\(\int \frac {x^4 (a+b \log (c x^n))^2}{(d+e x)^4} \, dx\) [113]

Optimal result

Integrand size = 23, antiderivative size = 398 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {2 a b n x}{e^4}+\frac {2 b^2 n^2 x}{e^4}-\frac {b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac {b^2 d n^2 \log (x)}{3 e^5}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac {5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {3 b^2 d n^2 \log (d+e x)}{e^5}-\frac {26 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {26 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^5} \]

[Out]

-2*a*b*n*x/e^4+2*b^2*n^2*x/e^4-1/3*b^2*d^2*n^2/e^5/(e*x+d)-1/3*b^2*d*n^2*ln(x)/e^5-2*b^2*n*x*ln(c*x^n)/e^4+1/3
*b*d^3*n*(a+b*ln(c*x^n))/e^5/(e*x+d)^2+10/3*b*d*n*x*(a+b*ln(c*x^n))/e^4/(e*x+d)-5/3*d*(a+b*ln(c*x^n))^2/e^5+x*
(a+b*ln(c*x^n))^2/e^4-1/3*d^4*(a+b*ln(c*x^n))^2/e^5/(e*x+d)^3+2*d^3*(a+b*ln(c*x^n))^2/e^5/(e*x+d)^2+6*d*x*(a+b
*ln(c*x^n))^2/e^4/(e*x+d)-3*b^2*d*n^2*ln(e*x+d)/e^5-26/3*b*d*n*(a+b*ln(c*x^n))*ln(1+e*x/d)/e^5-4*d*(a+b*ln(c*x
^n))^2*ln(1+e*x/d)/e^5-26/3*b^2*d*n^2*polylog(2,-e*x/d)/e^5-8*b*d*n*(a+b*ln(c*x^n))*polylog(2,-e*x/d)/e^5+8*b^
2*d*n^2*polylog(3,-e*x/d)/e^5

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.08, number of steps used = 27, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2395, 2333, 2332, 2356, 2389, 2379, 2438, 2351, 31, 46, 2355, 2354, 2421, 6724} \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}-\frac {8 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^5}+\frac {10 b d n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^5}-\frac {4 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^5}-\frac {12 b d n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^5}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {2 a b n x}{e^4}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}-\frac {b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac {10 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 e^5}-\frac {12 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^5}-\frac {b^2 d n^2 \log (x)}{3 e^5}-\frac {3 b^2 d n^2 \log (d+e x)}{e^5}+\frac {2 b^2 n^2 x}{e^4} \]

[In]

Int[(x^4*(a + b*Log[c*x^n])^2)/(d + e*x)^4,x]

[Out]

(-2*a*b*n*x)/e^4 + (2*b^2*n^2*x)/e^4 - (b^2*d^2*n^2)/(3*e^5*(d + e*x)) - (b^2*d*n^2*Log[x])/(3*e^5) - (2*b^2*n
*x*Log[c*x^n])/e^4 + (b*d^3*n*(a + b*Log[c*x^n]))/(3*e^5*(d + e*x)^2) + (10*b*d*n*x*(a + b*Log[c*x^n]))/(3*e^4
*(d + e*x)) + (10*b*d*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*e^5) + (x*(a + b*Log[c*x^n])^2)/e^4 - (d^4*(a
+ b*Log[c*x^n])^2)/(3*e^5*(d + e*x)^3) + (2*d^3*(a + b*Log[c*x^n])^2)/(e^5*(d + e*x)^2) + (6*d*x*(a + b*Log[c*
x^n])^2)/(e^4*(d + e*x)) - (3*b^2*d*n^2*Log[d + e*x])/e^5 - (12*b*d*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^5
- (4*d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^5 - (10*b^2*d*n^2*PolyLog[2, -(d/(e*x))])/(3*e^5) - (12*b^2*d
*n^2*PolyLog[2, -((e*x)/d)])/e^5 - (8*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^5 + (8*b^2*d*n^2*Poly
Log[3, -((e*x)/d)])/e^5

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 2351

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

Rule 2354

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

Rule 2355

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

Rule 2356

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

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2395

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2438

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

Rule 6724

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*} \text {integral}& = \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e^4}+\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^4}-\frac {4 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^3}+\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^2}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}\right ) \, dx \\ & = \frac {\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^4}-\frac {(4 d) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^4}+\frac {\left (6 d^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^4}-\frac {\left (4 d^3\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^4}+\frac {d^4 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{e^4} \\ & = \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}+\frac {(8 b d n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^5}-\frac {\left (4 b d^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^5}+\frac {\left (2 b d^4 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 e^5}-\frac {(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^4}-\frac {(12 b d n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^4} \\ & = -\frac {2 a b n x}{e^4}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {12 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}-\frac {\left (4 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^5}+\frac {\left (2 b d^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 e^5}-\frac {\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e^4}+\frac {\left (4 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^4}-\frac {\left (2 b d^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 e^4}+\frac {\left (8 b^2 d n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^5}+\frac {\left (12 b^2 d n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^5} \\ & = -\frac {2 a b n x}{e^4}+\frac {2 b^2 n^2 x}{e^4}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {4 b d n x \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)}+\frac {4 b d n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {12 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {12 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^5}+\frac {\left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 e^5}-\frac {\left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 e^4}-\frac {\left (4 b^2 d n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{e^5}-\frac {\left (b^2 d^3 n^2\right ) \int \frac {1}{x (d+e x)^2} \, dx}{3 e^5}-\frac {\left (4 b^2 d n^2\right ) \int \frac {1}{d+e x} \, dx}{e^4} \\ & = -\frac {2 a b n x}{e^4}+\frac {2 b^2 n^2 x}{e^4}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}+\frac {10 b d n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {4 b^2 d n^2 \log (d+e x)}{e^5}-\frac {12 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {4 b^2 d n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{e^5}-\frac {12 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^5}+\frac {\left (2 b^2 d n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{3 e^5}-\frac {\left (b^2 d^3 n^2\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{3 e^5}+\frac {\left (2 b^2 d n^2\right ) \int \frac {1}{d+e x} \, dx}{3 e^4} \\ & = -\frac {2 a b n x}{e^4}+\frac {2 b^2 n^2 x}{e^4}-\frac {b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac {b^2 d n^2 \log (x)}{3 e^5}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}+\frac {10 b d n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {3 b^2 d n^2 \log (d+e x)}{e^5}-\frac {12 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {10 b^2 d n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{3 e^5}-\frac {12 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.86 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {-\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {10 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-13 d \left (a+b \log \left (c x^n\right )\right )^2-3 e x \left (a+b \log \left (c x^n\right )\right )^2+\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}-\frac {6 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {18 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+6 b e n x \left (a-b n+b \log \left (c x^n\right )\right )-10 b^2 d n^2 (\log (x)-\log (d+e x))+\frac {b^2 d n^2 (d+(d+e x) \log (x)-(d+e x) \log (d+e x))}{d+e x}+26 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+12 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+26 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+24 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-24 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{3 e^5} \]

[In]

Integrate[(x^4*(a + b*Log[c*x^n])^2)/(d + e*x)^4,x]

[Out]

-1/3*(-((b*d^3*n*(a + b*Log[c*x^n]))/(d + e*x)^2) + (10*b*d^2*n*(a + b*Log[c*x^n]))/(d + e*x) - 13*d*(a + b*Lo
g[c*x^n])^2 - 3*e*x*(a + b*Log[c*x^n])^2 + (d^4*(a + b*Log[c*x^n])^2)/(d + e*x)^3 - (6*d^3*(a + b*Log[c*x^n])^
2)/(d + e*x)^2 + (18*d^2*(a + b*Log[c*x^n])^2)/(d + e*x) + 6*b*e*n*x*(a - b*n + b*Log[c*x^n]) - 10*b^2*d*n^2*(
Log[x] - Log[d + e*x]) + (b^2*d*n^2*(d + (d + e*x)*Log[x] - (d + e*x)*Log[d + e*x]))/(d + e*x) + 26*b*d*n*(a +
b*Log[c*x^n])*Log[1 + (e*x)/d] + 12*d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 26*b^2*d*n^2*PolyLog[2, -((e*x)
/d)] + 24*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] - 24*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^5

Maple [C] (warning: unable to verify)

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

Time = 0.56 (sec) , antiderivative size = 943, normalized size of antiderivative = 2.37

method	result	size
risch	\(\text {Expression too large to display}\)	\(943\)

[In]

int(x^4*(a+b*ln(c*x^n))^2/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-6*b^2*ln(x^n)^2/e^5*d^2/(e*x+d)+2*b^2*ln(x^n)^2/e^5*d^3/(e*x+d)^2-2*b^2*n*ln(x^n)*x/e^4-13/3*b^2/e^5*n^2*d*ln
(x)^2+26/3*b^2/e^5*n^2*dilog(-e*x/d)*d-1/3*b^2*ln(x^n)^2*d^4/e^5/(e*x+d)^3-4*b^2*ln(x^n)^2/e^5*d*ln(e*x+d)-8*b
^2/e^5*d*n^2*ln(x)*polylog(2,-e*x/d)+1/3*b^2*n*ln(x^n)/e^5*d^3/(e*x+d)^2-26/3*b^2*n*ln(x^n)/e^5*d*ln(e*x+d)-10
/3*b^2*n*ln(x^n)/e^5*d^2/(e*x+d)+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2
+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(ln(x^n)*x/e^4-1/3*ln(x^n)*d^4/e^5
/(e*x+d)^3-4*ln(x^n)/e^5*d*ln(e*x+d)-6*ln(x^n)/e^5*d^2/(e*x+d)+2*ln(x^n)/e^5*d^3/(e*x+d)^2-1/3*n*(1/e^5*(3*e*x
+3*d-1/2*d^3/(e*x+d)^2+13*d*ln(e*x+d)+5*d^2/(e*x+d)-13*d*ln(e*x))-12/e^5*d*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d)
)))-1/3*b^2*d^2*n^2/e^5/(e*x+d)+3*b^2*d*n^2*ln(x)/e^5-3*b^2*d*n^2*ln(e*x+d)/e^5+8*b^2*d*n^2*polylog(3,-e*x/d)/
e^5+1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(
I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*(x/e^4-1/3/e^5*d^4/(e*x+d)^3-4/e^5*d*ln(e*x+d)-6/e^5*d^2/(e
*x+d)+2/e^5*d^3/(e*x+d)^2)+b^2*ln(x^n)^2*x/e^4+26/3*b^2*n/e^5*ln(x)*ln(x^n)*d+26/3*b^2/e^5*n^2*ln(e*x+d)*ln(-e
*x/d)*d-8*b^2/e^5*d*ln(x)*dilog(-e*x/d)*n^2+8*b^2*n/e^5*d*ln(x^n)*dilog(-e*x/d)+4*b^2/e^5*d*n^2*ln(e*x+d)*ln(x
)^2-4*b^2/e^5*d*n^2*ln(x)^2*ln(1+e*x/d)-8*b^2/e^5*d*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2+8*b^2*n/e^5*d*ln(x^n)*ln(e*
x+d)*ln(-e*x/d)+2*b^2*n^2*x/e^4

Fricas [F]

\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{4}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x^4*(a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral((b^2*x^4*log(c*x^n)^2 + 2*a*b*x^4*log(c*x^n) + a^2*x^4)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^
3*e*x + d^4), x)

Sympy [F]

\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, dx \]

[In]

integrate(x**4*(a+b*ln(c*x**n))**2/(e*x+d)**4,x)

[Out]

Integral(x**4*(a + b*log(c*x**n))**2/(d + e*x)**4, x)

Maxima [F]

\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{4}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x^4*(a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/3*a^2*((18*d^2*e^2*x^2 + 30*d^3*e*x + 13*d^4)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) - 3*x/e^4 + 1
2*d*log(e*x + d)/e^5) + integrate((b^2*x^4*log(x^n)^2 + 2*(b^2*log(c) + a*b)*x^4*log(x^n) + (b^2*log(c)^2 + 2*
a*b*log(c))*x^4)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)

Giac [F]

\[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{4}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x^4*(a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2*x^4/(e*x + d)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {x^4\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

int((x^4*(a + b*log(c*x^n))^2)/(d + e*x)^4,x)

[Out]

int((x^4*(a + b*log(c*x^n))^2)/(d + e*x)^4, x)