3.70 \(\int x^3 (a+b \log (c x^n)) \log (d (e+f x)^m) \, dx\)

Optimal. Leaf size=283 \[ \frac{b e^4 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{4 f^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{e^4 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac{e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac{e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac{e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac{1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b n x^4 \log \left (d (e+f x)^m\right )+\frac{3 b e^2 m n x^2}{32 f^2}-\frac{5 b e^3 m n x}{16 f^3}+\frac{b e^4 m n \log (e+f x)}{16 f^4}+\frac{b e^4 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{4 f^4}-\frac{7 b e m n x^3}{144 f}+\frac{1}{32} b m n x^4 \]

[Out]

(-5*b*e^3*m*n*x)/(16*f^3) + (3*b*e^2*m*n*x^2)/(32*f^2) - (7*b*e*m*n*x^3)/(144*f) + (b*m*n*x^4)/32 + (e^3*m*x*(
a + b*Log[c*x^n]))/(4*f^3) - (e^2*m*x^2*(a + b*Log[c*x^n]))/(8*f^2) + (e*m*x^3*(a + b*Log[c*x^n]))/(12*f) - (m
*x^4*(a + b*Log[c*x^n]))/16 + (b*e^4*m*n*Log[e + f*x])/(16*f^4) + (b*e^4*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(4*
f^4) - (e^4*m*(a + b*Log[c*x^n])*Log[e + f*x])/(4*f^4) - (b*n*x^4*Log[d*(e + f*x)^m])/16 + (x^4*(a + b*Log[c*x
^n])*Log[d*(e + f*x)^m])/4 + (b*e^4*m*n*PolyLog[2, 1 + (f*x)/e])/(4*f^4)

________________________________________________________________________________________

Rubi [A]  time = 0.206262, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {2395, 43, 2376, 2394, 2315} \[ \frac{b e^4 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{4 f^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{e^4 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac{e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac{e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac{e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac{1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b n x^4 \log \left (d (e+f x)^m\right )+\frac{3 b e^2 m n x^2}{32 f^2}-\frac{5 b e^3 m n x}{16 f^3}+\frac{b e^4 m n \log (e+f x)}{16 f^4}+\frac{b e^4 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{4 f^4}-\frac{7 b e m n x^3}{144 f}+\frac{1}{32} b m n x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b*e^3*m*n*x)/(16*f^3) + (3*b*e^2*m*n*x^2)/(32*f^2) - (7*b*e*m*n*x^3)/(144*f) + (b*m*n*x^4)/32 + (e^3*m*x*(
a + b*Log[c*x^n]))/(4*f^3) - (e^2*m*x^2*(a + b*Log[c*x^n]))/(8*f^2) + (e*m*x^3*(a + b*Log[c*x^n]))/(12*f) - (m
*x^4*(a + b*Log[c*x^n]))/16 + (b*e^4*m*n*Log[e + f*x])/(16*f^4) + (b*e^4*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(4*
f^4) - (e^4*m*(a + b*Log[c*x^n])*Log[e + f*x])/(4*f^4) - (b*n*x^4*Log[d*(e + f*x)^m])/16 + (x^4*(a + b*Log[c*x
^n])*Log[d*(e + f*x)^m])/4 + (b*e^4*m*n*PolyLog[2, 1 + (f*x)/e])/(4*f^4)

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

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 2376

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

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx &=\frac{e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac{e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac{e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac{1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{4 f^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-(b n) \int \left (\frac{e^3 m}{4 f^3}-\frac{e^2 m x}{8 f^2}+\frac{e m x^2}{12 f}-\frac{m x^3}{16}-\frac{e^4 m \log (e+f x)}{4 f^4 x}+\frac{1}{4} x^3 \log \left (d (e+f x)^m\right )\right ) \, dx\\ &=-\frac{b e^3 m n x}{4 f^3}+\frac{b e^2 m n x^2}{16 f^2}-\frac{b e m n x^3}{36 f}+\frac{1}{64} b m n x^4+\frac{e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac{e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac{e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac{1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{4 f^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{1}{4} (b n) \int x^3 \log \left (d (e+f x)^m\right ) \, dx+\frac{\left (b e^4 m n\right ) \int \frac{\log (e+f x)}{x} \, dx}{4 f^4}\\ &=-\frac{b e^3 m n x}{4 f^3}+\frac{b e^2 m n x^2}{16 f^2}-\frac{b e m n x^3}{36 f}+\frac{1}{64} b m n x^4+\frac{e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac{e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac{e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac{1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{4 f^4}-\frac{e^4 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{4 f^4}-\frac{1}{16} b n x^4 \log \left (d (e+f x)^m\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{\left (b e^4 m n\right ) \int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx}{4 f^3}+\frac{1}{16} (b f m n) \int \frac{x^4}{e+f x} \, dx\\ &=-\frac{b e^3 m n x}{4 f^3}+\frac{b e^2 m n x^2}{16 f^2}-\frac{b e m n x^3}{36 f}+\frac{1}{64} b m n x^4+\frac{e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac{e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac{e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac{1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{4 f^4}-\frac{e^4 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{4 f^4}-\frac{1}{16} b n x^4 \log \left (d (e+f x)^m\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{b e^4 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{4 f^4}+\frac{1}{16} (b f m n) \int \left (-\frac{e^3}{f^4}+\frac{e^2 x}{f^3}-\frac{e x^2}{f^2}+\frac{x^3}{f}+\frac{e^4}{f^4 (e+f x)}\right ) \, dx\\ &=-\frac{5 b e^3 m n x}{16 f^3}+\frac{3 b e^2 m n x^2}{32 f^2}-\frac{7 b e m n x^3}{144 f}+\frac{1}{32} b m n x^4+\frac{e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac{e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac{e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac{1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 m n \log (e+f x)}{16 f^4}+\frac{b e^4 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{4 f^4}-\frac{e^4 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{4 f^4}-\frac{1}{16} b n x^4 \log \left (d (e+f x)^m\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{b e^4 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{4 f^4}\\ \end{align*}

Mathematica [A]  time = 0.222576, size = 290, normalized size = 1.02 \[ -\frac{72 b e^4 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right )-72 a f^4 x^4 \log \left (d (e+f x)^m\right )+36 a e^2 f^2 m x^2-72 a e^3 f m x+72 a e^4 m \log (e+f x)-24 a e f^3 m x^3+18 a f^4 m x^4+6 b \log \left (c x^n\right ) \left (-12 f^4 x^4 \log \left (d (e+f x)^m\right )+f m x \left (6 e^2 f x-12 e^3-4 e f^2 x^2+3 f^3 x^3\right )+12 e^4 m \log (e+f x)\right )+18 b f^4 n x^4 \log \left (d (e+f x)^m\right )-27 b e^2 f^2 m n x^2+90 b e^3 f m n x-18 b e^4 m n \log (e+f x)-72 b e^4 m n \log (x) \log (e+f x)+72 b e^4 m n \log (x) \log \left (\frac{f x}{e}+1\right )+14 b e f^3 m n x^3-9 b f^4 m n x^4}{288 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-72*a*e^3*f*m*x + 90*b*e^3*f*m*n*x + 36*a*e^2*f^2*m*x^2 - 27*b*e^2*f^2*m*n*x^2 - 24*a*e*f^3*m*x^3 + 14*b*e*f
^3*m*n*x^3 + 18*a*f^4*m*x^4 - 9*b*f^4*m*n*x^4 + 72*a*e^4*m*Log[e + f*x] - 18*b*e^4*m*n*Log[e + f*x] - 72*b*e^4
*m*n*Log[x]*Log[e + f*x] - 72*a*f^4*x^4*Log[d*(e + f*x)^m] + 18*b*f^4*n*x^4*Log[d*(e + f*x)^m] + 6*b*Log[c*x^n
]*(f*m*x*(-12*e^3 + 6*e^2*f*x - 4*e*f^2*x^2 + 3*f^3*x^3) + 12*e^4*m*Log[e + f*x] - 12*f^4*x^4*Log[d*(e + f*x)^
m]) + 72*b*e^4*m*n*Log[x]*Log[1 + (f*x)/e] + 72*b*e^4*m*n*PolyLog[2, -((f*x)/e)])/(288*f^4)

________________________________________________________________________________________

Maple [C]  time = 0.394, size = 2403, normalized size = 8.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/4*x^4*b*ln(x^n)+1/16*x^4*(-2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+
2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*b*Pi*csgn(I*c*x^n)^3+4*b*ln(c)-b*n+4*a))*ln((f*x+e)^m)-1/8/f^2*x^2*a*
e^2*m+1/4/f^3*a*e^3*m*x-1/4/f^4*e^4*m*ln(f*x+e)*a+1/16*b*e^4*m*n*ln(f*x+e)/f^4-7/144*b*e*m*n*x^3/f-5/16*b*e^3*
m*n*x/f^3+3/32*b*e^2*m*n*x^2/f^2+1/32*b*m*n*x^4+1/4*b*e^4*m*n*ln(-f*x/e)*ln(f*x+e)/f^4+1/12/f*x^3*a*e*m-1/16*m
*b*ln(x^n)*x^4+1/4*ln(d)*b*x^4*ln(x^n)-1/16*x^4*ln(c)*b*m-1/16*ln(d)*b*n*x^4+1/4*x^4*ln(c)*ln(d)*b-205/576*b*e
^4*m*n/f^4-1/16*x^4*a*m+1/4*x^4*ln(d)*a-1/16*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x^4*b*csgn(I*x^n)*cs
gn(I*c*x^n)^2-1/16*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^4*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/16*Pi^2*csgn(I*d)*cs
gn(I*d*(f*x+e)^m)^2*x^4*b*csgn(I*c)*csgn(I*c*x^n)^2-1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^4*b*csgn(I*x^n
)*csgn(I*c*x^n)^2-1/16*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x^4*b*csgn(I*c)*csgn(I*c*x^n)^2-1/32*I*x^4
*Pi*b*m*csgn(I*x^n)*csgn(I*c*x^n)^2-1/32*I*x^4*Pi*b*m*csgn(I*c)*csgn(I*c*x^n)^2-1/32*I*Pi*b*n*x^4*csgn(I*d)*cs
gn(I*d*(f*x+e)^m)^2-1/16*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*x^4*b*csgn(I*c*x^n)^3+1/8*I*x^4*
ln(c)*Pi*b*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/8*I*x^4*Pi*ln(d)*b*csgn(I*c)*csgn(I*c*x^n)^2+1/8*I*x^4*Pi
*ln(d)*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/8*I*Pi*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*b*x^4*ln(x^n)+1/8*I*Pi*csgn(I*(f
*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*b*x^4*ln(x^n)-1/8*I*x^4*Pi*a*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/
8*I*x^4*ln(c)*Pi*b*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/32*I*Pi*b*n*x^4*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1
/16*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^4*b*csgn(I*c*x^n)^3-1/8*I*x^4*Pi*a*csgn(I*d*(f*x+e)^m)^3+1/16*Pi^2*csgn(I*(f*
x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x^4*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/16*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*
csgn(I*d*(f*x+e)^m)*x^4*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^4*b*csgn(I*c
)*csgn(I*x^n)*csgn(I*c*x^n)-1/24*I/f*Pi*x^3*b*e*m*csgn(I*c*x^n)^3-1/8*I*Pi*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*
d*(f*x+e)^m)*b*x^4*ln(x^n)+1/8*I/f^4*e^4*m*ln(f*x+e)*Pi*b*csgn(I*c*x^n)^3-1/8*I/f^3*Pi*b*e^3*m*csgn(I*c*x^n)^3
*x+1/16*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*x^4*b*csgn(I*c)*csgn(I*c*x^n)^2-1/8*I*x^4*ln(c)*P
i*b*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)-1/8*I*x^4*Pi*ln(d)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1
/32*I*Pi*b*n*x^4*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/16*I/f^2*Pi*x^2*b*e^2*m*csgn(I*c*x^n)^3+1/3
2*I*x^4*Pi*b*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/8*I/f^4*e^4*m*ln(f*x+e)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2+
1/8*I/f^3*Pi*b*e^3*m*csgn(I*c)*csgn(I*c*x^n)^2*x+1/8*I/f^3*Pi*b*e^3*m*csgn(I*x^n)*csgn(I*c*x^n)^2*x+1/24*I/f*P
i*x^3*b*e*m*csgn(I*c)*csgn(I*c*x^n)^2-1/16*I/f^2*Pi*x^2*b*e^2*m*csgn(I*c)*csgn(I*c*x^n)^2-1/16*I/f^2*Pi*x^2*b*
e^2*m*csgn(I*x^n)*csgn(I*c*x^n)^2-1/8*I/f^4*e^4*m*ln(f*x+e)*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2-1/16*Pi^2*csgn(I*d)
*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*x^4*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/24*I/f*Pi*x^3*b*e*m*csgn(I*
x^n)*csgn(I*c*x^n)^2+1/16*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x^4*b*csgn(I*c*x^n)^3+1/12*m/f*b*ln(x^n
)*e*x^3-1/8*m/f^2*b*ln(x^n)*x^2*e^2+1/4*m/f^3*b*ln(x^n)*x*e^3-1/4*m/f^4*b*ln(x^n)*e^4*ln(f*x+e)-1/8*I*x^4*ln(c
)*Pi*b*csgn(I*d*(f*x+e)^m)^3-1/8*I*x^4*Pi*ln(d)*b*csgn(I*c*x^n)^3+1/32*I*Pi*b*n*x^4*csgn(I*d*(f*x+e)^m)^3-1/8*
I*Pi*csgn(I*d*(f*x+e)^m)^3*b*x^4*ln(x^n)+1/16*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^4*b*csgn(I*c*x^n)^3+1/16*
Pi^2*csgn(I*d*(f*x+e)^m)^3*x^4*b*csgn(I*c)*csgn(I*c*x^n)^2+1/16*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^4*b*csgn(I*x^n)*c
sgn(I*c*x^n)^2+1/32*I*x^4*Pi*b*m*csgn(I*c*x^n)^3+1/8*I*x^4*Pi*a*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2+1/8*I*x^4*Pi*a
*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/4/f^3*ln(c)*b*e^3*m*x+1/12/f*ln(c)*x^3*b*e*m-1/8/f^2*ln(c)*x^2*b*e^
2*m-1/4/f^4*e^4*m*ln(f*x+e)*b*ln(c)+1/4*n*b/f^4*e^4*m*dilog(-f*x/e)+1/16*I/f^2*Pi*x^2*b*e^2*m*csgn(I*c)*csgn(I
*x^n)*csgn(I*c*x^n)-1/8*I/f^3*Pi*b*e^3*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x+1/8*I/f^4*e^4*m*ln(f*x+e)*Pi*b*
csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/24*I/f*Pi*x^3*b*e*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)

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Maxima [A]  time = 1.86598, size = 514, normalized size = 1.82 \begin{align*} -\frac{{\left (\log \left (\frac{f x}{e} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{f x}{e}\right )\right )} b e^{4} m n}{4 \, f^{4}} - \frac{{\left (4 \, a e^{4} m -{\left (e^{4} m n - 4 \, e^{4} m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{16 \, f^{4}} + \frac{72 \, b e^{4} m n \log \left (f x + e\right ) \log \left (x\right ) - 9 \,{\left (2 \,{\left (f^{4} m - 4 \, f^{4} \log \left (d\right )\right )} a -{\left (f^{4} m n - 2 \, f^{4} n \log \left (d\right ) - 2 \,{\left (f^{4} m - 4 \, f^{4} \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{4} + 2 \,{\left (12 \, a e f^{3} m -{\left (7 \, e f^{3} m n - 12 \, e f^{3} m \log \left (c\right )\right )} b\right )} x^{3} - 9 \,{\left (4 \, a e^{2} f^{2} m -{\left (3 \, e^{2} f^{2} m n - 4 \, e^{2} f^{2} m \log \left (c\right )\right )} b\right )} x^{2} + 18 \,{\left (4 \, a e^{3} f m -{\left (5 \, e^{3} f m n - 4 \, e^{3} f m \log \left (c\right )\right )} b\right )} x + 18 \,{\left (4 \, b f^{4} x^{4} \log \left (x^{n}\right ) +{\left (4 \, a f^{4} -{\left (f^{4} n - 4 \, f^{4} \log \left (c\right )\right )} b\right )} x^{4}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) + 6 \,{\left (4 \, b e f^{3} m x^{3} - 6 \, b e^{2} f^{2} m x^{2} + 12 \, b e^{3} f m x - 12 \, b e^{4} m \log \left (f x + e\right ) - 3 \,{\left (f^{4} m - 4 \, f^{4} \log \left (d\right )\right )} b x^{4}\right )} \log \left (x^{n}\right )}{288 \, f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(log(f*x/e + 1)*log(x) + dilog(-f*x/e))*b*e^4*m*n/f^4 - 1/16*(4*a*e^4*m - (e^4*m*n - 4*e^4*m*log(c))*b)*l
og(f*x + e)/f^4 + 1/288*(72*b*e^4*m*n*log(f*x + e)*log(x) - 9*(2*(f^4*m - 4*f^4*log(d))*a - (f^4*m*n - 2*f^4*n
*log(d) - 2*(f^4*m - 4*f^4*log(d))*log(c))*b)*x^4 + 2*(12*a*e*f^3*m - (7*e*f^3*m*n - 12*e*f^3*m*log(c))*b)*x^3
- 9*(4*a*e^2*f^2*m - (3*e^2*f^2*m*n - 4*e^2*f^2*m*log(c))*b)*x^2 + 18*(4*a*e^3*f*m - (5*e^3*f*m*n - 4*e^3*f*m
*log(c))*b)*x + 18*(4*b*f^4*x^4*log(x^n) + (4*a*f^4 - (f^4*n - 4*f^4*log(c))*b)*x^4)*log((f*x + e)^m) + 6*(4*b
*e*f^3*m*x^3 - 6*b*e^2*f^2*m*x^2 + 12*b*e^3*f*m*x - 12*b*e^4*m*log(f*x + e) - 3*(f^4*m - 4*f^4*log(d))*b*x^4)*
log(x^n))/f^4

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

[Out]

integral((b*x^3*log(c*x^n) + a*x^3)*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**3*(a+b*ln(c*x**n))*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 )} x^{3} \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^3*(a+b*log(c*x^n))*log(d*(f*x+e)^m),x, algorithm="giac")

[Out]

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