3.1.71 \(\int x^2 (a+b \log (c x^n)) \log (d (e+f x)^m) \, dx\) [71]

Optimal. Leaf size=243 \[ \frac {4 b e^2 m n x}{9 f^2}-\frac {5 b e m n x^2}{36 f}+\frac {2}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b e^3 m n \log (e+f x)}{9 f^3}-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {b e^3 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{3 f^3} \]

[Out]

4/9*b*e^2*m*n*x/f^2-5/36*b*e*m*n*x^2/f+2/27*b*m*n*x^3-1/3*e^2*m*x*(a+b*ln(c*x^n))/f^2+1/6*e*m*x^2*(a+b*ln(c*x^
n))/f-1/9*m*x^3*(a+b*ln(c*x^n))-1/9*b*e^3*m*n*ln(f*x+e)/f^3-1/3*b*e^3*m*n*ln(-f*x/e)*ln(f*x+e)/f^3+1/3*e^3*m*(
a+b*ln(c*x^n))*ln(f*x+e)/f^3-1/9*b*n*x^3*ln(d*(f*x+e)^m)+1/3*x^3*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)-1/3*b*e^3*m*n
*polylog(2,1+f*x/e)/f^3

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Rubi [A]
time = 0.12, antiderivative size = 243, 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^3 m n \text {PolyLog}\left (2,\frac {f x}{e}+1\right )}{3 f^3}+\frac {1}{3} 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 (e+f x) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )-\frac {b e^3 m n \log (e+f x)}{9 f^3}-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {4 b e^2 m n x}{9 f^2}-\frac {5 b e m n x^2}{36 f}+\frac {2}{27} b m n x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*e^2*m*n*x)/(9*f^2) - (5*b*e*m*n*x^2)/(36*f) + (2*b*m*n*x^3)/27 - (e^2*m*x*(a + b*Log[c*x^n]))/(3*f^2) + (
e*m*x^2*(a + b*Log[c*x^n]))/(6*f) - (m*x^3*(a + b*Log[c*x^n]))/9 - (b*e^3*m*n*Log[e + f*x])/(9*f^3) - (b*e^3*m
*n*Log[-((f*x)/e)]*Log[e + f*x])/(3*f^3) + (e^3*m*(a + b*Log[c*x^n])*Log[e + f*x])/(3*f^3) - (b*n*x^3*Log[d*(e
+ f*x)^m])/9 + (x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/3 - (b*e^3*m*n*PolyLog[2, 1 + (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 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^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx &=-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-(b n) \int \left (-\frac {e^2 m}{3 f^2}+\frac {e m x}{6 f}-\frac {m x^2}{9}+\frac {e^3 m \log (e+f x)}{3 f^3 x}+\frac {1}{3} x^2 \log \left (d (e+f x)^m\right )\right ) \, dx\\ &=\frac {b e^2 m n x}{3 f^2}-\frac {b e m n x^2}{12 f}+\frac {1}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {1}{3} (b n) \int x^2 \log \left (d (e+f x)^m\right ) \, dx-\frac {\left (b e^3 m n\right ) \int \frac {\log (e+f x)}{x} \, dx}{3 f^3}\\ &=\frac {b e^2 m n x}{3 f^2}-\frac {b e m n x^2}{12 f}+\frac {1}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {\left (b e^3 m n\right ) \int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx}{3 f^2}+\frac {1}{9} (b f m n) \int \frac {x^3}{e+f x} \, dx\\ &=\frac {b e^2 m n x}{3 f^2}-\frac {b e m n x^2}{12 f}+\frac {1}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {b e^3 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{3 f^3}+\frac {1}{9} (b f m n) \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 {4 b e^2 m n x}{9 f^2}-\frac {5 b e m n x^2}{36 f}+\frac {2}{27} b m n x^3-\frac {e^2 m x \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b e^3 m n \log (e+f x)}{9 f^3}-\frac {b e^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 f^3}+\frac {e^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 f^3}-\frac {1}{9} b n x^3 \log \left (d (e+f x)^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {b e^3 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{3 f^3}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 252, normalized size = 1.04 \begin {gather*} \frac {-36 a e^2 f m x+48 b e^2 f m n x+18 a e f^2 m x^2-15 b e f^2 m n x^2-12 a f^3 m x^3+8 b f^3 m n x^3+36 a e^3 m \log (e+f x)-12 b e^3 m n \log (e+f x)-36 b e^3 m n \log (x) \log (e+f x)+36 a f^3 x^3 \log \left (d (e+f x)^m\right )-12 b f^3 n x^3 \log \left (d (e+f x)^m\right )-6 b \log \left (c x^n\right ) \left (f m x \left (6 e^2-3 e f x+2 f^2 x^2\right )-6 e^3 m \log (e+f x)-6 f^3 x^3 \log \left (d (e+f x)^m\right )\right )+36 b e^3 m n \log (x) \log \left (1+\frac {f x}{e}\right )+36 b e^3 m n \text {Li}_2\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])*Log[d*(e + f*x)^m],x]

[Out]

(-36*a*e^2*f*m*x + 48*b*e^2*f*m*n*x + 18*a*e*f^2*m*x^2 - 15*b*e*f^2*m*n*x^2 - 12*a*f^3*m*x^3 + 8*b*f^3*m*n*x^3
+ 36*a*e^3*m*Log[e + f*x] - 12*b*e^3*m*n*Log[e + f*x] - 36*b*e^3*m*n*Log[x]*Log[e + f*x] + 36*a*f^3*x^3*Log[d
*(e + f*x)^m] - 12*b*f^3*n*x^3*Log[d*(e + f*x)^m] - 6*b*Log[c*x^n]*(f*m*x*(6*e^2 - 3*e*f*x + 2*f^2*x^2) - 6*e^
3*m*Log[e + f*x] - 6*f^3*x^3*Log[d*(e + f*x)^m]) + 36*b*e^3*m*n*Log[x]*Log[1 + (f*x)/e] + 36*b*e^3*m*n*PolyLog
[2, -((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 = 0.39, size = 2222, normalized size = 9.14

method	result	size
risch	\(\text {Expression too large to display}\)	\(2222\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b*e^3*m*n*ln(-f*x/e)*ln(f*x+e)/f^3-1/9*b*e^3*m*n*ln(f*x+e)/f^3+2/27*b*m*n*x^3-1/18*I*Pi*b*n*x^3*csgn(I*(f
*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1/12*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*x^3*b*csgn(I*c*x^n)^3
-1/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^3*b*csgn(I*c)*csgn(I*c*x^n)^2+1/3*x^3*ln(d)*a+1/3/f^3*e^3*m*ln(f*
x+e)*a-1/3/f^2*a*e^2*m*x-1/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/12*Pi^2
*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x^3*b*csgn(I*c)*csgn(I*c*x^n)^2-1/12*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*
(f*x+e)^m)^2*x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/6/f*x^2*a*e*m+1/6/f*ln(c)*x^2*b*e*m+1/3/f^3*e^3*m*ln(f*x+e)*b
*ln(c)-1/3/f^2*ln(c)*b*e^2*m*x-1/9*x^3*a*m+49/108*b*e^3*m*n/f^3-1/12*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d
*(f*x+e)^m)*x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I/f^2*Pi*b*e^2*m*csgn(I*x^n)*csgn(I*c*x^n)^2*x+1/6*I
/f^3*e^3*m*ln(f*x+e)*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2+1/3*x^3*ln(d)*ln(c)*b-1/9*x^3*ln(c)*b*m-1/9*ln(d)*b*n*x^3-
1/6*I*x^3*Pi*ln(c)*b*csgn(I*d*(f*x+e)^m)^3+1/6*I*x^3*Pi*a*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/18*I*x^3*Pi*b*m*cs
gn(I*c)*csgn(I*c*x^n)^2-1/18*I*x^3*Pi*b*m*csgn(I*x^n)*csgn(I*c*x^n)^2-1/18*I*Pi*b*n*x^3*csgn(I*d)*csgn(I*d*(f*
x+e)^m)^2+1/6*I/f^2*Pi*b*e^2*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x-1/6*I/f^3*e^3*m*ln(f*x+e)*Pi*b*csgn(I*c)*
csgn(I*x^n)*csgn(I*c*x^n)-1/12*I/f*Pi*x^2*b*e*m*csgn(I*c*x^n)^3-1/6*I/f^3*e^3*m*ln(f*x+e)*Pi*b*csgn(I*c*x^n)^3
+1/12*I/f*Pi*x^2*b*e*m*csgn(I*x^n)*csgn(I*c*x^n)^2-1/6*I/f^2*Pi*b*e^2*m*csgn(I*c)*csgn(I*c*x^n)^2*x+1/12*Pi^2*
csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x^3*b*csgn(I*c*x^n)^3+1/12*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^3*b*csgn(I*c)*
csgn(I*c*x^n)^2+1/12*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/6*I/f^3*e^3*m*ln(f*x+e)*Pi
*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/12*I/f*Pi*x^2*b*e*m*csgn(I*c)*csgn(I*c*x^n)^2+1/6*I*x^3*Pi*ln(d)*b*csgn(I*x^n
)*csgn(I*c*x^n)^2+1/6*I*x^3*Pi*ln(c)*b*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2+1/6*I*x^3*Pi*ln(c)*b*csgn(I*(f*x+e)^m)*
csgn(I*d*(f*x+e)^m)^2+(1/3*x^3*b*ln(x^n)+1/18*x^3*(-3*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*I*b*Pi*csgn
(I*c)*csgn(I*c*x^n)^2+3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*c*x^n)^3+6*b*ln(c)-2*b*n+6*a))*ln((
f*x+e)^m)+1/12*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/18*I*P
i*b*n*x^3*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/18*I*x^3*Pi*b*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n
)+1/6*I/f^2*Pi*b*e^2*m*csgn(I*c*x^n)^3*x-1/3*n*b/f^3*e^3*m*dilog(-f*x/e)-1/12*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^3*b
*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I*x^3*Pi*a*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/6*I*x^3*
Pi*ln(d)*b*csgn(I*c)*csgn(I*c*x^n)^2-1/6*I*x^3*Pi*ln(d)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I*x^3*Pi*ln(
c)*b*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^3*b*csgn(I*c)
*csgn(I*x^n)*csgn(I*c*x^n)+1/12*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(
I*c*x^n)+1/12*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*x^3*b*csgn(I*c)*csgn(I*c*x^n)^2+1/6*I*Pi*cs
gn(I*d)*csgn(I*d*(f*x+e)^m)^2*b*x^3*ln(x^n)+1/6*I*Pi*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*b*x^3*ln(x^n)-1/6
*I*Pi*csgn(I*d*(f*x+e)^m)^3*b*x^3*ln(x^n)+1/18*I*Pi*b*n*x^3*csgn(I*d*(f*x+e)^m)^3+1/6*I*x^3*Pi*a*csgn(I*(f*x+e
)^m)*csgn(I*d*(f*x+e)^m)^2+1/18*I*x^3*Pi*b*m*csgn(I*c*x^n)^3-1/6*I*x^3*Pi*ln(d)*b*csgn(I*c*x^n)^3-1/6*I*Pi*csg
n(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*b*x^3*ln(x^n)-1/6*I*x^3*Pi*a*csgn(I*d*(f*x+e)^m)^3-1/12*Pi^2*csgn
(I*d*(f*x+e)^m)^3*x^3*b*csgn(I*c*x^n)^3+4/9*b*e^2*m*n*x/f^2-5/36*b*e*m*n*x^2/f-1/9*m*b*ln(x^n)*x^3+1/3*ln(d)*b
*x^3*ln(x^n)-1/12*I/f*Pi*x^2*b*e*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)
^2*x^3*b*csgn(I*c*x^n)^3+1/3*m/f^3*b*ln(x^n)*e^3*ln(f*x+e)+1/6*m/f*b*ln(x^n)*e*x^2-1/3*m/f^2*b*ln(x^n)*x*e^2

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Maxima [A]
time = 0.39, size = 311, normalized size = 1.28 \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^{3}}{3 \, f^{3}} - \frac {{\left ({\left (m n - 3 \, m \log \left (c\right )\right )} b - 3 \, a m\right )} e^{3} \log \left (f x + e\right )}{9 \, f^{3}} - \frac {36 \, b m n e^{3} \log \left (f x + e\right ) \log \left (x\right ) + 4 \, {\left (3 \, {\left (f^{3} m - 3 \, f^{3} \log \left (d\right )\right )} a - {\left (2 \, f^{3} m n - 3 \, f^{3} n \log \left (d\right ) - 3 \, {\left (f^{3} m - 3 \, f^{3} \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{3} - 3 \, {\left (6 \, a f^{2} m - {\left (5 \, f^{2} m n - 6 \, f^{2} m \log \left (c\right )\right )} b\right )} x^{2} e + 12 \, {\left (3 \, a f m - {\left (4 \, f m n - 3 \, f m \log \left (c\right )\right )} b\right )} x e^{2} - 12 \, {\left (3 \, b f^{3} x^{3} \log \left (x^{n}\right ) + {\left (3 \, a f^{3} - {\left (f^{3} n - 3 \, f^{3} \log \left (c\right )\right )} b\right )} x^{3}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - 6 \, {\left (3 \, b f^{2} m x^{2} e - 2 \, {\left (f^{3} m - 3 \, f^{3} \log \left (d\right )\right )} b x^{3} - 6 \, b f m x e^{2} + 6 \, b m e^{3} \log \left (f x + e\right )\right )} \log \left (x^{n}\right )}{108 \, f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(log(f*x*e^(-1) + 1)*log(x) + dilog(-f*x*e^(-1)))*b*m*n*e^3/f^3 - 1/9*((m*n - 3*m*log(c))*b - 3*a*m)*e^3*l
og(f*x + e)/f^3 - 1/108*(36*b*m*n*e^3*log(f*x + e)*log(x) + 4*(3*(f^3*m - 3*f^3*log(d))*a - (2*f^3*m*n - 3*f^3
*n*log(d) - 3*(f^3*m - 3*f^3*log(d))*log(c))*b)*x^3 - 3*(6*a*f^2*m - (5*f^2*m*n - 6*f^2*m*log(c))*b)*x^2*e + 1
2*(3*a*f*m - (4*f*m*n - 3*f*m*log(c))*b)*x*e^2 - 12*(3*b*f^3*x^3*log(x^n) + (3*a*f^3 - (f^3*n - 3*f^3*log(c))*
b)*x^3)*log((f*x + e)^m) - 6*(3*b*f^2*m*x^2*e - 2*(f^3*m - 3*f^3*log(d))*b*x^3 - 6*b*f*m*x*e^2 + 6*b*m*e^3*log
(f*x + e))*log(x^n))/f^3

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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))*log(d*(f*x+e)^m),x, algorithm="fricas")

[Out]

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

[Out]

integrate((b*log(c*x^n) + a)*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 ) \,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)),x)

[Out]

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

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