3.4.58 \(\int x^3 \log (f x^m) (a+b \log (c (d+e x)^n)) \, dx\) [358]

Optimal. Leaf size=232 \[ -\frac {5 b d^3 m n x}{16 e^3}+\frac {3 b d^2 m n x^2}{32 e^2}-\frac {7 b d m n x^3}{144 e}+\frac {1}{32} b m n x^4+\frac {b d^3 n x \log \left (f x^m\right )}{4 e^3}-\frac {b d^2 n x^2 \log \left (f x^m\right )}{8 e^2}+\frac {b d n x^3 \log \left (f x^m\right )}{12 e}-\frac {1}{16} b n x^4 \log \left (f x^m\right )+\frac {b d^4 m n \log (d+e x)}{16 e^4}-\frac {1}{16} \left (m x^4-4 x^4 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b d^4 n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{4 e^4}-\frac {b d^4 m n \text {Li}_2\left (-\frac {e x}{d}\right )}{4 e^4} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2473, 45, 2393, 2332, 2341, 2354, 2438} \begin {gather*} -\frac {b d^4 m n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{4 e^4}-\frac {1}{16} \left (m x^4-4 x^4 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b d^4 n \log \left (\frac {e x}{d}+1\right ) \log \left (f x^m\right )}{4 e^4}+\frac {b d^4 m n \log (d+e x)}{16 e^4}+\frac {b d^3 n x \log \left (f x^m\right )}{4 e^3}-\frac {5 b d^3 m n x}{16 e^3}-\frac {b d^2 n x^2 \log \left (f x^m\right )}{8 e^2}+\frac {3 b d^2 m n x^2}{32 e^2}+\frac {b d n x^3 \log \left (f x^m\right )}{12 e}-\frac {7 b d m n x^3}{144 e}-\frac {1}{16} b n x^4 \log \left (f x^m\right )+\frac {1}{32} b m n x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 2473

Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :
> Simp[(-(g*(q + 1))^(-1))*(m*((g*x)^(q + 1)/(q + 1)) - (g*x)^(q + 1)*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]),
x] + (-Dist[b*e*(n/(g*(q + 1))), Int[(g*x)^(q + 1)*(Log[f*x^m]/(d + e*x)), x], x] + Dist[b*e*m*(n/(g*(q + 1)^2
)), Int[(g*x)^(q + 1)/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[q, -1]

Rubi steps

\begin {align*} \int x^3 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=-\frac {1}{16} \left (m x^4-4 x^4 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{4} (b e n) \int \frac {x^4 \log \left (f x^m\right )}{d+e x} \, dx+\frac {1}{16} (b e m n) \int \frac {x^4}{d+e x} \, dx\\ &=-\frac {1}{16} \left (m x^4-4 x^4 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{4} (b e n) \int \left (-\frac {d^3 \log \left (f x^m\right )}{e^4}+\frac {d^2 x \log \left (f x^m\right )}{e^3}-\frac {d x^2 \log \left (f x^m\right )}{e^2}+\frac {x^3 \log \left (f x^m\right )}{e}+\frac {d^4 \log \left (f x^m\right )}{e^4 (d+e x)}\right ) \, dx+\frac {1}{16} (b e m n) \int \left (-\frac {d^3}{e^4}+\frac {d^2 x}{e^3}-\frac {d x^2}{e^2}+\frac {x^3}{e}+\frac {d^4}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac {b d^3 m n x}{16 e^3}+\frac {b d^2 m n x^2}{32 e^2}-\frac {b d m n x^3}{48 e}+\frac {1}{64} b m n x^4+\frac {b d^4 m n \log (d+e x)}{16 e^4}-\frac {1}{16} \left (m x^4-4 x^4 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{4} (b n) \int x^3 \log \left (f x^m\right ) \, dx+\frac {\left (b d^3 n\right ) \int \log \left (f x^m\right ) \, dx}{4 e^3}-\frac {\left (b d^4 n\right ) \int \frac {\log \left (f x^m\right )}{d+e x} \, dx}{4 e^3}-\frac {\left (b d^2 n\right ) \int x \log \left (f x^m\right ) \, dx}{4 e^2}+\frac {(b d n) \int x^2 \log \left (f x^m\right ) \, dx}{4 e}\\ &=-\frac {5 b d^3 m n x}{16 e^3}+\frac {3 b d^2 m n x^2}{32 e^2}-\frac {7 b d m n x^3}{144 e}+\frac {1}{32} b m n x^4+\frac {b d^3 n x \log \left (f x^m\right )}{4 e^3}-\frac {b d^2 n x^2 \log \left (f x^m\right )}{8 e^2}+\frac {b d n x^3 \log \left (f x^m\right )}{12 e}-\frac {1}{16} b n x^4 \log \left (f x^m\right )+\frac {b d^4 m n \log (d+e x)}{16 e^4}-\frac {1}{16} \left (m x^4-4 x^4 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b d^4 n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{4 e^4}+\frac {\left (b d^4 m n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{4 e^4}\\ &=-\frac {5 b d^3 m n x}{16 e^3}+\frac {3 b d^2 m n x^2}{32 e^2}-\frac {7 b d m n x^3}{144 e}+\frac {1}{32} b m n x^4+\frac {b d^3 n x \log \left (f x^m\right )}{4 e^3}-\frac {b d^2 n x^2 \log \left (f x^m\right )}{8 e^2}+\frac {b d n x^3 \log \left (f x^m\right )}{12 e}-\frac {1}{16} b n x^4 \log \left (f x^m\right )+\frac {b d^4 m n \log (d+e x)}{16 e^4}-\frac {1}{16} \left (m x^4-4 x^4 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b d^4 n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{4 e^4}-\frac {b d^4 m n \text {Li}_2\left (-\frac {e x}{d}\right )}{4 e^4}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 221, normalized size = 0.95 \begin {gather*} \frac {-6 \log \left (f x^m\right ) \left (-12 a e^4 x^4+b e n x \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+12 b d^4 n \log (d+e x)-12 b e^4 x^4 \log \left (c (d+e x)^n\right )\right )+m \left (-90 b d^3 e n x+27 b d^2 e^2 n x^2-14 b d e^3 n x^3-18 a e^4 x^4+9 b e^4 n x^4+18 b d^4 n (1+4 \log (x)) \log (d+e x)-18 b e^4 x^4 \log \left (c (d+e x)^n\right )-72 b d^4 n \log (x) \log \left (1+\frac {e x}{d}\right )\right )-72 b d^4 m n \text {Li}_2\left (-\frac {e x}{d}\right )}{288 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.18, size = 2330, normalized size = 10.04

method	result	size
risch	\(\text {Expression too large to display}\)	\(2330\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 223, normalized size = 0.96 \begin {gather*} \frac {1}{288} \, {\left (72 \, {\left (\log \left (x e + d\right ) \log \left (-\frac {x e + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {x e + d}{d}\right )\right )} b d^{4} n e^{\left (-4\right )} - {\left (14 \, b d n x^{3} e^{3} - 27 \, b d^{2} n x^{2} e^{2} + 90 \, b d^{3} n x e - 18 \, b d^{4} n \log \left (x e + d\right ) + 18 \, b x^{4} e^{4} \log \left ({\left (x e + d\right )}^{n}\right ) - 9 \, {\left (b {\left (n - 2 \, \log \left (c\right )\right )} - 2 \, a\right )} x^{4} e^{4}\right )} e^{\left (-4\right )}\right )} m + \frac {1}{48} \, {\left (12 \, b x^{4} \log \left ({\left (x e + d\right )}^{n} c\right ) + 12 \, a x^{4} - {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (x e + d\right ) + {\left (3 \, x^{4} e^{3} - 4 \, d x^{3} e^{2} + 6 \, d^{2} x^{2} e - 12 \, d^{3} x\right )} e^{\left (-4\right )}\right )} b n e\right )} \log \left (f x^{m}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(72*(log(x*e + d)*log(-(x*e + d)/d + 1) + dilog((x*e + d)/d))*b*d^4*n*e^(-4) - (14*b*d*n*x^3*e^3 - 27*b*
d^2*n*x^2*e^2 + 90*b*d^3*n*x*e - 18*b*d^4*n*log(x*e + d) + 18*b*x^4*e^4*log((x*e + d)^n) - 9*(b*(n - 2*log(c))
- 2*a)*x^4*e^4)*e^(-4))*m + 1/48*(12*b*x^4*log((x*e + d)^n*c) + 12*a*x^4 - (12*d^4*e^(-5)*log(x*e + d) + (3*x
^4*e^3 - 4*d*x^3*e^2 + 6*d^2*x^2*e - 12*d^3*x)*e^(-4))*b*n*e)*log(f*x^m)

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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