3.79 \(\int x (a+b \log (c x^n))^2 \log (d (e+f x)^m) \, dx\)

Optimal. Leaf size=373 \[ -\frac{b e^2 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{b^2 e^2 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{2 f^2}+\frac{b^2 e^2 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{f^2}-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}-\frac{e^2 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^2}+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}+\frac{1}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 a b e m n x}{2 f}-\frac{3 b^2 e m n x \log \left (c x^n\right )}{2 f}+\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac{7 b^2 e m n^2 x}{4 f}-\frac{3}{8} b^2 m n^2 x^2 \]

[Out]

(-3*a*b*e*m*n*x)/(2*f) + (7*b^2*e*m*n^2*x)/(4*f) - (3*b^2*m*n^2*x^2)/8 - (3*b^2*e*m*n*x*Log[c*x^n])/(2*f) + (b
*m*n*x^2*(a + b*Log[c*x^n]))/2 + (e*m*x*(a + b*Log[c*x^n])^2)/(2*f) - (m*x^2*(a + b*Log[c*x^n])^2)/4 - (b^2*e^
2*m*n^2*Log[e + f*x])/(4*f^2) + (b^2*n^2*x^2*Log[d*(e + f*x)^m])/4 - (b*n*x^2*(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 + (b*e^2*m*n*(a + b*Log[c*x^n])*Log[1 + (f*x)/e])/(
2*f^2) - (e^2*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(2*f^2) + (b^2*e^2*m*n^2*PolyLog[2, -((f*x)/e)])/(2*f^2
) - (b*e^2*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f^2 + (b^2*e^2*m*n^2*PolyLog[3, -((f*x)/e)])/f^2

________________________________________________________________________________________

Rubi [A]  time = 0.531206, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2305, 2304, 2378, 43, 2351, 2295, 2317, 2391, 2353, 2296, 2374, 6589} \[ -\frac{b e^2 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{b^2 e^2 m n^2 \text{PolyLog}\left (2,-\frac{f x}{e}\right )}{2 f^2}+\frac{b^2 e^2 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )}{f^2}-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}-\frac{e^2 m \log \left (\frac{f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^2}+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}+\frac{1}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 a b e m n x}{2 f}-\frac{3 b^2 e m n x \log \left (c x^n\right )}{2 f}+\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac{7 b^2 e m n^2 x}{4 f}-\frac{3}{8} b^2 m n^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*b*e*m*n*x)/(2*f) + (7*b^2*e*m*n^2*x)/(4*f) - (3*b^2*m*n^2*x^2)/8 - (3*b^2*e*m*n*x*Log[c*x^n])/(2*f) + (b
*m*n*x^2*(a + b*Log[c*x^n]))/2 + (e*m*x*(a + b*Log[c*x^n])^2)/(2*f) - (m*x^2*(a + b*Log[c*x^n])^2)/4 - (b^2*e^
2*m*n^2*Log[e + f*x])/(4*f^2) + (b^2*n^2*x^2*Log[d*(e + f*x)^m])/4 - (b*n*x^2*(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 + (b*e^2*m*n*(a + b*Log[c*x^n])*Log[1 + (f*x)/e])/(
2*f^2) - (e^2*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(2*f^2) + (b^2*e^2*m*n^2*PolyLog[2, -((f*x)/e)])/(2*f^2
) - (b*e^2*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f^2 + (b^2*e^2*m*n^2*PolyLog[3, -((f*x)/e)])/f^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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{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 2296

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 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 x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right ) \, dx &=\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac{b^2 n^2 x^2}{4 (e+f x)}-\frac{b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 (e+f x)}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 (e+f x)}\right ) \, dx\\ &=\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} (f m) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx+\frac{1}{2} (b f m n) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{e+f x} \, dx-\frac{1}{4} \left (b^2 f m n^2\right ) \int \frac{x^2}{e+f x} \, dx\\ &=\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} (f m) \int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^2 (e+f x)}\right ) \, dx+\frac{1}{2} (b f m n) \int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{f}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{f^2 (e+f x)}\right ) \, dx-\frac{1}{4} \left (b^2 f m n^2\right ) \int \left (-\frac{e}{f^2}+\frac{x}{f}+\frac{e^2}{f^2 (e+f x)}\right ) \, dx\\ &=\frac{b^2 e m n^2 x}{4 f}-\frac{1}{8} b^2 m n^2 x^2-\frac{b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} m \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac{(e m) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 f}-\frac{\left (e^2 m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{2 f}+\frac{1}{2} (b m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{(b e m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 f}+\frac{\left (b e^2 m n\right ) \int \frac{a+b \log \left (c x^n\right )}{e+f x} \, dx}{2 f}\\ &=-\frac{a b e m n x}{2 f}+\frac{b^2 e m n^2 x}{4 f}-\frac{1}{4} b^2 m n^2 x^2+\frac{1}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{2 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{2 f^2}+\frac{1}{2} (b m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (b e^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}{f^2}-\frac{(b e m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{f}-\frac{\left (b^2 e m n\right ) \int \log \left (c x^n\right ) \, dx}{2 f}-\frac{\left (b^2 e^2 m n^2\right ) \int \frac{\log \left (1+\frac{f x}{e}\right )}{x} \, dx}{2 f^2}\\ &=-\frac{3 a b e m n x}{2 f}+\frac{3 b^2 e m n^2 x}{4 f}-\frac{3}{8} b^2 m n^2 x^2-\frac{b^2 e m n x \log \left (c x^n\right )}{2 f}+\frac{1}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{2 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{2 f^2}+\frac{b^2 e^2 m n^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{2 f^2}-\frac{b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{f^2}-\frac{\left (b^2 e m n\right ) \int \log \left (c x^n\right ) \, dx}{f}+\frac{\left (b^2 e^2 m n^2\right ) \int \frac{\text{Li}_2\left (-\frac{f x}{e}\right )}{x} \, dx}{f^2}\\ &=-\frac{3 a b e m n x}{2 f}+\frac{7 b^2 e m n^2 x}{4 f}-\frac{3}{8} b^2 m n^2 x^2-\frac{3 b^2 e m n x \log \left (c x^n\right )}{2 f}+\frac{1}{2} b m n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{e m x \left (a+b \log \left (c x^n\right )\right )^2}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b^2 e^2 m n^2 \log (e+f x)}{4 f^2}+\frac{1}{4} b^2 n^2 x^2 \log \left (d (e+f x)^m\right )-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{f x}{e}\right )}{2 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{f x}{e}\right )}{2 f^2}+\frac{b^2 e^2 m n^2 \text{Li}_2\left (-\frac{f x}{e}\right )}{2 f^2}-\frac{b e^2 m n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f x}{e}\right )}{f^2}+\frac{b^2 e^2 m n^2 \text{Li}_3\left (-\frac{f x}{e}\right )}{f^2}\\ \end{align*}

Mathematica [A]  time = 0.275554, size = 674, normalized size = 1.81 \[ \frac{4 b e^2 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right ) \left (-2 a-2 b \log \left (c x^n\right )+b n\right )+8 b^2 e^2 m n^2 \text{PolyLog}\left (3,-\frac{f x}{e}\right )+4 a^2 f^2 x^2 \log \left (d (e+f x)^m\right )-4 a^2 e^2 m \log (e+f x)+4 a^2 e f m x-2 a^2 f^2 m x^2+8 a b f^2 x^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-8 a b e^2 m \log \left (c x^n\right ) \log (e+f x)+8 a b e f m x \log \left (c x^n\right )-4 a b f^2 m x^2 \log \left (c x^n\right )-4 a b f^2 n x^2 \log \left (d (e+f x)^m\right )+4 a b e^2 m n \log (e+f x)+8 a b e^2 m n \log (x) \log (e+f x)-8 a b e^2 m n \log (x) \log \left (\frac{f x}{e}+1\right )-12 a b e f m n x+4 a b f^2 m n x^2+4 b^2 f^2 x^2 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-4 b^2 f^2 n x^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-4 b^2 e^2 m \log ^2\left (c x^n\right ) \log (e+f x)+4 b^2 e^2 m n \log \left (c x^n\right ) \log (e+f x)+8 b^2 e^2 m n \log (x) \log \left (c x^n\right ) \log (e+f x)-8 b^2 e^2 m n \log (x) \log \left (c x^n\right ) \log \left (\frac{f x}{e}+1\right )+4 b^2 e f m x \log ^2\left (c x^n\right )-12 b^2 e f m n x \log \left (c x^n\right )-2 b^2 f^2 m x^2 \log ^2\left (c x^n\right )+4 b^2 f^2 m n x^2 \log \left (c x^n\right )+2 b^2 f^2 n^2 x^2 \log \left (d (e+f x)^m\right )-4 b^2 e^2 m n^2 \log ^2(x) \log (e+f x)+4 b^2 e^2 m n^2 \log ^2(x) \log \left (\frac{f x}{e}+1\right )-2 b^2 e^2 m n^2 \log (e+f x)-4 b^2 e^2 m n^2 \log (x) \log (e+f x)+4 b^2 e^2 m n^2 \log (x) \log \left (\frac{f x}{e}+1\right )+14 b^2 e f m n^2 x-3 b^2 f^2 m n^2 x^2}{8 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^2*e*f*m*x - 12*a*b*e*f*m*n*x + 14*b^2*e*f*m*n^2*x - 2*a^2*f^2*m*x^2 + 4*a*b*f^2*m*n*x^2 - 3*b^2*f^2*m*n^2
*x^2 + 8*a*b*e*f*m*x*Log[c*x^n] - 12*b^2*e*f*m*n*x*Log[c*x^n] - 4*a*b*f^2*m*x^2*Log[c*x^n] + 4*b^2*f^2*m*n*x^2
*Log[c*x^n] + 4*b^2*e*f*m*x*Log[c*x^n]^2 - 2*b^2*f^2*m*x^2*Log[c*x^n]^2 - 4*a^2*e^2*m*Log[e + f*x] + 4*a*b*e^2
*m*n*Log[e + f*x] - 2*b^2*e^2*m*n^2*Log[e + f*x] + 8*a*b*e^2*m*n*Log[x]*Log[e + f*x] - 4*b^2*e^2*m*n^2*Log[x]*
Log[e + f*x] - 4*b^2*e^2*m*n^2*Log[x]^2*Log[e + f*x] - 8*a*b*e^2*m*Log[c*x^n]*Log[e + f*x] + 4*b^2*e^2*m*n*Log
[c*x^n]*Log[e + f*x] + 8*b^2*e^2*m*n*Log[x]*Log[c*x^n]*Log[e + f*x] - 4*b^2*e^2*m*Log[c*x^n]^2*Log[e + f*x] +
4*a^2*f^2*x^2*Log[d*(e + f*x)^m] - 4*a*b*f^2*n*x^2*Log[d*(e + f*x)^m] + 2*b^2*f^2*n^2*x^2*Log[d*(e + f*x)^m] +
8*a*b*f^2*x^2*Log[c*x^n]*Log[d*(e + f*x)^m] - 4*b^2*f^2*n*x^2*Log[c*x^n]*Log[d*(e + f*x)^m] + 4*b^2*f^2*x^2*L
og[c*x^n]^2*Log[d*(e + f*x)^m] - 8*a*b*e^2*m*n*Log[x]*Log[1 + (f*x)/e] + 4*b^2*e^2*m*n^2*Log[x]*Log[1 + (f*x)/
e] + 4*b^2*e^2*m*n^2*Log[x]^2*Log[1 + (f*x)/e] - 8*b^2*e^2*m*n*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 4*b*e^2*m*
n*(-2*a + b*n - 2*b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] + 8*b^2*e^2*m*n^2*PolyLog[3, -((f*x)/e)])/(8*f^2)

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Maple [F]  time = 2.312, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( d \left ( fx+e \right ) ^{m} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*x*log(c*x^n)^2 + 2*a*b*x*log(c*x^n) + a^2*x)*log((f*x + e)^m*d), 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(x*(a+b*ln(c*x**n))**2*ln(d*(f*x+e)**m),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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