3.4.79 \(\int x^2 (a+b \log (c (d+e x)^n)) (f+g \log (c (d+e x)^n)) \, dx\) [379]

Optimal. Leaf size=258 \[ \frac {2 b d^2 g n^2 x}{e^2}-\frac {b d g n^2 (d+e x)^2}{2 e^3}+\frac {2 b g n^2 (d+e x)^3}{27 e^3}-\frac {b d^3 g n^2 \log ^2(d+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {d^2 n (d+e x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{e^3}+\frac {d n (d+e x)^2 \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{2 e^3}-\frac {n (d+e x)^3 \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{9 e^3}+\frac {d^3 n \log (d+e x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{3 e^3} \]

[Out]

2*b*d^2*g*n^2*x/e^2-1/2*b*d*g*n^2*(e*x+d)^2/e^3+2/27*b*g*n^2*(e*x+d)^3/e^3-1/3*b*d^3*g*n^2*ln(e*x+d)^2/e^3+1/3
*x^3*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(c*(e*x+d)^n))-d^2*n*(e*x+d)*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))/e^3+1/2*d*n*(e*
x+d)^2*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))/e^3-1/9*n*(e*x+d)^3*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))/e^3+1/3*d^3*n*ln(e*
x+d)*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))/e^3

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2483, 2458, 45, 2372, 12, 14, 2338} \begin {gather*} \frac {d^3 n \log (d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{3 e^3}-\frac {d^2 n (d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{e^3}+\frac {d n (d+e x)^2 \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{2 e^3}-\frac {n (d+e x)^3 \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{9 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-\frac {b d^3 g n^2 \log ^2(d+e x)}{3 e^3}+\frac {2 b d^2 g n^2 x}{e^2}-\frac {b d g n^2 (d+e x)^2}{2 e^3}+\frac {2 b g n^2 (d+e x)^3}{27 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*d^2*g*n^2*x)/e^2 - (b*d*g*n^2*(d + e*x)^2)/(2*e^3) + (2*b*g*n^2*(d + e*x)^3)/(27*e^3) - (b*d^3*g*n^2*Log[
d + e*x]^2)/(3*e^3) + (x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/3 - (d^2*n*(d + e*x)*(b*f +
a*g + 2*b*g*Log[c*(d + e*x)^n]))/e^3 + (d*n*(d + e*x)^2*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(2*e^3) - (n*(
d + e*x)^3*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(9*e^3) + (d^3*n*Log[d + e*x]*(b*f + a*g + 2*b*g*Log[c*(d +
e*x)^n]))/(3*e^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2483

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

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {1}{3} (b e n) \int \frac {x^3 \left (f+g \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-\frac {1}{3} (e g n) \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {1}{3} (b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (f+g \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )-\frac {1}{3} (g n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )\\ &=-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+2 \left (\frac {1}{3} \left (b g n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e x\right )\right )\\ &=-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+2 \frac {\left (b g n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e x\right )}{18 e^3}\\ &=-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+2 \frac {\left (b g n^2\right ) \text {Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{18 e^3}\\ &=-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+2 \left (\frac {b d^2 g n^2 x}{e^2}-\frac {b d g n^2 (d+e x)^2}{4 e^3}+\frac {b g n^2 (d+e x)^3}{27 e^3}-\frac {\left (b d^3 g n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x\right )}{3 e^3}\right )\\ &=2 \left (\frac {b d^2 g n^2 x}{e^2}-\frac {b d g n^2 (d+e x)^2}{4 e^3}+\frac {b g n^2 (d+e x)^3}{27 e^3}-\frac {b d^3 g n^2 \log ^2(d+e x)}{6 e^3}\right )-\frac {1}{18} g n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {1}{18} b n \left (\frac {18 d^2 (d+e x)}{e^3}-\frac {9 d (d+e x)^2}{e^3}+\frac {2 (d+e x)^3}{e^3}-\frac {6 d^3 \log (d+e x)}{e^3}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.15, size = 226, normalized size = 0.88 \begin {gather*} \frac {-18 b d^3 g n^2 \log ^2(d+e x)+6 d^3 n \log (d+e x) \left (3 b f+3 a g-11 b g n+6 b g \log \left (c (d+e x)^n\right )\right )+e x \left (3 a \left (-6 d^2 g n+3 d e g n x+2 e^2 (3 f-g n) x^2\right )+b n \left (d^2 (-18 f+66 g n)+3 d e (3 f-5 g n) x+2 e^2 (-3 f+2 g n) x^2\right )-6 \left (-3 a e^2 g x^2+b \left (6 d^2 g n-3 d e g n x+e^2 (-3 f+2 g n) x^2\right )\right ) \log \left (c (d+e x)^n\right )+18 b e^2 g x^2 \log ^2\left (c (d+e x)^n\right )\right )}{54 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*b*d^3*g*n^2*Log[d + e*x]^2 + 6*d^3*n*Log[d + e*x]*(3*b*f + 3*a*g - 11*b*g*n + 6*b*g*Log[c*(d + e*x)^n]) +
e*x*(3*a*(-6*d^2*g*n + 3*d*e*g*n*x + 2*e^2*(3*f - g*n)*x^2) + b*n*(d^2*(-18*f + 66*g*n) + 3*d*e*(3*f - 5*g*n)
*x + 2*e^2*(-3*f + 2*g*n)*x^2) - 6*(-3*a*e^2*g*x^2 + b*(6*d^2*g*n - 3*d*e*g*n*x + e^2*(-3*f + 2*g*n)*x^2))*Log
[c*(d + e*x)^n] + 18*b*e^2*g*x^2*Log[c*(d + e*x)^n]^2))/(54*e^3)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.49, size = 1785, normalized size = 6.92

method	result	size
risch	\(\text {Expression too large to display}\)	\(1785\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/9*b*d^2*g*n^2*x/e^2+2/27*b*g*n^2*x^3-1/9*n*a*g*x^3-1/9*n*b*f*x^3-1/3*I/e^3*ln(e*x+d)*Pi*b*d^3*g*n*csgn(I*c*
(e*x+d)^n)^3-1/6*I/e*Pi*b*d*g*n*x^2*csgn(I*c*(e*x+d)^n)^3+1/3*I/e^2*Pi*b*d^2*g*n*x*csgn(I*c*(e*x+d)^n)^3+1/3*x
^3*a*f-1/3*Pi^2*b*g*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4-1/12*Pi^2*b*g*x^3*csgn(I*c)^2*csgn(I
*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+1/6*Pi^2*b*g*x^3*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^3+1/6*P
i^2*b*g*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3+1/3/e*ln(c)*b*d*g*n*x^2-2/3/e^2*ln(c)*b*d^2*g*
n*x+2/3/e^3*ln(c)*ln(e*x+d)*b*d^3*g*n+1/9*I*n*Pi*b*g*x^3*csgn(I*c*(e*x+d)^n)^3+1/6*I*Pi*a*g*x^3*csgn(I*c)*csgn
(I*c*(e*x+d)^n)^2+1/3*I/e^3*ln(e*x+d)*Pi*b*d^3*g*n*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/3*I/e^3*ln(e*x+d)*Pi*b*d^
3*g*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/6*I/e*Pi*b*d*g*n*x^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1
/3*I/e^2*Pi*b*d^2*g*n*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/3*I/e^2*Pi*b*d^2*g*n*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e
*x+d)^n)^2-1/3/e^2*a*d^2*g*n*x+1/3*I*ln(c)*Pi*b*g*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/6*I*Pi*a*g*x^3
*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/6*I*Pi*b*f*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)
^n)+1/3*I*ln(c)*Pi*b*g*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/9*I*n*Pi*b*g*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-
1/9*I*n*Pi*b*g*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/9*(-3*I*Pi*b*e^3*g*x^3*csgn(I*c)*csgn(I*(e*x+d)^n
)*csgn(I*c*(e*x+d)^n)+3*I*Pi*b*e^3*g*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3*I*Pi*b*e^3*g*x^3*csgn(I*(e*x+d)^n)*
csgn(I*c*(e*x+d)^n)^2-3*I*Pi*b*e^3*g*x^3*csgn(I*c*(e*x+d)^n)^3+6*ln(c)*b*e^3*g*x^3-2*b*e^3*g*n*x^3+3*a*e^3*g*x
^3+3*b*d*e^2*g*n*x^2+3*b*e^3*f*x^3+6*ln(e*x+d)*b*d^3*g*n-6*b*d^2*e*g*n*x)/e^3*ln((e*x+d)^n)+1/3*x^3*b*g*ln((e*
x+d)^n)^2+1/6*I/e*Pi*b*d*g*n*x^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/12*Pi^2*b*g*x^3*csgn(I*c)^2*csgn(I*c*(e*x+d
)^n)^4+1/6*Pi^2*b*g*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^5-1/6*I*Pi*a*g*x^3*csgn(I*c*(e*x+d)^n)^3-1/6*I*Pi*b*f*x^
3*csgn(I*c*(e*x+d)^n)^3+1/6/e*a*d*g*n*x^2+1/6/e*b*d*f*n*x^2-5/18/e*b*d*g*n^2*x^2+1/3/e^3*ln(e*x+d)*a*d^3*g*n+1
/9*I*n*Pi*b*g*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/3*I*ln(c)*Pi*b*g*x^3*csgn(I*c)*csgn(I*(e*x
+d)^n)*csgn(I*c*(e*x+d)^n)-2/9*n*ln(c)*b*g*x^3-1/12*Pi^2*b*g*x^3*csgn(I*c*(e*x+d)^n)^6+1/3/e^3*ln(e*x+d)*b*d^3
*f*n-1/12*Pi^2*b*g*x^3*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+1/6*Pi^2*b*g*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(
e*x+d)^n)^5-1/6*I/e*Pi*b*d*g*n*x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/3*I/e^2*Pi*b*d^2*g*n*x*cs
gn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/3*I/e^3*ln(e*x+d)*Pi*b*d^3*g*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csg
n(I*c*(e*x+d)^n)+1/6*I*Pi*b*f*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/6*I*Pi*b*f*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*
(e*x+d)^n)^2+1/6*I*Pi*a*g*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/3*I*ln(c)*Pi*b*g*x^3*csgn(I*c*(e*x+d)^
n)^3-11/9*b*d^3*g*n^2/e^3*ln(e*x+d)+1/3*ln(c)^2*b*g*x^3+1/3*ln(c)*a*g*x^3+1/3*ln(c)*b*f*x^3-1/3*b*d^3*g*n^2*ln
(e*x+d)^2/e^3-1/3*b*d^2*f*n*x/e^2

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 278, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, b g x^{3} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + \frac {1}{3} \, b f x^{3} \log \left ({\left (x e + d\right )}^{n} c\right ) + \frac {1}{3} \, a g x^{3} \log \left ({\left (x e + d\right )}^{n} c\right ) + \frac {1}{3} \, a f x^{3} + \frac {1}{18} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} b f n e + \frac {1}{18} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} a g n e - \frac {1}{54} \, {\left ({\left (18 \, d^{3} \log \left (x e + d\right )^{2} - 4 \, x^{3} e^{3} + 15 \, d x^{2} e^{2} - 66 \, d^{2} x e + 66 \, d^{3} \log \left (x e + d\right )\right )} n^{2} e^{\left (-3\right )} - 6 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x e + d\right ) - {\left (2 \, x^{3} e^{2} - 3 \, d x^{2} e + 6 \, d^{2} x\right )} e^{\left (-3\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} b g \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*g*x^3*log((x*e + d)^n*c)^2 + 1/3*b*f*x^3*log((x*e + d)^n*c) + 1/3*a*g*x^3*log((x*e + d)^n*c) + 1/3*a*f*x
^3 + 1/18*(6*d^3*e^(-4)*log(x*e + d) - (2*x^3*e^2 - 3*d*x^2*e + 6*d^2*x)*e^(-3))*b*f*n*e + 1/18*(6*d^3*e^(-4)*
log(x*e + d) - (2*x^3*e^2 - 3*d*x^2*e + 6*d^2*x)*e^(-3))*a*g*n*e - 1/54*((18*d^3*log(x*e + d)^2 - 4*x^3*e^3 +
15*d*x^2*e^2 - 66*d^2*x*e + 66*d^3*log(x*e + d))*n^2*e^(-3) - 6*(6*d^3*e^(-4)*log(x*e + d) - (2*x^3*e^2 - 3*d*
x^2*e + 6*d^2*x)*e^(-3))*n*e*log((x*e + d)^n*c))*b*g

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 295, normalized size = 1.14 \begin {gather*} \frac {1}{54} \, {\left (18 \, b g x^{3} e^{3} \log \left (c\right )^{2} + 2 \, {\left (2 \, b g n^{2} + 9 \, a f - 3 \, {\left (b f + a g\right )} n\right )} x^{3} e^{3} - 3 \, {\left (5 \, b d g n^{2} - 3 \, {\left (b d f + a d g\right )} n\right )} x^{2} e^{2} + 6 \, {\left (11 \, b d^{2} g n^{2} - 3 \, {\left (b d^{2} f + a d^{2} g\right )} n\right )} x e + 18 \, {\left (b g n^{2} x^{3} e^{3} + b d^{3} g n^{2}\right )} \log \left (x e + d\right )^{2} + 6 \, {\left (3 \, b d g n^{2} x^{2} e^{2} - 6 \, b d^{2} g n^{2} x e - 11 \, b d^{3} g n^{2} - {\left (2 \, b g n^{2} - 3 \, {\left (b f + a g\right )} n\right )} x^{3} e^{3} + 3 \, {\left (b d^{3} f + a d^{3} g\right )} n + 6 \, {\left (b g n x^{3} e^{3} + b d^{3} g n\right )} \log \left (c\right )\right )} \log \left (x e + d\right ) + 6 \, {\left (3 \, b d g n x^{2} e^{2} - 6 \, b d^{2} g n x e - {\left (2 \, b g n - 3 \, b f - 3 \, a g\right )} x^{3} e^{3}\right )} \log \left (c\right )\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(18*b*g*x^3*e^3*log(c)^2 + 2*(2*b*g*n^2 + 9*a*f - 3*(b*f + a*g)*n)*x^3*e^3 - 3*(5*b*d*g*n^2 - 3*(b*d*f +
a*d*g)*n)*x^2*e^2 + 6*(11*b*d^2*g*n^2 - 3*(b*d^2*f + a*d^2*g)*n)*x*e + 18*(b*g*n^2*x^3*e^3 + b*d^3*g*n^2)*log(
x*e + d)^2 + 6*(3*b*d*g*n^2*x^2*e^2 - 6*b*d^2*g*n^2*x*e - 11*b*d^3*g*n^2 - (2*b*g*n^2 - 3*(b*f + a*g)*n)*x^3*e
^3 + 3*(b*d^3*f + a*d^3*g)*n + 6*(b*g*n*x^3*e^3 + b*d^3*g*n)*log(c))*log(x*e + d) + 6*(3*b*d*g*n*x^2*e^2 - 6*b
*d^2*g*n*x*e - (2*b*g*n - 3*b*f - 3*a*g)*x^3*e^3)*log(c))*e^(-3)

________________________________________________________________________________________

Sympy [A]
time = 1.20, size = 384, normalized size = 1.49 \begin {gather*} \begin {cases} \frac {a d^{3} g \log {\left (c \left (d + e x\right )^{n} \right )}}{3 e^{3}} - \frac {a d^{2} g n x}{3 e^{2}} + \frac {a d g n x^{2}}{6 e} + \frac {a f x^{3}}{3} - \frac {a g n x^{3}}{9} + \frac {a g x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{3} + \frac {b d^{3} f \log {\left (c \left (d + e x\right )^{n} \right )}}{3 e^{3}} - \frac {11 b d^{3} g n \log {\left (c \left (d + e x\right )^{n} \right )}}{9 e^{3}} + \frac {b d^{3} g \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{3 e^{3}} - \frac {b d^{2} f n x}{3 e^{2}} + \frac {11 b d^{2} g n^{2} x}{9 e^{2}} - \frac {2 b d^{2} g n x \log {\left (c \left (d + e x\right )^{n} \right )}}{3 e^{2}} + \frac {b d f n x^{2}}{6 e} - \frac {5 b d g n^{2} x^{2}}{18 e} + \frac {b d g n x^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{3 e} - \frac {b f n x^{3}}{9} + \frac {b f x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{3} + \frac {2 b g n^{2} x^{3}}{27} - \frac {2 b g n x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{9} + \frac {b g x^{3} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{3} & \text {for}\: e \neq 0 \\\frac {x^{3} \left (a + b \log {\left (c d^{n} \right )}\right ) \left (f + g \log {\left (c d^{n} \right )}\right )}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*(e*x+d)**n))*(f+g*ln(c*(e*x+d)**n)),x)

[Out]

Piecewise((a*d**3*g*log(c*(d + e*x)**n)/(3*e**3) - a*d**2*g*n*x/(3*e**2) + a*d*g*n*x**2/(6*e) + a*f*x**3/3 - a
*g*n*x**3/9 + a*g*x**3*log(c*(d + e*x)**n)/3 + b*d**3*f*log(c*(d + e*x)**n)/(3*e**3) - 11*b*d**3*g*n*log(c*(d
+ e*x)**n)/(9*e**3) + b*d**3*g*log(c*(d + e*x)**n)**2/(3*e**3) - b*d**2*f*n*x/(3*e**2) + 11*b*d**2*g*n**2*x/(9
*e**2) - 2*b*d**2*g*n*x*log(c*(d + e*x)**n)/(3*e**2) + b*d*f*n*x**2/(6*e) - 5*b*d*g*n**2*x**2/(18*e) + b*d*g*n
*x**2*log(c*(d + e*x)**n)/(3*e) - b*f*n*x**3/9 + b*f*x**3*log(c*(d + e*x)**n)/3 + 2*b*g*n**2*x**3/27 - 2*b*g*n
*x**3*log(c*(d + e*x)**n)/9 + b*g*x**3*log(c*(d + e*x)**n)**2/3, Ne(e, 0)), (x**3*(a + b*log(c*d**n))*(f + g*l
og(c*d**n))/3, True))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 756 vs. \(2 (249) = 498\).
time = 3.62, size = 756, normalized size = 2.93 \begin {gather*} \frac {1}{3} \, {\left (x e + d\right )}^{3} b g n^{2} e^{\left (-3\right )} \log \left (x e + d\right )^{2} - {\left (x e + d\right )}^{2} b d g n^{2} e^{\left (-3\right )} \log \left (x e + d\right )^{2} + {\left (x e + d\right )} b d^{2} g n^{2} e^{\left (-3\right )} \log \left (x e + d\right )^{2} - \frac {2}{9} \, {\left (x e + d\right )}^{3} b g n^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x e + d\right )}^{2} b d g n^{2} e^{\left (-3\right )} \log \left (x e + d\right ) - 2 \, {\left (x e + d\right )} b d^{2} g n^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {2}{3} \, {\left (x e + d\right )}^{3} b g n e^{\left (-3\right )} \log \left (x e + d\right ) \log \left (c\right ) - 2 \, {\left (x e + d\right )}^{2} b d g n e^{\left (-3\right )} \log \left (x e + d\right ) \log \left (c\right ) + 2 \, {\left (x e + d\right )} b d^{2} g n e^{\left (-3\right )} \log \left (x e + d\right ) \log \left (c\right ) + \frac {2}{27} \, {\left (x e + d\right )}^{3} b g n^{2} e^{\left (-3\right )} - \frac {1}{2} \, {\left (x e + d\right )}^{2} b d g n^{2} e^{\left (-3\right )} + 2 \, {\left (x e + d\right )} b d^{2} g n^{2} e^{\left (-3\right )} + \frac {1}{3} \, {\left (x e + d\right )}^{3} b f n e^{\left (-3\right )} \log \left (x e + d\right ) - {\left (x e + d\right )}^{2} b d f n e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x e + d\right )} b d^{2} f n e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {1}{3} \, {\left (x e + d\right )}^{3} a g n e^{\left (-3\right )} \log \left (x e + d\right ) - {\left (x e + d\right )}^{2} a d g n e^{\left (-3\right )} \log \left (x e + d\right ) + {\left (x e + d\right )} a d^{2} g n e^{\left (-3\right )} \log \left (x e + d\right ) - \frac {2}{9} \, {\left (x e + d\right )}^{3} b g n e^{\left (-3\right )} \log \left (c\right ) + {\left (x e + d\right )}^{2} b d g n e^{\left (-3\right )} \log \left (c\right ) - 2 \, {\left (x e + d\right )} b d^{2} g n e^{\left (-3\right )} \log \left (c\right ) + \frac {1}{3} \, {\left (x e + d\right )}^{3} b g e^{\left (-3\right )} \log \left (c\right )^{2} - {\left (x e + d\right )}^{2} b d g e^{\left (-3\right )} \log \left (c\right )^{2} + {\left (x e + d\right )} b d^{2} g e^{\left (-3\right )} \log \left (c\right )^{2} - \frac {1}{9} \, {\left (x e + d\right )}^{3} b f n e^{\left (-3\right )} + \frac {1}{2} \, {\left (x e + d\right )}^{2} b d f n e^{\left (-3\right )} - {\left (x e + d\right )} b d^{2} f n e^{\left (-3\right )} - \frac {1}{9} \, {\left (x e + d\right )}^{3} a g n e^{\left (-3\right )} + \frac {1}{2} \, {\left (x e + d\right )}^{2} a d g n e^{\left (-3\right )} - {\left (x e + d\right )} a d^{2} g n e^{\left (-3\right )} + \frac {1}{3} \, {\left (x e + d\right )}^{3} b f e^{\left (-3\right )} \log \left (c\right ) - {\left (x e + d\right )}^{2} b d f e^{\left (-3\right )} \log \left (c\right ) + {\left (x e + d\right )} b d^{2} f e^{\left (-3\right )} \log \left (c\right ) + \frac {1}{3} \, {\left (x e + d\right )}^{3} a g e^{\left (-3\right )} \log \left (c\right ) - {\left (x e + d\right )}^{2} a d g e^{\left (-3\right )} \log \left (c\right ) + {\left (x e + d\right )} a d^{2} g e^{\left (-3\right )} \log \left (c\right ) + \frac {1}{3} \, {\left (x e + d\right )}^{3} a f e^{\left (-3\right )} - {\left (x e + d\right )}^{2} a d f e^{\left (-3\right )} + {\left (x e + d\right )} a d^{2} f e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x*e + d)^3*b*g*n^2*e^(-3)*log(x*e + d)^2 - (x*e + d)^2*b*d*g*n^2*e^(-3)*log(x*e + d)^2 + (x*e + d)*b*d^2*
g*n^2*e^(-3)*log(x*e + d)^2 - 2/9*(x*e + d)^3*b*g*n^2*e^(-3)*log(x*e + d) + (x*e + d)^2*b*d*g*n^2*e^(-3)*log(x
*e + d) - 2*(x*e + d)*b*d^2*g*n^2*e^(-3)*log(x*e + d) + 2/3*(x*e + d)^3*b*g*n*e^(-3)*log(x*e + d)*log(c) - 2*(
x*e + d)^2*b*d*g*n*e^(-3)*log(x*e + d)*log(c) + 2*(x*e + d)*b*d^2*g*n*e^(-3)*log(x*e + d)*log(c) + 2/27*(x*e +
d)^3*b*g*n^2*e^(-3) - 1/2*(x*e + d)^2*b*d*g*n^2*e^(-3) + 2*(x*e + d)*b*d^2*g*n^2*e^(-3) + 1/3*(x*e + d)^3*b*f
*n*e^(-3)*log(x*e + d) - (x*e + d)^2*b*d*f*n*e^(-3)*log(x*e + d) + (x*e + d)*b*d^2*f*n*e^(-3)*log(x*e + d) + 1
/3*(x*e + d)^3*a*g*n*e^(-3)*log(x*e + d) - (x*e + d)^2*a*d*g*n*e^(-3)*log(x*e + d) + (x*e + d)*a*d^2*g*n*e^(-3
)*log(x*e + d) - 2/9*(x*e + d)^3*b*g*n*e^(-3)*log(c) + (x*e + d)^2*b*d*g*n*e^(-3)*log(c) - 2*(x*e + d)*b*d^2*g
*n*e^(-3)*log(c) + 1/3*(x*e + d)^3*b*g*e^(-3)*log(c)^2 - (x*e + d)^2*b*d*g*e^(-3)*log(c)^2 + (x*e + d)*b*d^2*g
*e^(-3)*log(c)^2 - 1/9*(x*e + d)^3*b*f*n*e^(-3) + 1/2*(x*e + d)^2*b*d*f*n*e^(-3) - (x*e + d)*b*d^2*f*n*e^(-3)
- 1/9*(x*e + d)^3*a*g*n*e^(-3) + 1/2*(x*e + d)^2*a*d*g*n*e^(-3) - (x*e + d)*a*d^2*g*n*e^(-3) + 1/3*(x*e + d)^3
*b*f*e^(-3)*log(c) - (x*e + d)^2*b*d*f*e^(-3)*log(c) + (x*e + d)*b*d^2*f*e^(-3)*log(c) + 1/3*(x*e + d)^3*a*g*e
^(-3)*log(c) - (x*e + d)^2*a*d*g*e^(-3)*log(c) + (x*e + d)*a*d^2*g*e^(-3)*log(c) + 1/3*(x*e + d)^3*a*f*e^(-3)
- (x*e + d)^2*a*d*f*e^(-3) + (x*e + d)*a*d^2*f*e^(-3)

________________________________________________________________________________________

Mupad [B]
time = 0.40, size = 323, normalized size = 1.25 \begin {gather*} \ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {x^3\,\left (a\,g+b\,f-\frac {2\,b\,g\,n}{3}\right )}{3}+\frac {x^2\,\left (\frac {3\,d\,\left (a\,g+b\,f\right )}{2\,e}-\frac {d\,\left (9\,a\,g+9\,b\,f-6\,b\,g\,n\right )}{6\,e}\right )}{3}-\frac {d\,x\,\left (\frac {9\,d\,\left (a\,g+b\,f\right )}{e}-\frac {d\,\left (9\,a\,g+9\,b\,f-6\,b\,g\,n\right )}{e}\right )}{9\,e}\right )+x^2\,\left (\frac {d\,\left (3\,a\,f-b\,g\,n^2\right )}{6\,e}-\frac {d\,\left (a\,f-\frac {a\,g\,n}{3}-\frac {b\,f\,n}{3}+\frac {2\,b\,g\,n^2}{9}\right )}{2\,e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {b\,g\,x^3}{3}+\frac {b\,d^3\,g}{3\,e^3}\right )-x\,\left (\frac {d\,\left (\frac {d\,\left (3\,a\,f-b\,g\,n^2\right )}{3\,e}-\frac {d\,\left (a\,f-\frac {a\,g\,n}{3}-\frac {b\,f\,n}{3}+\frac {2\,b\,g\,n^2}{9}\right )}{e}\right )}{e}-\frac {2\,b\,d^2\,g\,n^2}{3\,e^2}\right )+x^3\,\left (\frac {a\,f}{3}-\frac {a\,g\,n}{9}-\frac {b\,f\,n}{9}+\frac {2\,b\,g\,n^2}{27}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,a\,d^3\,g\,n+3\,b\,d^3\,f\,n-11\,b\,d^3\,g\,n^2\right )}{9\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*(d + e*x)^n)*((x^3*(a*g + b*f - (2*b*g*n)/3))/3 + (x^2*((3*d*(a*g + b*f))/(2*e) - (d*(9*a*g + 9*b*f - 6*
b*g*n))/(6*e)))/3 - (d*x*((9*d*(a*g + b*f))/e - (d*(9*a*g + 9*b*f - 6*b*g*n))/e))/(9*e)) + x^2*((d*(3*a*f - b*
g*n^2))/(6*e) - (d*(a*f - (a*g*n)/3 - (b*f*n)/3 + (2*b*g*n^2)/9))/(2*e)) + log(c*(d + e*x)^n)^2*((b*g*x^3)/3 +
(b*d^3*g)/(3*e^3)) - x*((d*((d*(3*a*f - b*g*n^2))/(3*e) - (d*(a*f - (a*g*n)/3 - (b*f*n)/3 + (2*b*g*n^2)/9))/e
))/e - (2*b*d^2*g*n^2)/(3*e^2)) + x^3*((a*f)/3 - (a*g*n)/9 - (b*f*n)/9 + (2*b*g*n^2)/27) + (log(d + e*x)*(3*a*
d^3*g*n + 3*b*d^3*f*n - 11*b*d^3*g*n^2))/(9*e^3)

________________________________________________________________________________________