3.88 \(\int \frac{(a+b \log (c x^n))^3 \log (d (e+f x)^m)}{x^2} \, dx\)

Optimal. Leaf size=411 \[ \frac{6 b^2 f m n^2 \text{PolyLog}\left (2,-\frac{e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{6 b^2 f m n^2 \text{PolyLog}\left (3,-\frac{e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{3 b f m n \text{PolyLog}\left (2,-\frac{e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac{6 b^3 f m n^3 \text{PolyLog}\left (2,-\frac{e}{f x}\right )}{e}+\frac{6 b^3 f m n^3 \text{PolyLog}\left (3,-\frac{e}{f x}\right )}{e}+\frac{6 b^3 f m n^3 \text{PolyLog}\left (4,-\frac{e}{f x}\right )}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 f m n^2 \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac{3 b f m n \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{f m \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e}-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}+\frac{6 b^3 f m n^3 \log (x)}{e}-\frac{6 b^3 f m n^3 \log (e+f x)}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.695843, antiderivative size = 459, normalized size of antiderivative = 1.12, number of steps used = 22, number of rules used = 15, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.577, Rules used = {2305, 2304, 2378, 36, 29, 31, 2344, 2301, 2317, 2391, 2302, 30, 2374, 6589, 2383} \[ -\frac{6 b^2 f m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{6 b^2 f m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{3 b f m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{6 b^3 f m n^3 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{e}+\frac{6 b^3 f m n^3 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{e}-\frac{6 b^3 f m n^3 \text{PolyLog}\left (4,-\frac{f x}{e}\right )}{e}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 f m n^2 \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}+\frac{f m \left (a+b \log \left (c x^n\right )\right )^4}{4 b e n}+\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{e}-\frac{f m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e}+\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{3 b f m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}+\frac{6 b^3 f m n^3 \log (x)}{e}-\frac{6 b^3 f m n^3 \log (e+f x)}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2378

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.),
x_Symbol] :> With[{u = IntHide[(g*x)^q*(a + b*Log[c*x^n])^p, x]}, Dist[Log[d*(e + f*x^m)^r], u, x] - Dist[f*m
*r, Int[Dist[x^(m - 1)/(e + f*x^m), u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && IGtQ[p, 0
] && RationalQ[m] && RationalQ[q]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 2344

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

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 2317

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 2391

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 2302

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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]

Rule 2383

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x^2} \, dx &=-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-(f m) \int \left (-\frac{6 b^3 n^3}{x (e+f x)}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x (e+f x)}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x (e+f x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{x (e+f x)}\right ) \, dx\\ &=-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}+(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x (e+f x)} \, dx+(3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (e+f x)} \, dx+\left (6 b^2 f m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x (e+f x)} \, dx+\left (6 b^3 f m n^3\right ) \int \frac{1}{x (e+f x)} \, dx\\ &=-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}+\frac{(f m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{e}-\frac{\left (f^2 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x} \, dx}{e}+\frac{(3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{e}-\frac{\left (3 b f^2 m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{e}+\frac{\left (6 b^2 f m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{e}-\frac{\left (6 b^2 f^2 m n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x} \, dx}{e}+\frac{\left (6 b^3 f m n^3\right ) \int \frac{1}{x} \, dx}{e}-\frac{\left (6 b^3 f^2 m n^3\right ) \int \frac{1}{e+f x} \, dx}{e}\\ &=\frac{6 b^3 f m n^3 \log (x)}{e}+\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{6 b^3 f m n^3 \log (e+f x)}{e}-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{e}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{e}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{e}+\frac{(3 f m) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{e}+\frac{(f m) \operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b e n}+\frac{(3 b f m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{x} \, dx}{e}+\frac{\left (6 b^2 f m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{x} \, dx}{e}+\frac{\left (6 b^3 f m n^3\right ) \int \frac{\log \left (1+\frac{f x}{e}\right )}{x} \, dx}{e}\\ &=\frac{6 b^3 f m n^3 \log (x)}{e}+\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{e}+\frac{f m \left (a+b \log \left (c x^n\right )\right )^4}{4 b e n}-\frac{6 b^3 f m n^3 \log (e+f x)}{e}-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{e}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{e}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{e}-\frac{6 b^3 f m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{e}-\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{e}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{e}+\frac{\left (6 b^2 f m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{e}+\frac{\left (6 b^3 f m n^3\right ) \int \frac{\text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{e}\\ &=\frac{6 b^3 f m n^3 \log (x)}{e}+\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{e}+\frac{f m \left (a+b \log \left (c x^n\right )\right )^4}{4 b e n}-\frac{6 b^3 f m n^3 \log (e+f x)}{e}-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{e}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{e}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{e}-\frac{6 b^3 f m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{e}-\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{e}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{e}+\frac{6 b^3 f m n^3 \text{Li}_3\left (-\frac{f x}{e}\right )}{e}+\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{e}-\frac{\left (6 b^3 f m n^3\right ) \int \frac{\text{Li}_3\left (-\frac{f x}{e}\right )}{x} \, dx}{e}\\ &=\frac{6 b^3 f m n^3 \log (x)}{e}+\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac{f m \left (a+b \log \left (c x^n\right )\right )^3}{e}+\frac{f m \left (a+b \log \left (c x^n\right )\right )^4}{4 b e n}-\frac{6 b^3 f m n^3 \log (e+f x)}{e}-\frac{6 b^3 n^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x}-\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{e}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{e}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{f x}{e}\right )}{e}-\frac{6 b^3 f m n^3 \text{Li}_2\left (-\frac{f x}{e}\right )}{e}-\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{e}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{e}+\frac{6 b^3 f m n^3 \text{Li}_3\left (-\frac{f x}{e}\right )}{e}+\frac{6 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f x}{e}\right )}{e}-\frac{6 b^3 f m n^3 \text{Li}_4\left (-\frac{f x}{e}\right )}{e}\\ \end{align*}

Mathematica [B]  time = 0.664673, size = 1347, normalized size = 3.28 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-4*a^3*f*m*x*Log[x] - 12*a^2*b*f*m*n*x*Log[x] - 24*a*b^2*f*m*n^2*x*Log[x] - 24*b^3*f*m*n^3*x*Log[x] + 6*a^2*
b*f*m*n*x*Log[x]^2 + 12*a*b^2*f*m*n^2*x*Log[x]^2 + 12*b^3*f*m*n^3*x*Log[x]^2 - 4*a*b^2*f*m*n^2*x*Log[x]^3 - 4*
b^3*f*m*n^3*x*Log[x]^3 + b^3*f*m*n^3*x*Log[x]^4 - 12*a^2*b*f*m*x*Log[x]*Log[c*x^n] - 24*a*b^2*f*m*n*x*Log[x]*L
og[c*x^n] - 24*b^3*f*m*n^2*x*Log[x]*Log[c*x^n] + 12*a*b^2*f*m*n*x*Log[x]^2*Log[c*x^n] + 12*b^3*f*m*n^2*x*Log[x
]^2*Log[c*x^n] - 4*b^3*f*m*n^2*x*Log[x]^3*Log[c*x^n] - 12*a*b^2*f*m*x*Log[x]*Log[c*x^n]^2 - 12*b^3*f*m*n*x*Log
[x]*Log[c*x^n]^2 + 6*b^3*f*m*n*x*Log[x]^2*Log[c*x^n]^2 - 4*b^3*f*m*x*Log[x]*Log[c*x^n]^3 + 4*a^3*f*m*x*Log[e +
f*x] + 12*a^2*b*f*m*n*x*Log[e + f*x] + 24*a*b^2*f*m*n^2*x*Log[e + f*x] + 24*b^3*f*m*n^3*x*Log[e + f*x] - 12*a
^2*b*f*m*n*x*Log[x]*Log[e + f*x] - 24*a*b^2*f*m*n^2*x*Log[x]*Log[e + f*x] - 24*b^3*f*m*n^3*x*Log[x]*Log[e + f*
x] + 12*a*b^2*f*m*n^2*x*Log[x]^2*Log[e + f*x] + 12*b^3*f*m*n^3*x*Log[x]^2*Log[e + f*x] - 4*b^3*f*m*n^3*x*Log[x
]^3*Log[e + f*x] + 12*a^2*b*f*m*x*Log[c*x^n]*Log[e + f*x] + 24*a*b^2*f*m*n*x*Log[c*x^n]*Log[e + f*x] + 24*b^3*
f*m*n^2*x*Log[c*x^n]*Log[e + f*x] - 24*a*b^2*f*m*n*x*Log[x]*Log[c*x^n]*Log[e + f*x] - 24*b^3*f*m*n^2*x*Log[x]*
Log[c*x^n]*Log[e + f*x] + 12*b^3*f*m*n^2*x*Log[x]^2*Log[c*x^n]*Log[e + f*x] + 12*a*b^2*f*m*x*Log[c*x^n]^2*Log[
e + f*x] + 12*b^3*f*m*n*x*Log[c*x^n]^2*Log[e + f*x] - 12*b^3*f*m*n*x*Log[x]*Log[c*x^n]^2*Log[e + f*x] + 4*b^3*
f*m*x*Log[c*x^n]^3*Log[e + f*x] + 4*a^3*e*Log[d*(e + f*x)^m] + 12*a^2*b*e*n*Log[d*(e + f*x)^m] + 24*a*b^2*e*n^
2*Log[d*(e + f*x)^m] + 24*b^3*e*n^3*Log[d*(e + f*x)^m] + 12*a^2*b*e*Log[c*x^n]*Log[d*(e + f*x)^m] + 24*a*b^2*e
*n*Log[c*x^n]*Log[d*(e + f*x)^m] + 24*b^3*e*n^2*Log[c*x^n]*Log[d*(e + f*x)^m] + 12*a*b^2*e*Log[c*x^n]^2*Log[d*
(e + f*x)^m] + 12*b^3*e*n*Log[c*x^n]^2*Log[d*(e + f*x)^m] + 4*b^3*e*Log[c*x^n]^3*Log[d*(e + f*x)^m] + 12*a^2*b
*f*m*n*x*Log[x]*Log[1 + (f*x)/e] + 24*a*b^2*f*m*n^2*x*Log[x]*Log[1 + (f*x)/e] + 24*b^3*f*m*n^3*x*Log[x]*Log[1
+ (f*x)/e] - 12*a*b^2*f*m*n^2*x*Log[x]^2*Log[1 + (f*x)/e] - 12*b^3*f*m*n^3*x*Log[x]^2*Log[1 + (f*x)/e] + 4*b^3
*f*m*n^3*x*Log[x]^3*Log[1 + (f*x)/e] + 24*a*b^2*f*m*n*x*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 24*b^3*f*m*n^2*x*
Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] - 12*b^3*f*m*n^2*x*Log[x]^2*Log[c*x^n]*Log[1 + (f*x)/e] + 12*b^3*f*m*n*x*Lo
g[x]*Log[c*x^n]^2*Log[1 + (f*x)/e] + 12*b*f*m*n*x*(a^2 + 2*a*b*n + 2*b^2*n^2 + 2*b*(a + b*n)*Log[c*x^n] + b^2*
Log[c*x^n]^2)*PolyLog[2, -((f*x)/e)] - 24*b^2*f*m*n^2*x*(a + b*n + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)] + 24*b
^3*f*m*n^3*x*PolyLog[4, -((f*x)/e)])/(4*e*x)

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Maple [C]  time = 1.706, size = 42181, normalized size = 102.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m)/x^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^3*f*m*x*log(f*x + e) - b^3*f*m*x*log(x) + b^3*e*log(d))*log(x^n)^3 + (b^3*e*log(x^n)^3 + 3*(e*n + e*log(c
))*a^2*b + 3*(2*e*n^2 + 2*e*n*log(c) + e*log(c)^2)*a*b^2 + (6*e*n^3 + 6*e*n^2*log(c) + 3*e*n*log(c)^2 + e*log(
c)^3)*b^3 + a^3*e + 3*((e*n + e*log(c))*b^3 + a*b^2*e)*log(x^n)^2 + 3*(2*(e*n + e*log(c))*a*b^2 + (2*e*n^2 + 2
*e*n*log(c) + e*log(c)^2)*b^3 + a^2*b*e)*log(x^n))*log((f*x + e)^m))/(e*x) + integrate((b^3*e^2*log(c)^3*log(d
) + 3*a*b^2*e^2*log(c)^2*log(d) + 3*a^2*b*e^2*log(c)*log(d) + a^3*e^2*log(d) + 3*(a*b^2*e^2*log(d) + (e^2*n*lo
g(d) + e^2*log(c)*log(d))*b^3 + ((e*f*m + e*f*log(d))*a*b^2 + (e*f*m*n + e*f*n*log(d) + (e*f*m + e*f*log(d))*l
og(c))*b^3)*x + (b^3*f^2*m*n*x^2 + b^3*e*f*m*n*x)*log(f*x + e) - (b^3*f^2*m*n*x^2 + b^3*e*f*m*n*x)*log(x))*log
(x^n)^2 + ((e*f*m + e*f*log(d))*a^3 + 3*(e*f*m*n + (e*f*m + e*f*log(d))*log(c))*a^2*b + 3*(2*e*f*m*n^2 + 2*e*f
*m*n*log(c) + (e*f*m + e*f*log(d))*log(c)^2)*a*b^2 + (6*e*f*m*n^3 + 6*e*f*m*n^2*log(c) + 3*e*f*m*n*log(c)^2 +
(e*f*m + e*f*log(d))*log(c)^3)*b^3)*x + 3*(b^3*e^2*log(c)^2*log(d) + 2*a*b^2*e^2*log(c)*log(d) + a^2*b*e^2*log
(d) + ((e*f*m + e*f*log(d))*a^2*b + 2*(e*f*m*n + (e*f*m + e*f*log(d))*log(c))*a*b^2 + (2*e*f*m*n^2 + 2*e*f*m*n
*log(c) + (e*f*m + e*f*log(d))*log(c)^2)*b^3)*x)*log(x^n))/(e*f*x^3 + e^2*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a^3)*log((f*x + e)^m*d)/x^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((a+b*ln(c*x**n))**3*ln(d*(f*x+e)**m)/x**2,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 (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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