3.1.78 \(\int x^2 (a+b \log (c x^n))^2 \log (d (e+f x)^m) \, dx\) [78]

Optimal. Leaf size=452 \[ \frac {8 a b e^2 m n x}{9 f^2}-\frac {26 b^2 e^2 m n^2 x}{27 f^2}+\frac {19 b^2 e m n^2 x^2}{108 f}-\frac {2}{27} b^2 m n^2 x^3+\frac {8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}-\frac {5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{9 f^3}+\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{3 f^3} \]

[Out]

8/9*a*b*e^2*m*n*x/f^2-26/27*b^2*e^2*m*n^2*x/f^2+19/108*b^2*e*m*n^2*x^2/f-2/27*b^2*m*n^2*x^3+8/9*b^2*e^2*m*n*x*
ln(c*x^n)/f^2-5/18*b*e*m*n*x^2*(a+b*ln(c*x^n))/f+4/27*b*m*n*x^3*(a+b*ln(c*x^n))-1/3*e^2*m*x*(a+b*ln(c*x^n))^2/
f^2+1/6*e*m*x^2*(a+b*ln(c*x^n))^2/f-1/9*m*x^3*(a+b*ln(c*x^n))^2+2/27*b^2*e^3*m*n^2*ln(f*x+e)/f^3+2/27*b^2*n^2*
x^3*ln(d*(f*x+e)^m)-2/9*b*n*x^3*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)+1/3*x^3*(a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m)-2/9*
b*e^3*m*n*(a+b*ln(c*x^n))*ln(1+f*x/e)/f^3+1/3*e^3*m*(a+b*ln(c*x^n))^2*ln(1+f*x/e)/f^3-2/9*b^2*e^3*m*n^2*polylo
g(2,-f*x/e)/f^3+2/3*b*e^3*m*n*(a+b*ln(c*x^n))*polylog(2,-f*x/e)/f^3-2/3*b^2*e^3*m*n^2*polylog(3,-f*x/e)/f^3

________________________________________________________________________________________

Rubi [A]
time = 0.45, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2342, 2341, 2425, 45, 2393, 2332, 2354, 2438, 2395, 2333, 2421, 6724} \begin {gather*} \frac {2 b e^3 m n \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {PolyLog}\left (2,-\frac {f x}{e}\right )}{9 f^3}-\frac {2 b^2 e^3 m n^2 \text {PolyLog}\left (3,-\frac {f x}{e}\right )}{3 f^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {e^3 m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^3}-\frac {2 b e^3 m n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {8 a b e^2 m n x}{9 f^2}+\frac {8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}-\frac {26 b^2 e^2 m n^2 x}{27 f^2}+\frac {19 b^2 e m n^2 x^2}{108 f}-\frac {2}{27} b^2 m n^2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*b*e^2*m*n*x)/(9*f^2) - (26*b^2*e^2*m*n^2*x)/(27*f^2) + (19*b^2*e*m*n^2*x^2)/(108*f) - (2*b^2*m*n^2*x^3)/2
7 + (8*b^2*e^2*m*n*x*Log[c*x^n])/(9*f^2) - (5*b*e*m*n*x^2*(a + b*Log[c*x^n]))/(18*f) + (4*b*m*n*x^3*(a + b*Log
[c*x^n]))/27 - (e^2*m*x*(a + b*Log[c*x^n])^2)/(3*f^2) + (e*m*x^2*(a + b*Log[c*x^n])^2)/(6*f) - (m*x^3*(a + b*L
og[c*x^n])^2)/9 + (2*b^2*e^3*m*n^2*Log[e + f*x])/(27*f^3) + (2*b^2*n^2*x^3*Log[d*(e + f*x)^m])/27 - (2*b*n*x^3
*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/9 + (x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/3 - (2*b*e^3*m*n*(a
+ b*Log[c*x^n])*Log[1 + (f*x)/e])/(9*f^3) + (e^3*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(3*f^3) - (2*b^2*e^3
*m*n^2*PolyLog[2, -((f*x)/e)])/(9*f^3) + (2*b*e^3*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/(3*f^3) - (2*
b^2*e^3*m*n^2*PolyLog[3, -((f*x)/e)])/(3*f^3)

Rule 45

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 2332

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

Rule 2333

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 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 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 2393

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 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 2425

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 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*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right ) \, dx &=\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-(f m) \int \left (\frac {2 b^2 n^2 x^3}{27 (e+f x)}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 (e+f x)}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 (e+f x)}\right ) \, dx\\ &=\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{3} (f m) \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx+\frac {1}{9} (2 b f m n) \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{e+f x} \, dx-\frac {1}{27} \left (2 b^2 f m n^2\right ) \int \frac {x^3}{e+f x} \, dx\\ &=\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{3} (f m) \int \left (\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^3}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )^2}{f^3 (e+f x)}\right ) \, dx+\frac {1}{9} (2 b f m n) \int \left (\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{f^3}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{f^3 (e+f x)}\right ) \, dx-\frac {1}{27} \left (2 b^2 f m n^2\right ) \int \left (\frac {e^2}{f^3}-\frac {e x}{f^2}+\frac {x^2}{f}-\frac {e^3}{f^3 (e+f x)}\right ) \, dx\\ &=-\frac {2 b^2 e^2 m n^2 x}{27 f^2}+\frac {b^2 e m n^2 x^2}{27 f}-\frac {2}{81} b^2 m n^2 x^3+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {1}{3} m \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {\left (e^2 m\right ) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f^2}+\frac {\left (e^3 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x} \, dx}{3 f^2}+\frac {(e m) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f}+\frac {1}{9} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (2 b e^2 m n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f^2}-\frac {\left (2 b e^3 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x} \, dx}{9 f^2}-\frac {(2 b e m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f}\\ &=\frac {2 a b e^2 m n x}{9 f^2}-\frac {2 b^2 e^2 m n^2 x}{27 f^2}+\frac {5 b^2 e m n^2 x^2}{54 f}-\frac {4}{81} b^2 m n^2 x^3-\frac {b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{9 f}+\frac {2}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 f^3}+\frac {1}{9} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {\left (2 b e^3 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}{3 f^3}+\frac {\left (2 b e^2 m n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f^2}+\frac {\left (2 b^2 e^2 m n\right ) \int \log \left (c x^n\right ) \, dx}{9 f^2}-\frac {(b e m n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}+\frac {\left (2 b^2 e^3 m n^2\right ) \int \frac {\log \left (1+\frac {f x}{e}\right )}{x} \, dx}{9 f^3}\\ &=\frac {8 a b e^2 m n x}{9 f^2}-\frac {8 b^2 e^2 m n^2 x}{27 f^2}+\frac {19 b^2 e m n^2 x^2}{108 f}-\frac {2}{27} b^2 m n^2 x^3+\frac {2 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}-\frac {5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{9 f^3}+\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{3 f^3}+\frac {\left (2 b^2 e^2 m n\right ) \int \log \left (c x^n\right ) \, dx}{3 f^2}-\frac {\left (2 b^2 e^3 m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx}{3 f^3}\\ &=\frac {8 a b e^2 m n x}{9 f^2}-\frac {26 b^2 e^2 m n^2 x}{27 f^2}+\frac {19 b^2 e m n^2 x^2}{108 f}-\frac {2}{27} b^2 m n^2 x^3+\frac {8 b^2 e^2 m n x \log \left (c x^n\right )}{9 f^2}-\frac {5 b e m n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 e^3 m n^2 \log (e+f x)}{27 f^3}+\frac {2}{27} b^2 n^2 x^3 \log \left (d (e+f x)^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )-\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{9 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{9 f^3}+\frac {2 b e^3 m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )}{3 f^3}-\frac {2 b^2 e^3 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{3 f^3}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 788, normalized size = 1.74 \begin {gather*} \frac {-36 a^2 e^2 f m x+96 a b e^2 f m n x-104 b^2 e^2 f m n^2 x+18 a^2 e f^2 m x^2-30 a b e f^2 m n x^2+19 b^2 e f^2 m n^2 x^2-12 a^2 f^3 m x^3+16 a b f^3 m n x^3-8 b^2 f^3 m n^2 x^3-72 a b e^2 f m x \log \left (c x^n\right )+96 b^2 e^2 f m n x \log \left (c x^n\right )+36 a b e f^2 m x^2 \log \left (c x^n\right )-30 b^2 e f^2 m n x^2 \log \left (c x^n\right )-24 a b f^3 m x^3 \log \left (c x^n\right )+16 b^2 f^3 m n x^3 \log \left (c x^n\right )-36 b^2 e^2 f m x \log ^2\left (c x^n\right )+18 b^2 e f^2 m x^2 \log ^2\left (c x^n\right )-12 b^2 f^3 m x^3 \log ^2\left (c x^n\right )+36 a^2 e^3 m \log (e+f x)-24 a b e^3 m n \log (e+f x)+8 b^2 e^3 m n^2 \log (e+f x)-72 a b e^3 m n \log (x) \log (e+f x)+24 b^2 e^3 m n^2 \log (x) \log (e+f x)+36 b^2 e^3 m n^2 \log ^2(x) \log (e+f x)+72 a b e^3 m \log \left (c x^n\right ) \log (e+f x)-24 b^2 e^3 m n \log \left (c x^n\right ) \log (e+f x)-72 b^2 e^3 m n \log (x) \log \left (c x^n\right ) \log (e+f x)+36 b^2 e^3 m \log ^2\left (c x^n\right ) \log (e+f x)+36 a^2 f^3 x^3 \log \left (d (e+f x)^m\right )-24 a b f^3 n x^3 \log \left (d (e+f x)^m\right )+8 b^2 f^3 n^2 x^3 \log \left (d (e+f x)^m\right )+72 a b f^3 x^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-24 b^2 f^3 n x^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+36 b^2 f^3 x^3 \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+72 a b e^3 m n \log (x) \log \left (1+\frac {f x}{e}\right )-24 b^2 e^3 m n^2 \log (x) \log \left (1+\frac {f x}{e}\right )-36 b^2 e^3 m n^2 \log ^2(x) \log \left (1+\frac {f x}{e}\right )+72 b^2 e^3 m n \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+24 b e^3 m n \left (3 a-b n+3 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x}{e}\right )-72 b^2 e^3 m n^2 \text {Li}_3\left (-\frac {f x}{e}\right )}{108 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-36*a^2*e^2*f*m*x + 96*a*b*e^2*f*m*n*x - 104*b^2*e^2*f*m*n^2*x + 18*a^2*e*f^2*m*x^2 - 30*a*b*e*f^2*m*n*x^2 +
19*b^2*e*f^2*m*n^2*x^2 - 12*a^2*f^3*m*x^3 + 16*a*b*f^3*m*n*x^3 - 8*b^2*f^3*m*n^2*x^3 - 72*a*b*e^2*f*m*x*Log[c*
x^n] + 96*b^2*e^2*f*m*n*x*Log[c*x^n] + 36*a*b*e*f^2*m*x^2*Log[c*x^n] - 30*b^2*e*f^2*m*n*x^2*Log[c*x^n] - 24*a*
b*f^3*m*x^3*Log[c*x^n] + 16*b^2*f^3*m*n*x^3*Log[c*x^n] - 36*b^2*e^2*f*m*x*Log[c*x^n]^2 + 18*b^2*e*f^2*m*x^2*Lo
g[c*x^n]^2 - 12*b^2*f^3*m*x^3*Log[c*x^n]^2 + 36*a^2*e^3*m*Log[e + f*x] - 24*a*b*e^3*m*n*Log[e + f*x] + 8*b^2*e
^3*m*n^2*Log[e + f*x] - 72*a*b*e^3*m*n*Log[x]*Log[e + f*x] + 24*b^2*e^3*m*n^2*Log[x]*Log[e + f*x] + 36*b^2*e^3
*m*n^2*Log[x]^2*Log[e + f*x] + 72*a*b*e^3*m*Log[c*x^n]*Log[e + f*x] - 24*b^2*e^3*m*n*Log[c*x^n]*Log[e + f*x] -
72*b^2*e^3*m*n*Log[x]*Log[c*x^n]*Log[e + f*x] + 36*b^2*e^3*m*Log[c*x^n]^2*Log[e + f*x] + 36*a^2*f^3*x^3*Log[d
*(e + f*x)^m] - 24*a*b*f^3*n*x^3*Log[d*(e + f*x)^m] + 8*b^2*f^3*n^2*x^3*Log[d*(e + f*x)^m] + 72*a*b*f^3*x^3*Lo
g[c*x^n]*Log[d*(e + f*x)^m] - 24*b^2*f^3*n*x^3*Log[c*x^n]*Log[d*(e + f*x)^m] + 36*b^2*f^3*x^3*Log[c*x^n]^2*Log
[d*(e + f*x)^m] + 72*a*b*e^3*m*n*Log[x]*Log[1 + (f*x)/e] - 24*b^2*e^3*m*n^2*Log[x]*Log[1 + (f*x)/e] - 36*b^2*e
^3*m*n^2*Log[x]^2*Log[1 + (f*x)/e] + 72*b^2*e^3*m*n*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 24*b*e^3*m*n*(3*a - b
*n + 3*b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] - 72*b^2*e^3*m*n^2*PolyLog[3, -((f*x)/e)])/(108*f^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.58, size = 12902, normalized size = 28.54

method	result	size
risch	\(\text {Expression too large to display}\)	\(12902\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(3*(3*b^2*f^2*m*x^2*e - 2*(f^3*m - 3*f^3*log(d))*b^2*x^3 - 6*b^2*f*m*x*e^2 + 6*b^2*m*e^3*log(f*x + e))*lo
g(x^n)^2 + 2*(9*b^2*f^3*x^3*log(x^n)^2 + 6*(3*a*b*f^3 - (f^3*n - 3*f^3*log(c))*b^2)*x^3*log(x^n) + (9*a^2*f^3
- 6*(f^3*n - 3*f^3*log(c))*a*b + (2*f^3*n^2 - 6*f^3*n*log(c) + 9*f^3*log(c)^2)*b^2)*x^3)*log((f*x + e)^m))/f^3
- integrate(1/27*((9*(f^4*m - 3*f^4*log(d))*a^2 - 6*(f^4*m*n - 3*(f^4*m - 3*f^4*log(d))*log(c))*a*b + (2*f^4*
m*n^2 - 6*f^4*m*n*log(c) + 9*(f^4*m - 3*f^4*log(d))*log(c)^2)*b^2)*x^4 - 27*(b^2*f^3*log(c)^2*log(d) + 2*a*b*f
^3*log(c)*log(d) + a^2*f^3*log(d))*x^3*e - 3*(3*b^2*f^2*m*n*x^2*e^2 + 6*b^2*f*m*n*x*e^3 - 2*(3*(f^4*m - 3*f^4*
log(d))*a*b - (2*f^4*m*n - 3*f^4*n*log(d) - 3*(f^4*m - 3*f^4*log(d))*log(c))*b^2)*x^4 + (18*a*b*f^3*log(d) - (
f^3*m*n + 6*f^3*n*log(d) - 18*f^3*log(c)*log(d))*b^2)*x^3*e - 6*(b^2*f*m*n*x*e^3 + b^2*m*n*e^4)*log(f*x + e))*
log(x^n))/(f^4*x^2 + f^3*x*e), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(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^2*log((f*x + e)^m*d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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