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

Optimal. Leaf size=344 \[ -\frac{b f^2 m n \text{PolyLog}\left (2,-\frac{e}{f x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{b^2 f^2 m n^2 \text{PolyLog}\left (2,-\frac{e}{f x}\right )}{2 e^2}-\frac{b^2 f^2 m n^2 \text{PolyLog}\left (3,-\frac{e}{f x}\right )}{e^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}+\frac{f^2 m \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac{b f^2 m n \log \left (\frac{e}{f x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{b^2 f^2 m n^2 \log (x)}{4 e^2}+\frac{b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac{7 b^2 f m n^2}{4 e x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.586247, antiderivative size = 385, normalized size of antiderivative = 1.12, number of steps used = 19, number of rules used = 13, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2305, 2304, 2378, 44, 2351, 2301, 2317, 2391, 2353, 2302, 30, 2374, 6589} \[ \frac{b f^2 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{b^2 f^2 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{2 e^2}-\frac{b^2 f^2 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{e^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )}{2 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}+\frac{f^2 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac{b f^2 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{f m \left (a+b \log \left (c x^n\right )\right )^2}{2 e x}-\frac{3 b f m n \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac{b^2 n^2 \log \left (d (e+f x)^m\right )}{4 x^2}-\frac{b^2 f^2 m n^2 \log (x)}{4 e^2}+\frac{b^2 f^2 m n^2 \log (e+f x)}{4 e^2}-\frac{7 b^2 f m n^2}{4 e x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 44

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] && L
tQ[m + n + 2, 0])

Rule 2351

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

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 2353

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 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]

Rubi steps

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

Mathematica [B]  time = 0.381332, size = 796, normalized size = 2.31 \[ -\frac{2 b^2 f^2 m n^2 x^2 \log ^3(x)-3 b^2 f^2 m n^2 x^2 \log ^2(x)-6 a b f^2 m n x^2 \log ^2(x)-6 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log ^2(x)-6 b^2 f^2 m n^2 x^2 \log (e+f x) \log ^2(x)+6 b^2 f^2 m n^2 x^2 \log \left (\frac{f x}{e}+1\right ) \log ^2(x)+3 b^2 f^2 m n^2 x^2 \log (x)+6 a^2 f^2 m x^2 \log (x)+6 a b f^2 m n x^2 \log (x)+6 b^2 f^2 m x^2 \log ^2\left (c x^n\right ) \log (x)+12 a b f^2 m x^2 \log \left (c x^n\right ) \log (x)+6 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log (x)+6 b^2 f^2 m n^2 x^2 \log (e+f x) \log (x)+12 a b f^2 m n x^2 \log (e+f x) \log (x)+12 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log (e+f x) \log (x)-6 b^2 f^2 m n^2 x^2 \log \left (\frac{f x}{e}+1\right ) \log (x)-12 a b f^2 m n x^2 \log \left (\frac{f x}{e}+1\right ) \log (x)-12 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right ) \log (x)+6 b^2 e f m x \log ^2\left (c x^n\right )+21 b^2 e f m n^2 x+6 a^2 e f m x+18 a b e f m n x+12 a b e f m x \log \left (c x^n\right )+18 b^2 e f m n x \log \left (c x^n\right )-3 b^2 f^2 m n^2 x^2 \log (e+f x)-6 a^2 f^2 m x^2 \log (e+f x)-6 a b f^2 m n x^2 \log (e+f x)-6 b^2 f^2 m x^2 \log ^2\left (c x^n\right ) \log (e+f x)-12 a b f^2 m x^2 \log \left (c x^n\right ) \log (e+f x)-6 b^2 f^2 m n x^2 \log \left (c x^n\right ) \log (e+f x)+6 a^2 e^2 \log \left (d (e+f x)^m\right )+3 b^2 e^2 n^2 \log \left (d (e+f x)^m\right )+6 b^2 e^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 a b e^2 n \log \left (d (e+f x)^m\right )+12 a b e^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b^2 e^2 n \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b f^2 m n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{f x}{e}\right )+12 b^2 f^2 m n^2 x^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{12 e^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*a^2*e*f*m*x + 18*a*b*e*f*m*n*x + 21*b^2*e*f*m*n^2*x + 6*a^2*f^2*m*x^2*Log[x] + 6*a*b*f^2*m*n*x^2*Log[x] +
3*b^2*f^2*m*n^2*x^2*Log[x] - 6*a*b*f^2*m*n*x^2*Log[x]^2 - 3*b^2*f^2*m*n^2*x^2*Log[x]^2 + 2*b^2*f^2*m*n^2*x^2*L
og[x]^3 + 12*a*b*e*f*m*x*Log[c*x^n] + 18*b^2*e*f*m*n*x*Log[c*x^n] + 12*a*b*f^2*m*x^2*Log[x]*Log[c*x^n] + 6*b^2
*f^2*m*n*x^2*Log[x]*Log[c*x^n] - 6*b^2*f^2*m*n*x^2*Log[x]^2*Log[c*x^n] + 6*b^2*e*f*m*x*Log[c*x^n]^2 + 6*b^2*f^
2*m*x^2*Log[x]*Log[c*x^n]^2 - 6*a^2*f^2*m*x^2*Log[e + f*x] - 6*a*b*f^2*m*n*x^2*Log[e + f*x] - 3*b^2*f^2*m*n^2*
x^2*Log[e + f*x] + 12*a*b*f^2*m*n*x^2*Log[x]*Log[e + f*x] + 6*b^2*f^2*m*n^2*x^2*Log[x]*Log[e + f*x] - 6*b^2*f^
2*m*n^2*x^2*Log[x]^2*Log[e + f*x] - 12*a*b*f^2*m*x^2*Log[c*x^n]*Log[e + f*x] - 6*b^2*f^2*m*n*x^2*Log[c*x^n]*Lo
g[e + f*x] + 12*b^2*f^2*m*n*x^2*Log[x]*Log[c*x^n]*Log[e + f*x] - 6*b^2*f^2*m*x^2*Log[c*x^n]^2*Log[e + f*x] + 6
*a^2*e^2*Log[d*(e + f*x)^m] + 6*a*b*e^2*n*Log[d*(e + f*x)^m] + 3*b^2*e^2*n^2*Log[d*(e + f*x)^m] + 12*a*b*e^2*L
og[c*x^n]*Log[d*(e + f*x)^m] + 6*b^2*e^2*n*Log[c*x^n]*Log[d*(e + f*x)^m] + 6*b^2*e^2*Log[c*x^n]^2*Log[d*(e + f
*x)^m] - 12*a*b*f^2*m*n*x^2*Log[x]*Log[1 + (f*x)/e] - 6*b^2*f^2*m*n^2*x^2*Log[x]*Log[1 + (f*x)/e] + 6*b^2*f^2*
m*n^2*x^2*Log[x]^2*Log[1 + (f*x)/e] - 12*b^2*f^2*m*n*x^2*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] - 6*b*f^2*m*n*x^2*
(2*a + b*n + 2*b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] + 12*b^2*f^2*m*n^2*x^2*PolyLog[3, -((f*x)/e)])/(12*e^2*x^2
)

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Maple [F]  time = 1.829, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( fx+e \right ) ^{m} \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (b^{2} f^{2} m x^{2} \log \left (f x + e\right ) - b^{2} f^{2} m x^{2} \log \left (x\right ) - b^{2} e f m x - b^{2} e^{2} \log \left (d\right )\right )} \log \left (x^{n}\right )^{2} -{\left (2 \, b^{2} e^{2} \log \left (x^{n}\right )^{2} + 2 \, a^{2} e^{2} + 2 \,{\left (e^{2} n + 2 \, e^{2} \log \left (c\right )\right )} a b +{\left (e^{2} n^{2} + 2 \, e^{2} n \log \left (c\right ) + 2 \, e^{2} \log \left (c\right )^{2}\right )} b^{2} + 2 \,{\left (2 \, a b e^{2} +{\left (e^{2} n + 2 \, e^{2} \log \left (c\right )\right )} b^{2}\right )} \log \left (x^{n}\right )\right )} \log \left ({\left (f x + e\right )}^{m}\right )}{4 \, e^{2} x^{2}} - \int -\frac{4 \, b^{2} e^{3} \log \left (c\right )^{2} \log \left (d\right ) + 8 \, a b e^{3} \log \left (c\right ) \log \left (d\right ) + 4 \, a^{2} e^{3} \log \left (d\right ) +{\left (2 \,{\left (e^{2} f m + 2 \, e^{2} f \log \left (d\right )\right )} a^{2} + 2 \,{\left (e^{2} f m n + 2 \,{\left (e^{2} f m + 2 \, e^{2} f \log \left (d\right )\right )} \log \left (c\right )\right )} a b +{\left (e^{2} f m n^{2} + 2 \, e^{2} f m n \log \left (c\right ) + 2 \,{\left (e^{2} f m + 2 \, e^{2} f \log \left (d\right )\right )} \log \left (c\right )^{2}\right )} b^{2}\right )} x + 2 \,{\left (2 \, b^{2} e f^{2} m n x^{2} + 4 \, a b e^{3} \log \left (d\right ) + 2 \,{\left (e^{3} n \log \left (d\right ) + 2 \, e^{3} \log \left (c\right ) \log \left (d\right )\right )} b^{2} +{\left (2 \,{\left (e^{2} f m + 2 \, e^{2} f \log \left (d\right )\right )} a b +{\left (3 \, e^{2} f m n + 2 \, e^{2} f n \log \left (d\right ) + 2 \,{\left (e^{2} f m + 2 \, e^{2} f \log \left (d\right )\right )} \log \left (c\right )\right )} b^{2}\right )} x - 2 \,{\left (b^{2} f^{3} m n x^{3} + b^{2} e f^{2} m n x^{2}\right )} \log \left (f x + e\right ) + 2 \,{\left (b^{2} f^{3} m n x^{3} + b^{2} e f^{2} m n x^{2}\right )} \log \left (x\right )\right )} \log \left (x^{n}\right )}{4 \,{\left (e^{2} f x^{4} + e^{3} x^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(b^2*f^2*m*x^2*log(f*x + e) - b^2*f^2*m*x^2*log(x) - b^2*e*f*m*x - b^2*e^2*log(d))*log(x^n)^2 - (2*b^2*
e^2*log(x^n)^2 + 2*a^2*e^2 + 2*(e^2*n + 2*e^2*log(c))*a*b + (e^2*n^2 + 2*e^2*n*log(c) + 2*e^2*log(c)^2)*b^2 +
2*(2*a*b*e^2 + (e^2*n + 2*e^2*log(c))*b^2)*log(x^n))*log((f*x + e)^m))/(e^2*x^2) - integrate(-1/4*(4*b^2*e^3*l
og(c)^2*log(d) + 8*a*b*e^3*log(c)*log(d) + 4*a^2*e^3*log(d) + (2*(e^2*f*m + 2*e^2*f*log(d))*a^2 + 2*(e^2*f*m*n
+ 2*(e^2*f*m + 2*e^2*f*log(d))*log(c))*a*b + (e^2*f*m*n^2 + 2*e^2*f*m*n*log(c) + 2*(e^2*f*m + 2*e^2*f*log(d))
*log(c)^2)*b^2)*x + 2*(2*b^2*e*f^2*m*n*x^2 + 4*a*b*e^3*log(d) + 2*(e^3*n*log(d) + 2*e^3*log(c)*log(d))*b^2 + (
2*(e^2*f*m + 2*e^2*f*log(d))*a*b + (3*e^2*f*m*n + 2*e^2*f*n*log(d) + 2*(e^2*f*m + 2*e^2*f*log(d))*log(c))*b^2)
*x - 2*(b^2*f^3*m*n*x^3 + b^2*e*f^2*m*n*x^2)*log(f*x + e) + 2*(b^2*f^3*m*n*x^3 + b^2*e*f^2*m*n*x^2)*log(x))*lo
g(x^n))/(e^2*f*x^4 + e^3*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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