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

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

[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)

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Rubi [A]  time = 0.220241, antiderivative size = 232, normalized size of antiderivative = 1., 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 = {2426, 43, 2351, 2295, 2304, 2317, 2391} \[ -\frac{b d^4 m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{4 e^4}-\frac{1}{16} \left (m x^4-4 x^4 \log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b d^4 n \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{4 e^4}+\frac{b d^3 n x \log \left (f x^m\right )}{4 e^3}-\frac{b d^2 n x^2 \log \left (f x^m\right )}{8 e^2}+\frac{3 b d^2 m n x^2}{32 e^2}-\frac{5 b d^3 m n x}{16 e^3}+\frac{b d^4 m n \log (d+e x)}{16 e^4}+\frac{b d n x^3 \log \left (f x^m\right )}{12 e}-\frac{7 b d m n x^3}{144 e}-\frac{1}{16} b n x^4 \log \left (f x^m\right )+\frac{1}{32} b m n x^4 \]

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 2426

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

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*}

Mathematica [A]  time = 0.190901, size = 221, normalized size = 0.95 \[ \frac{-72 b d^4 m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )-6 \log \left (f x^m\right ) \left (-12 a e^4 x^4-12 b e^4 x^4 \log \left (c (d+e x)^n\right )+b e n x \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+12 b d^4 n \log (d+e x)\right )+m \left (-18 a e^4 x^4-18 b e^4 x^4 \log \left (c (d+e x)^n\right )+27 b d^2 e^2 n x^2-90 b d^3 e n x+18 b d^4 n (4 \log (x)+1) \log (d+e x)-72 b d^4 n \log (x) \log \left (\frac{e x}{d}+1\right )-14 b d e^3 n x^3+9 b e^4 n x^4\right )}{288 e^4} \]

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]  time = 1.115, size = 2330, normalized size = 10. \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.26021, size = 312, normalized size = 1.34 \begin{align*} \frac{1}{288} \,{\left (\frac{72 \,{\left (\log \left (e x + d\right ) \log \left (-\frac{e x + d}{d} + 1\right ) +{\rm Li}_2\left (\frac{e x + d}{d}\right )\right )} b d^{4} n}{e^{4}} - \frac{18 \, b e^{4} x^{4} \log \left ({\left (e x + d\right )}^{n}\right ) + 14 \, b d e^{3} n x^{3} - 27 \, b d^{2} e^{2} n x^{2} + 90 \, b d^{3} e n x - 18 \, b d^{4} n \log \left (e x + d\right ) + 9 \,{\left (2 \, a e^{4} -{\left (e^{4} n - 2 \, e^{4} \log \left (c\right )\right )} b\right )} x^{4}}{e^{4}}\right )} m + \frac{1}{48} \,{\left (12 \, b x^{4} \log \left ({\left (e x + d\right )}^{n} c\right ) + 12 \, a x^{4} - b e n{\left (\frac{12 \, d^{4} \log \left (e x + d\right )}{e^{5}} + \frac{3 \, e^{3} x^{4} - 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} - 12 \, d^{3} x}{e^{4}}\right )}\right )} \log \left (f x^{m}\right ) \end{align*}

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

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

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

[Out]

Timed out

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

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