n^2 x}{e g^2}-\frac{2 b^2 d^3 n^2 x}{e^3 g}-\frac{b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac{3 b^2 d^2 n^2 (d+e x)^2}{4 e^4 g}-\frac{2 b^2 d n^2 (d+e x)^3}{9 e^4 g}+\frac{b^2 n^2 (d+e x)^4}{32 e^4 g}+\frac{b^2 d^4 n^2 \log ^2(d+e x)}{4 e^4 g}-\frac{2 b^2 d f n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}+\frac{b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac{b n \left (\frac{48 d^3 (d+e x)}{e^4}-\frac{36 d^2 (d+e x)^2}{e^4}+\frac{16 d (d+e x)^3}{e^4}-\frac{3 (d+e x)^4}{e^4}-\frac{12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac{d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac{f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^3}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^3}+\frac{b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}+\frac{b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^3}-\frac{b^2 f^2 n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^3}-\frac{b^2 f^2 n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^3}\\ \end{align*}

Mathematica [C]  time = 1.01959, size = 862, normalized size = 1.04 \[ \frac{72 g^2 x^4 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 e^4-144 f g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 e^4+144 f^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (g x^2+f\right ) e^4-12 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-24 f^2 \left (\log (d+e x) \log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )\right ) e^4-24 f^2 \left (\log (d+e x) \log \left (1-\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right )\right ) e^4+12 f g \left (e x (2 d-e x)-2 \left (d^2-e^2 x^2\right ) \log (d+e x)\right ) e^2+g^2 \left (e x \left (-12 d^3+6 e x d^2-4 e^2 x^2 d+3 e^3 x^3\right )+12 \left (d^4-e^4 x^4\right ) \log (d+e x)\right )\right )+b^2 n^2 \left (144 f^2 \left (\log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )\right ) e^4+144 f^2 \left (\log \left (1-\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right ) \log ^2(d+e x)+2 \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right ) \log (d+e x)-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{\sqrt{g} d+i e \sqrt{f}}\right )\right ) e^4-72 f g \left (-2 \left (d^2-e^2 x^2\right ) \log ^2(d+e x)+\left (6 d^2+4 e x d-2 e^2 x^2\right ) \log (d+e x)+e x (e x-6 d)\right ) e^2-g^2 \left (72 \left (d^4-e^4 x^4\right ) \log ^2(d+e x)-12 \left (25 d^4+12 e x d^3-6 e^2 x^2 d^2+4 e^3 x^3 d-3 e^4 x^4\right ) \log (d+e x)+e x \left (300 d^3-78 e x d^2+28 e^2 x^2 d-9 e^3 x^3\right )\right )\right )}{288 e^4 g^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(a + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2),x]

[Out]

(-144*e^4*f*g*x^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 72*e^4*g^2*x^4*(a - b*n*Log[d + e*x] + b*L
og[c*(d + e*x)^n])^2 + 144*e^4*f^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x^2] - 12*b*n*(a
- b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(12*e^2*f*g*(e*x*(2*d - e*x) - 2*(d^2 - e^2*x^2)*Log[d + e*x]) + g^
2*(e*x*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 12*(d^4 - e^4*x^4)*Log[d + e*x]) - 24*e^4*f^2*(Log[d
+ e*x]*Log[1 - (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt
[f] + d*Sqrt[g])]) - 24*e^4*f^2*(Log[d + e*x]*Log[1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + PolyLog
[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])) + b^2*n^2*(-72*e^2*f*g*(e*x*(-6*d + e*x) + (6*d^2 + 4*d*e
*x - 2*e^2*x^2)*Log[d + e*x] - 2*(d^2 - e^2*x^2)*Log[d + e*x]^2) - g^2*(e*x*(300*d^3 - 78*d^2*e*x + 28*d*e^2*x
^2 - 9*e^3*x^3) - 12*(25*d^4 + 12*d^3*e*x - 6*d^2*e^2*x^2 + 4*d*e^3*x^3 - 3*e^4*x^4)*Log[d + e*x] + 72*(d^4 -
e^4*x^4)*Log[d + e*x]^2) + 144*e^4*f^2*(Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g]
)] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog[3, (Sqrt[g]*(d +
e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])]) + 144*e^4*f^2*(Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] +
d*Sqrt[g])] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog[3, (Sqrt[g]
*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])))/(288*e^4*g^3)

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Maple [F]  time = 1.493, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{g{x}^{2}+f}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f),x)

[Out]

int(x^5*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a^{2}{\left (\frac{2 \, f^{2} \log \left (g x^{2} + f\right )}{g^{3}} + \frac{g x^{4} - 2 \, f x^{2}}{g^{2}}\right )} + \int \frac{b^{2} x^{5} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} x^{5} \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} x^{5}}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f),x, algorithm="maxima")

[Out]

1/4*a^2*(2*f^2*log(g*x^2 + f)/g^3 + (g*x^4 - 2*f*x^2)/g^2) + integrate((b^2*x^5*log((e*x + d)^n)^2 + 2*(b^2*lo
g(c) + a*b)*x^5*log((e*x + d)^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x^5)/(g*x^2 + f), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{5} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b x^{5} \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} x^{5}}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f),x, algorithm="fricas")

[Out]

integral((b^2*x^5*log((e*x + d)^n*c)^2 + 2*a*b*x^5*log((e*x + d)^n*c) + a^2*x^5)/(g*x^2 + f), 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(x**5*(a+b*ln(c*(e*x+d)**n))**2/(g*x**2+f),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{5}}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f),x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^2*x^5/(g*x^2 + f), x)