\(\int \frac {x^3 (a+b \log (c x^n))^2}{(d+e x)^4} \, dx\) [114]

Optimal result

Integrand size = 23, antiderivative size = 333 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\frac {b^2 d n^2}{3 e^4 (d+e x)}+\frac {b^2 n^2 \log (x)}{3 e^4}-\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^2}-\frac {7 b n x \left (a+b \log \left (c x^n\right )\right )}{3 e^3 (d+e x)}+\frac {7 \left (a+b \log \left (c x^n\right )\right )^2}{6 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^4 (d+e x)^3}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac {2 b^2 n^2 \log (d+e x)}{e^4}+\frac {11 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 e^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {11 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 e^4}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4} \]

[Out]

1/3*b^2*d*n^2/e^4/(e*x+d)+1/3*b^2*n^2*ln(x)/e^4-1/3*b*d^2*n*(a+b*ln(c*x^n))/e^4/(e*x+d)^2-7/3*b*n*x*(a+b*ln(c*
x^n))/e^3/(e*x+d)+7/6*(a+b*ln(c*x^n))^2/e^4+1/3*d^3*(a+b*ln(c*x^n))^2/e^4/(e*x+d)^3-3/2*d^2*(a+b*ln(c*x^n))^2/
e^4/(e*x+d)^2-3*x*(a+b*ln(c*x^n))^2/e^3/(e*x+d)+2*b^2*n^2*ln(e*x+d)/e^4+11/3*b*n*(a+b*ln(c*x^n))*ln(1+e*x/d)/e
^4+(a+b*ln(c*x^n))^2*ln(1+e*x/d)/e^4+11/3*b^2*n^2*polylog(2,-e*x/d)/e^4+2*b*n*(a+b*ln(c*x^n))*polylog(2,-e*x/d
)/e^4-2*b^2*n^2*polylog(3,-e*x/d)/e^4

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.09, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2395, 2356, 2389, 2379, 2438, 2351, 31, 46, 2355, 2354, 2421, 6724} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^4 (d+e x)^3}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}-\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^2}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {7 b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^4}+\frac {6 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}-\frac {7 b n x \left (a+b \log \left (c x^n\right )\right )}{3 e^3 (d+e x)}+\frac {7 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 e^4}+\frac {6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4}+\frac {b^2 d n^2}{3 e^4 (d+e x)}+\frac {2 b^2 n^2 \log (d+e x)}{e^4}+\frac {b^2 n^2 \log (x)}{3 e^4} \]

[In]

Int[(x^3*(a + b*Log[c*x^n])^2)/(d + e*x)^4,x]

[Out]

(b^2*d*n^2)/(3*e^4*(d + e*x)) + (b^2*n^2*Log[x])/(3*e^4) - (b*d^2*n*(a + b*Log[c*x^n]))/(3*e^4*(d + e*x)^2) -
(7*b*n*x*(a + b*Log[c*x^n]))/(3*e^3*(d + e*x)) - (7*b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*e^4) + (d^3*(a
+ b*Log[c*x^n])^2)/(3*e^4*(d + e*x)^3) - (3*d^2*(a + b*Log[c*x^n])^2)/(2*e^4*(d + e*x)^2) - (3*x*(a + b*Log[c
*x^n])^2)/(e^3*(d + e*x)) + (2*b^2*n^2*Log[d + e*x])/e^4 + (6*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 + (
(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^4 + (7*b^2*n^2*PolyLog[2, -(d/(e*x))])/(3*e^4) + (6*b^2*n^2*PolyLog[2
, -((e*x)/d)])/e^4 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^4 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)]
)/e^4

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 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 {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^4}+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^3}-\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^3}-\frac {(3 d) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^3}+\frac {\left (3 d^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^3}-\frac {d^3 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{e^3} \\ & = \frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^4 (d+e x)^3}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4}+\frac {\left (3 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (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 )}{x (d+e x)^3} \, dx}{3 e^4}+\frac {(6 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3} \\ & = \frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^4 (d+e x)^3}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac {6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {(3 b d n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^4}-\frac {\left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 e^4}-\frac {(3 b d n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^3}+\frac {\left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 e^3}-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^4}-\frac {\left (6 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4} \\ & = -\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^2}-\frac {3 b n x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}-\frac {3 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^4 (d+e x)^3}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac {6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {6 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^4}-\frac {(2 b d n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 e^4}+\frac {(2 b d n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 e^3}+\frac {\left (3 b^2 n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{e^4}+\frac {\left (b^2 d^2 n^2\right ) \int \frac {1}{x (d+e x)^2} \, dx}{3 e^4}+\frac {\left (3 b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{e^3} \\ & = -\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^2}-\frac {7 b n x \left (a+b \log \left (c x^n\right )\right )}{3 e^3 (d+e x)}-\frac {7 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^4 (d+e x)^3}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac {3 b^2 n^2 \log (d+e x)}{e^4}+\frac {6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {3 b^2 n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{e^4}+\frac {6 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^4}-\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{3 e^4}+\frac {\left (b^2 d^2 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^4}-\frac {\left (2 b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{3 e^3} \\ & = \frac {b^2 d n^2}{3 e^4 (d+e x)}+\frac {b^2 n^2 \log (x)}{3 e^4}-\frac {b d^2 n \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)^2}-\frac {7 b n x \left (a+b \log \left (c x^n\right )\right )}{3 e^3 (d+e x)}-\frac {7 b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^4 (d+e x)^3}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac {2 b^2 n^2 \log (d+e x)}{e^4}+\frac {6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {7 b^2 n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{3 e^4}+\frac {6 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\frac {-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {14 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-11 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}-\frac {9 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {18 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-14 b^2 n^2 (\log (x)-\log (d+e x))+\frac {2 b^2 n^2 (d+(d+e x) \log (x)-(d+e x) \log (d+e x))}{d+e x}+22 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+6 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+22 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+12 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{6 e^4} \]

[In]

Integrate[(x^3*(a + b*Log[c*x^n])^2)/(d + e*x)^4,x]

[Out]

((-2*b*d^2*n*(a + b*Log[c*x^n]))/(d + e*x)^2 + (14*b*d*n*(a + b*Log[c*x^n]))/(d + e*x) - 11*(a + b*Log[c*x^n])
^2 + (2*d^3*(a + b*Log[c*x^n])^2)/(d + e*x)^3 - (9*d^2*(a + b*Log[c*x^n])^2)/(d + e*x)^2 + (18*d*(a + b*Log[c*
x^n])^2)/(d + e*x) - 14*b^2*n^2*(Log[x] - Log[d + e*x]) + (2*b^2*n^2*(d + (d + e*x)*Log[x] - (d + e*x)*Log[d +
e*x]))/(d + e*x) + 22*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] + 6*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 22*
b^2*n^2*PolyLog[2, -((e*x)/d)] + 12*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] - 12*b^2*n^2*PolyLog[3, -((e
*x)/d)])/(6*e^4)

Maple [C] (warning: unable to verify)

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

Time = 0.60 (sec) , antiderivative size = 854, normalized size of antiderivative = 2.56

method	result	size
risch	\(\text {Expression too large to display}\)	\(854\)

[In]

int(x^3*(a+b*ln(c*x^n))^2/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/3*b^2*ln(x^n)^2/e^4*d^3/(e*x+d)^3+b^2*ln(x^n)^2/e^4*ln(e*x+d)+3*b^2*ln(x^n)^2/e^4*d/(e*x+d)-3/2*b^2*ln(x^n)^
2/e^4*d^2/(e*x+d)^2+7/3*b^2*n*ln(x^n)/e^4*d/(e*x+d)-1/3*b^2*n*ln(x^n)/e^4*d^2/(e*x+d)^2+11/3*b^2*n*ln(x^n)/e^4
*ln(e*x+d)-11/3*b^2*n/e^4*ln(x^n)*ln(x)+11/6*b^2/e^4*n^2*ln(x)^2-11/3*b^2/e^4*n^2*ln(e*x+d)*ln(-e*x/d)-11/3*b^
2/e^4*n^2*dilog(-e*x/d)+1/3*b^2*d*n^2/e^4/(e*x+d)+2*b^2*n^2*ln(e*x+d)/e^4-2*b^2*n^2*ln(x)/e^4+2*b^2/e^4*ln(x)*
ln(e*x+d)*ln(-e*x/d)*n^2+2*b^2/e^4*ln(x)*dilog(-e*x/d)*n^2-2*b^2*n/e^4*ln(x^n)*ln(e*x+d)*ln(-e*x/d)-2*b^2*n/e^
4*ln(x^n)*dilog(-e*x/d)-b^2/e^4*n^2*ln(e*x+d)*ln(x)^2+b^2/e^4*n^2*ln(x)^2*ln(1+e*x/d)+2*b^2/e^4*n^2*ln(x)*poly
log(2,-e*x/d)-2*b^2*n^2*polylog(3,-e*x/d)/e^4+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*cs
gn(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)/e^4*d^3/
(e*x+d)^3+ln(x^n)/e^4*ln(e*x+d)+3*ln(x^n)/e^4*d/(e*x+d)-3/2*ln(x^n)/e^4*d^2/(e*x+d)^2-1/6*n*(-7/e^4*d/(e*x+d)-
11/e^4*ln(e*x+d)+1/e^4*d^2/(e*x+d)^2+11/e^4*ln(e*x)+6/e^4*ln(e*x+d)*ln(-e*x/d)+6/e^4*dilog(-e*x/d)))+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*(1/3/e^4*d^3/(e*x+d)^3+1/e^4*ln(e*x+d)+3/e^4*d/(e*x+d)-3/2/e^4*d^2/(e*x+
d)^2)

Fricas [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral((b^2*x^3*log(c*x^n)^2 + 2*a*b*x^3*log(c*x^n) + a^2*x^3)/(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]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, dx \]

[In]

integrate(x**3*(a+b*ln(c*x**n))**2/(e*x+d)**4,x)

[Out]

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

Maxima [F]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*a^2*((18*d*e^2*x^2 + 27*d^2*e*x + 11*d^3)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4) + 6*log(e*x + d)
/e^4) + integrate((b^2*x^3*log(x^n)^2 + 2*(b^2*log(c) + a*b)*x^3*log(x^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x^3)
/(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]

\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {x^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

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

[Out]

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