\(\int x (a+b \log (c (d+e \sqrt [3]{x})^n))^2 \, dx\) [451]

Optimal result

Integrand size = 22, antiderivative size = 480 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac {6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6}-\frac {6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac {6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac {6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \]

[Out]

15/4*b^2*d^4*n^2*(d+e*x^(1/3))^2/e^6-20/9*b^2*d^3*n^2*(d+e*x^(1/3))^3/e^6+15/16*b^2*d^2*n^2*(d+e*x^(1/3))^4/e^
6-6/25*b^2*d*n^2*(d+e*x^(1/3))^5/e^6+1/36*b^2*n^2*(d+e*x^(1/3))^6/e^6-6*b^2*d^5*n^2*x^(1/3)/e^5+1/2*b^2*d^6*n^
2*ln(d+e*x^(1/3))^2/e^6+6*b*d^5*n*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/e^6-15/2*b*d^4*n*(d+e*x^(1/3))^2*(
a+b*ln(c*(d+e*x^(1/3))^n))/e^6+20/3*b*d^3*n*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/3))^n))/e^6-15/4*b*d^2*n*(d+e*
x^(1/3))^4*(a+b*ln(c*(d+e*x^(1/3))^n))/e^6+6/5*b*d*n*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3))^n))/e^6-1/6*b*n*(
d+e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))/e^6-b*d^6*n*ln(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/e^6+1/2*x^2
*(a+b*ln(c*(d+e*x^(1/3))^n))^2

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2504, 2445, 2458, 45, 2372, 12, 14, 2338} \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac {6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac {6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}-\frac {6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac {6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6} \]

[In]

Int[x*(a + b*Log[c*(d + e*x^(1/3))^n])^2,x]

[Out]

(15*b^2*d^4*n^2*(d + e*x^(1/3))^2)/(4*e^6) - (20*b^2*d^3*n^2*(d + e*x^(1/3))^3)/(9*e^6) + (15*b^2*d^2*n^2*(d +
e*x^(1/3))^4)/(16*e^6) - (6*b^2*d*n^2*(d + e*x^(1/3))^5)/(25*e^6) + (b^2*n^2*(d + e*x^(1/3))^6)/(36*e^6) - (6
*b^2*d^5*n^2*x^(1/3))/e^5 + (b^2*d^6*n^2*Log[d + e*x^(1/3)]^2)/(2*e^6) + (6*b*d^5*n*(d + e*x^(1/3))*(a + b*Log
[c*(d + e*x^(1/3))^n]))/e^6 - (15*b*d^4*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*e^6) + (20*b*
d^3*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^6) - (15*b*d^2*n*(d + e*x^(1/3))^4*(a + b*Log[c
*(d + e*x^(1/3))^n]))/(4*e^6) + (6*b*d*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(5*e^6) - (b*n*(d
+ e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(6*e^6) - (b*d^6*n*Log[d + e*x^(1/3)]*(a + b*Log[c*(d + e*x^
(1/3))^n]))/e^6 + (x^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/2

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 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 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 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(b e n) \text {Subst}\left (\int \frac {x^6 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^6 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right ) \\ & = \frac {6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac {6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{60 e^6 x} \, dx,x,d+e \sqrt [3]{x}\right ) \\ & = \frac {6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac {6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{60 e^6} \\ & = \frac {6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac {6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5+\frac {60 d^6 \log (x)}{x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{60 e^6} \\ & = \frac {15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac {6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6}-\frac {6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac {6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 d^6 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{e^6} \\ & = \frac {15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac {6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6}-\frac {6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac {6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac {6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}-\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.66 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {e \sqrt [3]{x} \left (1800 a^2 e^5 x^{5/3}+60 a b n \left (60 d^5-30 d^4 e \sqrt [3]{x}+20 d^3 e^2 x^{2/3}-15 d^2 e^3 x+12 d e^4 x^{4/3}-10 e^5 x^{5/3}\right )+b^2 n^2 \left (-8820 d^5+2610 d^4 e \sqrt [3]{x}-1140 d^3 e^2 x^{2/3}+555 d^2 e^3 x-264 d e^4 x^{4/3}+100 e^5 x^{5/3}\right )\right )+180 b d^6 n (-20 a+49 b n) \log \left (d+e \sqrt [3]{x}\right )-60 b e \sqrt [3]{x} \left (-60 a e^5 x^{5/3}+b n \left (-60 d^5+30 d^4 e \sqrt [3]{x}-20 d^3 e^2 x^{2/3}+15 d^2 e^3 x-12 d e^4 x^{4/3}+10 e^5 x^{5/3}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-1800 b^2 \left (d^6-e^6 x^2\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3600 e^6} \]

[In]

Integrate[x*(a + b*Log[c*(d + e*x^(1/3))^n])^2,x]

[Out]

(e*x^(1/3)*(1800*a^2*e^5*x^(5/3) + 60*a*b*n*(60*d^5 - 30*d^4*e*x^(1/3) + 20*d^3*e^2*x^(2/3) - 15*d^2*e^3*x + 1
2*d*e^4*x^(4/3) - 10*e^5*x^(5/3)) + b^2*n^2*(-8820*d^5 + 2610*d^4*e*x^(1/3) - 1140*d^3*e^2*x^(2/3) + 555*d^2*e
^3*x - 264*d*e^4*x^(4/3) + 100*e^5*x^(5/3))) + 180*b*d^6*n*(-20*a + 49*b*n)*Log[d + e*x^(1/3)] - 60*b*e*x^(1/3
)*(-60*a*e^5*x^(5/3) + b*n*(-60*d^5 + 30*d^4*e*x^(1/3) - 20*d^3*e^2*x^(2/3) + 15*d^2*e^3*x - 12*d*e^4*x^(4/3)
+ 10*e^5*x^(5/3)))*Log[c*(d + e*x^(1/3))^n] - 1800*b^2*(d^6 - e^6*x^2)*Log[c*(d + e*x^(1/3))^n]^2)/(3600*e^6)

Maple [F]

\[\int x {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{2}d x\]

[In]

int(x*(a+b*ln(c*(d+e*x^(1/3))^n))^2,x)

[Out]

int(x*(a+b*ln(c*(d+e*x^(1/3))^n))^2,x)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.01 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {1800 \, b^{2} e^{6} x^{2} \log \left (c\right )^{2} + 100 \, {\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n + 18 \, a^{2} e^{6}\right )} x^{2} + 1800 \, {\left (b^{2} e^{6} n^{2} x^{2} - b^{2} d^{6} n^{2}\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 60 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x + 60 \, {\left (20 \, b^{2} d^{3} e^{3} n^{2} x + 147 \, b^{2} d^{6} n^{2} - 60 \, a b d^{6} n - 10 \, {\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n\right )} x^{2} + 60 \, {\left (b^{2} e^{6} n x^{2} - b^{2} d^{6} n\right )} \log \left (c\right ) + 6 \, {\left (2 \, b^{2} d e^{5} n^{2} x - 5 \, b^{2} d^{4} e^{2} n^{2}\right )} x^{\frac {2}{3}} - 15 \, {\left (b^{2} d^{2} e^{4} n^{2} x - 4 \, b^{2} d^{5} e n^{2}\right )} x^{\frac {1}{3}}\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 600 \, {\left (2 \, b^{2} d^{3} e^{3} n x - {\left (b^{2} e^{6} n - 6 \, a b e^{6}\right )} x^{2}\right )} \log \left (c\right ) + 6 \, {\left (435 \, b^{2} d^{4} e^{2} n^{2} - 300 \, a b d^{4} e^{2} n - 4 \, {\left (11 \, b^{2} d e^{5} n^{2} - 30 \, a b d e^{5} n\right )} x + 60 \, {\left (2 \, b^{2} d e^{5} n x - 5 \, b^{2} d^{4} e^{2} n\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} - 15 \, {\left (588 \, b^{2} d^{5} e n^{2} - 240 \, a b d^{5} e n - {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x + 60 \, {\left (b^{2} d^{2} e^{4} n x - 4 \, b^{2} d^{5} e n\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}}{3600 \, e^{6}} \]

[In]

integrate(x*(a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="fricas")

[Out]

1/3600*(1800*b^2*e^6*x^2*log(c)^2 + 100*(b^2*e^6*n^2 - 6*a*b*e^6*n + 18*a^2*e^6)*x^2 + 1800*(b^2*e^6*n^2*x^2 -
b^2*d^6*n^2)*log(e*x^(1/3) + d)^2 - 60*(19*b^2*d^3*e^3*n^2 - 20*a*b*d^3*e^3*n)*x + 60*(20*b^2*d^3*e^3*n^2*x +
147*b^2*d^6*n^2 - 60*a*b*d^6*n - 10*(b^2*e^6*n^2 - 6*a*b*e^6*n)*x^2 + 60*(b^2*e^6*n*x^2 - b^2*d^6*n)*log(c) +
6*(2*b^2*d*e^5*n^2*x - 5*b^2*d^4*e^2*n^2)*x^(2/3) - 15*(b^2*d^2*e^4*n^2*x - 4*b^2*d^5*e*n^2)*x^(1/3))*log(e*x
^(1/3) + d) + 600*(2*b^2*d^3*e^3*n*x - (b^2*e^6*n - 6*a*b*e^6)*x^2)*log(c) + 6*(435*b^2*d^4*e^2*n^2 - 300*a*b*
d^4*e^2*n - 4*(11*b^2*d*e^5*n^2 - 30*a*b*d*e^5*n)*x + 60*(2*b^2*d*e^5*n*x - 5*b^2*d^4*e^2*n)*log(c))*x^(2/3) -
15*(588*b^2*d^5*e*n^2 - 240*a*b*d^5*e*n - (37*b^2*d^2*e^4*n^2 - 60*a*b*d^2*e^4*n)*x + 60*(b^2*d^2*e^4*n*x - 4
*b^2*d^5*e*n)*log(c))*x^(1/3))/e^6

Sympy [F]

\[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\int x \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}\, dx \]

[In]

integrate(x*(a+b*ln(c*(d+e*x**(1/3))**n))**2,x)

[Out]

Integral(x*(a + b*log(c*(d + e*x**(1/3))**n))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.67 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} - \frac {1}{60} \, a b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} + a b x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{2} x^{2} - \frac {1}{3600} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 264 \, d e^{5} x^{\frac {5}{3}} + 555 \, d^{2} e^{4} x^{\frac {4}{3}} - 1140 \, d^{3} e^{3} x + 8820 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right ) + 2610 \, d^{4} e^{2} x^{\frac {2}{3}} - 8820 \, d^{5} e x^{\frac {1}{3}}\right )} n^{2}}{e^{6}}\right )} b^{2} \]

[In]

integrate(x*(a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="maxima")

[Out]

1/2*b^2*x^2*log((e*x^(1/3) + d)^n*c)^2 - 1/60*a*b*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*
x^(5/3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60*d^5*x^(1/3))/e^6) + a*b*x^2*log((e*x^(1/3)
+ d)^n*c) + 1/2*a^2*x^2 - 1/3600*(60*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*x^(5/3) + 15
*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60*d^5*x^(1/3))/e^6)*log((e*x^(1/3) + d)^n*c) - (100*e^6*
x^2 + 1800*d^6*log(e*x^(1/3) + d)^2 - 264*d*e^5*x^(5/3) + 555*d^2*e^4*x^(4/3) - 1140*d^3*e^3*x + 8820*d^6*log(
e*x^(1/3) + d) + 2610*d^4*e^2*x^(2/3) - 8820*d^5*e*x^(1/3))*n^2/e^6)*b^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 930 vs. \(2 (412) = 824\).

Time = 0.32 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.94 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate(x*(a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="giac")

[Out]

1/3600*(1800*b^2*e*x^2*log(c)^2 + 3600*a*b*e*x^2*log(c) + (1800*(e*x^(1/3) + d)^6*log(e*x^(1/3) + d)^2/e^5 - 1
0800*(e*x^(1/3) + d)^5*d*log(e*x^(1/3) + d)^2/e^5 + 27000*(e*x^(1/3) + d)^4*d^2*log(e*x^(1/3) + d)^2/e^5 - 360
00*(e*x^(1/3) + d)^3*d^3*log(e*x^(1/3) + d)^2/e^5 + 27000*(e*x^(1/3) + d)^2*d^4*log(e*x^(1/3) + d)^2/e^5 - 108
00*(e*x^(1/3) + d)*d^5*log(e*x^(1/3) + d)^2/e^5 - 600*(e*x^(1/3) + d)^6*log(e*x^(1/3) + d)/e^5 + 4320*(e*x^(1/
3) + d)^5*d*log(e*x^(1/3) + d)/e^5 - 13500*(e*x^(1/3) + d)^4*d^2*log(e*x^(1/3) + d)/e^5 + 24000*(e*x^(1/3) + d
)^3*d^3*log(e*x^(1/3) + d)/e^5 - 27000*(e*x^(1/3) + d)^2*d^4*log(e*x^(1/3) + d)/e^5 + 21600*(e*x^(1/3) + d)*d^
5*log(e*x^(1/3) + d)/e^5 + 100*(e*x^(1/3) + d)^6/e^5 - 864*(e*x^(1/3) + d)^5*d/e^5 + 3375*(e*x^(1/3) + d)^4*d^
2/e^5 - 8000*(e*x^(1/3) + d)^3*d^3/e^5 + 13500*(e*x^(1/3) + d)^2*d^4/e^5 - 21600*(e*x^(1/3) + d)*d^5/e^5)*b^2*
n^2 + 1800*a^2*e*x^2 + 60*(60*(e*x^(1/3) + d)^6*log(e*x^(1/3) + d)/e^5 - 360*(e*x^(1/3) + d)^5*d*log(e*x^(1/3)
+ d)/e^5 + 900*(e*x^(1/3) + d)^4*d^2*log(e*x^(1/3) + d)/e^5 - 1200*(e*x^(1/3) + d)^3*d^3*log(e*x^(1/3) + d)/e
^5 + 900*(e*x^(1/3) + d)^2*d^4*log(e*x^(1/3) + d)/e^5 - 360*(e*x^(1/3) + d)*d^5*log(e*x^(1/3) + d)/e^5 - 10*(e
*x^(1/3) + d)^6/e^5 + 72*(e*x^(1/3) + d)^5*d/e^5 - 225*(e*x^(1/3) + d)^4*d^2/e^5 + 400*(e*x^(1/3) + d)^3*d^3/e
^5 - 450*(e*x^(1/3) + d)^2*d^4/e^5 + 360*(e*x^(1/3) + d)*d^5/e^5)*b^2*n*log(c) + 60*(60*(e*x^(1/3) + d)^6*log(
e*x^(1/3) + d)/e^5 - 360*(e*x^(1/3) + d)^5*d*log(e*x^(1/3) + d)/e^5 + 900*(e*x^(1/3) + d)^4*d^2*log(e*x^(1/3)
+ d)/e^5 - 1200*(e*x^(1/3) + d)^3*d^3*log(e*x^(1/3) + d)/e^5 + 900*(e*x^(1/3) + d)^2*d^4*log(e*x^(1/3) + d)/e^
5 - 360*(e*x^(1/3) + d)*d^5*log(e*x^(1/3) + d)/e^5 - 10*(e*x^(1/3) + d)^6/e^5 + 72*(e*x^(1/3) + d)^5*d/e^5 - 2
25*(e*x^(1/3) + d)^4*d^2/e^5 + 400*(e*x^(1/3) + d)^3*d^3/e^5 - 450*(e*x^(1/3) + d)^2*d^4/e^5 + 360*(e*x^(1/3)
+ d)*d^5/e^5)*a*b*n)/e

Mupad [B] (verification not implemented)

Time = 3.01 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.90 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {a^2\,x^2}{2}+\frac {b^2\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2}+\frac {b^2\,n^2\,x^2}{36}+a\,b\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )-\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^6}-\frac {a\,b\,n\,x^2}{6}-\frac {b^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{6}+\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+e\,x^{1/3}\right )}{20\,e^6}+\frac {37\,b^2\,d^2\,n^2\,x^{4/3}}{240\,e^2}+\frac {29\,b^2\,d^4\,n^2\,x^{2/3}}{40\,e^4}-\frac {49\,b^2\,d^5\,n^2\,x^{1/3}}{20\,e^5}-\frac {19\,b^2\,d^3\,n^2\,x}{60\,e^3}-\frac {11\,b^2\,d\,n^2\,x^{5/3}}{150\,e}-\frac {b^2\,d^2\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{4\,e^2}-\frac {b^2\,d^4\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2\,e^4}+\frac {b^2\,d^5\,n\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^5}+\frac {a\,b\,d^3\,n\,x}{3\,e^3}+\frac {a\,b\,d\,n\,x^{5/3}}{5\,e}-\frac {a\,b\,d^6\,n\,\ln \left (d+e\,x^{1/3}\right )}{e^6}+\frac {b^2\,d^3\,n\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^3}+\frac {b^2\,d\,n\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{5\,e}-\frac {a\,b\,d^2\,n\,x^{4/3}}{4\,e^2}-\frac {a\,b\,d^4\,n\,x^{2/3}}{2\,e^4}+\frac {a\,b\,d^5\,n\,x^{1/3}}{e^5} \]

[In]

int(x*(a + b*log(c*(d + e*x^(1/3))^n))^2,x)

[Out]

(a^2*x^2)/2 + (b^2*x^2*log(c*(d + e*x^(1/3))^n)^2)/2 + (b^2*n^2*x^2)/36 + a*b*x^2*log(c*(d + e*x^(1/3))^n) - (
b^2*d^6*log(c*(d + e*x^(1/3))^n)^2)/(2*e^6) - (a*b*n*x^2)/6 - (b^2*n*x^2*log(c*(d + e*x^(1/3))^n))/6 + (49*b^2
*d^6*n^2*log(d + e*x^(1/3)))/(20*e^6) + (37*b^2*d^2*n^2*x^(4/3))/(240*e^2) + (29*b^2*d^4*n^2*x^(2/3))/(40*e^4)
- (49*b^2*d^5*n^2*x^(1/3))/(20*e^5) - (19*b^2*d^3*n^2*x)/(60*e^3) - (11*b^2*d*n^2*x^(5/3))/(150*e) - (b^2*d^2
*n*x^(4/3)*log(c*(d + e*x^(1/3))^n))/(4*e^2) - (b^2*d^4*n*x^(2/3)*log(c*(d + e*x^(1/3))^n))/(2*e^4) + (b^2*d^5
*n*x^(1/3)*log(c*(d + e*x^(1/3))^n))/e^5 + (a*b*d^3*n*x)/(3*e^3) + (a*b*d*n*x^(5/3))/(5*e) - (a*b*d^6*n*log(d
+ e*x^(1/3)))/e^6 + (b^2*d^3*n*x*log(c*(d + e*x^(1/3))^n))/(3*e^3) + (b^2*d*n*x^(5/3)*log(c*(d + e*x^(1/3))^n)
)/(5*e) - (a*b*d^2*n*x^(4/3))/(4*e^2) - (a*b*d^4*n*x^(2/3))/(2*e^4) + (a*b*d^5*n*x^(1/3))/e^5