3.1.70 \(\int x^3 (a+b \log (c x^n)) \log (d (e+f x)^m) \, dx\) [70]

Optimal. Leaf size=283 \[ -\frac {5 b e^3 m n x}{16 f^3}+\frac {3 b e^2 m n x^2}{32 f^2}-\frac {7 b e m n x^3}{144 f}+\frac {1}{32} b m n x^4+\frac {e^3 m x \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {e^2 m x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {e m x^3 \left (a+b \log \left (c x^n\right )\right )}{12 f}-\frac {1}{16} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 m n \log (e+f x)}{16 f^4}+\frac {b e^4 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{4 f^4}-\frac {e^4 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{4 f^4}-\frac {1}{16} b n x^4 \log \left (d (e+f x)^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {b e^4 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{4 f^4} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 283, normalized size of antiderivative = 1.00, 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 = {2442, 45, 2423, 2441, 2352} \begin {gather*} \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 {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 {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 \end {gather*}

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 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 2352

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

Rule 2423

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 2441

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 2442

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]

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*}

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Mathematica [A]
time = 0.16, size = 290, normalized size = 1.02 \begin {gather*} -\frac {-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 \left (d (e+f x)^m\right )+18 b f^4 n x^4 \log \left (d (e+f x)^m\right )+6 b \log \left (c x^n\right ) \left (f m x \left (-12 e^3+6 e^2 f x-4 e f^2 x^2+3 f^3 x^3\right )+12 e^4 m \log (e+f x)-12 f^4 x^4 \log \left (d (e+f x)^m\right )\right )+72 b e^4 m n \log (x) \log \left (1+\frac {f x}{e}\right )+72 b e^4 m n \text {Li}_2\left (-\frac {f x}{e}\right )}{288 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/288*(-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)])/f^4

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.43, size = 2403, normalized size = 8.49

method	result	size
risch	\(\text {Expression too large to display}\)	\(2403\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.39, size = 356, normalized size = 1.26 \begin {gather*} -\frac {{\left (\log \left (f x e^{\left (-1\right )} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-f x e^{\left (-1\right )}\right )\right )} b m n e^{4}}{4 \, f^{4}} + \frac {{\left ({\left (m n - 4 \, m \log \left (c\right )\right )} b - 4 \, a m\right )} e^{4} \log \left (f x + e\right )}{16 \, f^{4}} + \frac {72 \, b m n e^{4} \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 f^{3} m - {\left (7 \, f^{3} m n - 12 \, f^{3} m \log \left (c\right )\right )} b\right )} x^{3} e - 9 \, {\left (4 \, a f^{2} m - {\left (3 \, f^{2} m n - 4 \, f^{2} m \log \left (c\right )\right )} b\right )} x^{2} e^{2} + 18 \, {\left (4 \, a f m - {\left (5 \, f m n - 4 \, f m \log \left (c\right )\right )} b\right )} x e^{3} + 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 f^{3} m x^{3} e - 6 \, b f^{2} m x^{2} e^{2} - 3 \, {\left (f^{4} m - 4 \, f^{4} \log \left (d\right )\right )} b x^{4} + 12 \, b f m x e^{3} - 12 \, b m e^{4} \log \left (f x + e\right )\right )} \log \left (x^{n}\right )}{288 \, f^{4}} \end {gather*}

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

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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*(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)] 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*(a+b*ln(c*x**n))*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^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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