\(\int \frac {(a+b \log (c x^n))^2}{x^2 (d+e x)^4} \, dx\) [119]

Optimal result

Integrand size = 23, antiderivative size = 420 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=-\frac {2 b^2 n^2}{d^4 x}-\frac {b^2 e n^2}{3 d^4 (d+e x)}-\frac {b^2 e n^2 \log (x)}{3 d^5}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac {3 b^2 e n^2 \log (d+e x)}{d^5}-\frac {26 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {26 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^5}+\frac {8 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.03, number of steps used = 26, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2395, 2342, 2341, 2356, 2389, 2379, 2438, 2351, 31, 46, 2355, 2354, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}-\frac {8 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac {4 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^5}-\frac {8 b e n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^5}-\frac {6 b e n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}+\frac {8 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 d^5}-\frac {6 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^5}-\frac {b^2 e n^2 \log (x)}{3 d^5}+\frac {3 b^2 e n^2 \log (d+e x)}{d^5}-\frac {b^2 e n^2}{3 d^4 (d+e x)}-\frac {2 b^2 n^2}{d^4 x} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=-\frac {\frac {6 b^2 d n^2}{x}+\frac {6 b d n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b d^2 e n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {8 b d e n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-13 e \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {d^3 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}+\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {9 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}+8 b^2 e n^2 (\log (x)-\log (d+e x))+\frac {b^2 e n^2 (d+(d+e x) \log (x)-(d+e x) \log (d+e x))}{d+e x}+26 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-12 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+26 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-24 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+24 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{3 d^5} \]

[In]

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

[Out]

-1/3*((6*b^2*d*n^2)/x + (6*b*d*n*(a + b*Log[c*x^n]))/x - (b*d^2*e*n*(a + b*Log[c*x^n]))/(d + e*x)^2 - (8*b*d*e
*n*(a + b*Log[c*x^n]))/(d + e*x) - 13*e*(a + b*Log[c*x^n])^2 + (3*d*(a + b*Log[c*x^n])^2)/x + (d^3*e*(a + b*Lo
g[c*x^n])^2)/(d + e*x)^3 + (3*d^2*e*(a + b*Log[c*x^n])^2)/(d + e*x)^2 + (9*d*e*(a + b*Log[c*x^n])^2)/(d + e*x)
+ (4*e*(a + b*Log[c*x^n])^3)/(b*n) + 8*b^2*e*n^2*(Log[x] - Log[d + e*x]) + (b^2*e*n^2*(d + (d + e*x)*Log[x] -
(d + e*x)*Log[d + e*x]))/(d + e*x) + 26*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 12*e*(a + b*Log[c*x^n])^2
*Log[1 + (e*x)/d] + 26*b^2*e*n^2*PolyLog[2, -((e*x)/d)] - 24*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] +
24*b^2*e*n^2*PolyLog[3, -((e*x)/d)])/d^5

Maple [C] (warning: unable to verify)

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

Time = 0.66 (sec) , antiderivative size = 1015, normalized size of antiderivative = 2.42

method	result	size
risch	\(\text {Expression too large to display}\)	\(1015\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^4*x^6 + 4*d*e^3*x^5 + 6*d^2*e^2*x^4 + 4*d^3*e*x^3 + d^
4*x^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/3*a^2*((12*e^3*x^3 + 30*d*e^2*x^2 + 22*d^2*e*x + 3*d^3)/(d^4*e^3*x^4 + 3*d^5*e^2*x^3 + 3*d^6*e*x^2 + d^7*x)
- 12*e*log(e*x + d)/d^5 + 12*e*log(x)/d^5) + integrate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2
*log(c) + a*b)*log(x^n))/(e^4*x^6 + 4*d*e^3*x^5 + 6*d^2*e^2*x^4 + 4*d^3*e*x^3 + d^4*x^2), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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