\(\int x (a+b \log (c (d+e x^{2/3})^n))^2 \, dx\) [472]

Optimal result

Integrand size = 22, antiderivative size = 275 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=-\frac {3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e x^{2/3}\right )}{2 e^3}-\frac {3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac {b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \]

[Out]

-3/4*b^2*d*n^2*(d+e*x^(2/3))^2/e^3+1/9*b^2*n^2*(d+e*x^(2/3))^3/e^3+3*b^2*d^2*n^2*x^(2/3)/e^2-1/2*b^2*d^3*n^2*l
n(d+e*x^(2/3))^2/e^3-3*b*d^2*n*(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))/e^3+3/2*b*d*n*(d+e*x^(2/3))^2*(a+b*ln
(c*(d+e*x^(2/3))^n))/e^3-1/3*b*n*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e*x^(2/3))^n))/e^3+b*d^3*n*ln(d+e*x^(2/3))*(a+b*
ln(c*(d+e*x^(2/3))^n))/e^3+1/2*x^2*(a+b*ln(c*(d+e*x^(2/3))^n))^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 275, 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} \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}-\frac {3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {b^2 d^3 n^2 \log ^2\left (d+e x^{2/3}\right )}{2 e^3}+\frac {3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3} \]

[In]

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

[Out]

(-3*b^2*d*n^2*(d + e*x^(2/3))^2)/(4*e^3) + (b^2*n^2*(d + e*x^(2/3))^3)/(9*e^3) + (3*b^2*d^2*n^2*x^(2/3))/e^2 -
(b^2*d^3*n^2*Log[d + e*x^(2/3)]^2)/(2*e^3) - (3*b*d^2*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^3
+ (3*b*d*n*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*e^3) - (b*n*(d + e*x^(2/3))^3*(a + b*Log[c*
(d + e*x^(2/3))^n]))/(3*e^3) + (b*d^3*n*Log[d + e*x^(2/3)]*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^3 + (x^2*(a + b
*Log[c*(d + e*x^(2/3))^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}& = \frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,x^{2/3}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(b e n) \text {Subst}\left (\int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,x^{2/3}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(b 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^{2/3}\right ) \\ & = -\frac {3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac {b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\left (b^2 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^{2/3}\right ) \\ & = -\frac {3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac {b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {\left (b^2 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^{2/3}\right )}{6 e^3} \\ & = -\frac {3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac {b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {\left (b^2 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^{2/3}\right )}{6 e^3} \\ & = -\frac {3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac {b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {\left (b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x^{2/3}\right )}{e^3} \\ & = -\frac {3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e x^{2/3}\right )}{2 e^3}-\frac {3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac {b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.87 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {18 a^2 d^3-36 a b d^2 e n x^{2/3}+66 b^2 d^2 e n^2 x^{2/3}+18 a b d e^2 n x^{4/3}-15 b^2 d e^2 n^2 x^{4/3}+18 a^2 e^3 x^2-12 a b e^3 n x^2+4 b^2 e^3 n^2 x^2-30 b^2 d^3 n^2 \log \left (d+e x^{2/3}\right )+6 b \left (6 a \left (d^3+e^3 x^2\right )-b n \left (6 d^3+6 d^2 e x^{2/3}-3 d e^2 x^{4/3}+2 e^3 x^2\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+18 b^2 \left (d^3+e^3 x^2\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )}{36 e^3} \]

[In]

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

[Out]

(18*a^2*d^3 - 36*a*b*d^2*e*n*x^(2/3) + 66*b^2*d^2*e*n^2*x^(2/3) + 18*a*b*d*e^2*n*x^(4/3) - 15*b^2*d*e^2*n^2*x^
(4/3) + 18*a^2*e^3*x^2 - 12*a*b*e^3*n*x^2 + 4*b^2*e^3*n^2*x^2 - 30*b^2*d^3*n^2*Log[d + e*x^(2/3)] + 6*b*(6*a*(
d^3 + e^3*x^2) - b*n*(6*d^3 + 6*d^2*e*x^(2/3) - 3*d*e^2*x^(4/3) + 2*e^3*x^2))*Log[c*(d + e*x^(2/3))^n] + 18*b^
2*(d^3 + e^3*x^2)*Log[c*(d + e*x^(2/3))^n]^2)/(36*e^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.11 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {18 \, b^{2} e^{3} x^{2} \log \left (c\right )^{2} - 12 \, {\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} x^{2} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + 9 \, a^{2} e^{3}\right )} x^{2} + 18 \, {\left (b^{2} e^{3} n^{2} x^{2} + b^{2} d^{3} n^{2}\right )} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 6 \, {\left (3 \, b^{2} d e^{2} n^{2} x^{\frac {4}{3}} - 6 \, b^{2} d^{2} e n^{2} x^{\frac {2}{3}} - 11 \, b^{2} d^{3} n^{2} + 6 \, a b d^{3} n - 2 \, {\left (b^{2} e^{3} n^{2} - 3 \, a b e^{3} n\right )} x^{2} + 6 \, {\left (b^{2} e^{3} n x^{2} + b^{2} d^{3} n\right )} \log \left (c\right )\right )} \log \left (e x^{\frac {2}{3}} + d\right ) + 6 \, {\left (11 \, b^{2} d^{2} e n^{2} - 6 \, b^{2} d^{2} e n \log \left (c\right ) - 6 \, a b d^{2} e n\right )} x^{\frac {2}{3}} + 3 \, {\left (6 \, b^{2} d e^{2} n x \log \left (c\right ) - {\left (5 \, b^{2} d e^{2} n^{2} - 6 \, a b d e^{2} n\right )} x\right )} x^{\frac {1}{3}}}{36 \, e^{3}} \]

[In]

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

[Out]

1/36*(18*b^2*e^3*x^2*log(c)^2 - 12*(b^2*e^3*n - 3*a*b*e^3)*x^2*log(c) + 2*(2*b^2*e^3*n^2 - 6*a*b*e^3*n + 9*a^2
*e^3)*x^2 + 18*(b^2*e^3*n^2*x^2 + b^2*d^3*n^2)*log(e*x^(2/3) + d)^2 + 6*(3*b^2*d*e^2*n^2*x^(4/3) - 6*b^2*d^2*e
*n^2*x^(2/3) - 11*b^2*d^3*n^2 + 6*a*b*d^3*n - 2*(b^2*e^3*n^2 - 3*a*b*e^3*n)*x^2 + 6*(b^2*e^3*n*x^2 + b^2*d^3*n
)*log(c))*log(e*x^(2/3) + d) + 6*(11*b^2*d^2*e*n^2 - 6*b^2*d^2*e*n*log(c) - 6*a*b*d^2*e*n)*x^(2/3) + 3*(6*b^2*
d*e^2*n*x*log(c) - (5*b^2*d*e^2*n^2 - 6*a*b*d*e^2*n)*x)*x^(1/3))/e^3

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.84 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + \frac {1}{6} \, a b e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{2} - 3 \, d e x^{\frac {4}{3}} + 6 \, d^{2} x^{\frac {2}{3}}}{e^{3}}\right )} + a b x^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{36} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{2} - 3 \, d e x^{\frac {4}{3}} + 6 \, d^{2} x^{\frac {2}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + \frac {{\left (4 \, e^{3} x^{2} - 18 \, d^{3} \log \left (e x^{\frac {2}{3}} + d\right )^{2} - 15 \, d e^{2} x^{\frac {4}{3}} - 66 \, d^{3} \log \left (e x^{\frac {2}{3}} + d\right ) + 66 \, d^{2} e x^{\frac {2}{3}}\right )} n^{2}}{e^{3}}\right )} b^{2} \]

[In]

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

[Out]

1/2*b^2*x^2*log((e*x^(2/3) + d)^n*c)^2 + 1/6*a*b*e*n*(6*d^3*log(e*x^(2/3) + d)/e^4 - (2*e^2*x^2 - 3*d*e*x^(4/3
) + 6*d^2*x^(2/3))/e^3) + a*b*x^2*log((e*x^(2/3) + d)^n*c) + 1/2*a^2*x^2 + 1/36*(6*e*n*(6*d^3*log(e*x^(2/3) +
d)/e^4 - (2*e^2*x^2 - 3*d*e*x^(4/3) + 6*d^2*x^(2/3))/e^3)*log((e*x^(2/3) + d)^n*c) + (4*e^3*x^2 - 18*d^3*log(e
*x^(2/3) + d)^2 - 15*d*e^2*x^(4/3) - 66*d^3*log(e*x^(2/3) + d) + 66*d^2*e*x^(2/3))*n^2/e^3)*b^2

Giac [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.14 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} + \frac {1}{36} \, {\left (18 \, x^{2} \log \left (e x^{\frac {2}{3}} + d\right )^{2} - {\left (6 \, {\left (\frac {2 \, {\left (e x^{\frac {2}{3}} + d\right )}^{3}}{e^{4}} - \frac {9 \, {\left (e x^{\frac {2}{3}} + d\right )}^{2} d}{e^{4}} + \frac {18 \, {\left (e x^{\frac {2}{3}} + d\right )} d^{2}}{e^{4}}\right )} \log \left (e x^{\frac {2}{3}} + d\right ) - \frac {18 \, d^{3} \log \left (e x^{\frac {2}{3}} + d\right )^{2}}{e^{4}} - \frac {4 \, {\left (e x^{\frac {2}{3}} + d\right )}^{3}}{e^{4}} + \frac {27 \, {\left (e x^{\frac {2}{3}} + d\right )}^{2} d}{e^{4}} - \frac {108 \, {\left (e x^{\frac {2}{3}} + d\right )} d^{2}}{e^{4}}\right )} e\right )} b^{2} n^{2} + \frac {1}{6} \, {\left (6 \, x^{2} \log \left (e x^{\frac {2}{3}} + d\right ) + e {\left (\frac {6 \, d^{3} \log \left ({\left | e x^{\frac {2}{3}} + d \right |}\right )}{e^{4}} - \frac {2 \, e^{2} x^{2} - 3 \, d e x^{\frac {4}{3}} + 6 \, d^{2} x^{\frac {2}{3}}}{e^{3}}\right )}\right )} b^{2} n \log \left (c\right ) + a b x^{2} \log \left (c\right ) + \frac {1}{6} \, {\left (6 \, x^{2} \log \left (e x^{\frac {2}{3}} + d\right ) + e {\left (\frac {6 \, d^{3} \log \left ({\left | e x^{\frac {2}{3}} + d \right |}\right )}{e^{4}} - \frac {2 \, e^{2} x^{2} - 3 \, d e x^{\frac {4}{3}} + 6 \, d^{2} x^{\frac {2}{3}}}{e^{3}}\right )}\right )} a b n + \frac {1}{2} \, a^{2} x^{2} \]

[In]

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

[Out]

1/2*b^2*x^2*log(c)^2 + 1/36*(18*x^2*log(e*x^(2/3) + d)^2 - (6*(2*(e*x^(2/3) + d)^3/e^4 - 9*(e*x^(2/3) + d)^2*d
/e^4 + 18*(e*x^(2/3) + d)*d^2/e^4)*log(e*x^(2/3) + d) - 18*d^3*log(e*x^(2/3) + d)^2/e^4 - 4*(e*x^(2/3) + d)^3/
e^4 + 27*(e*x^(2/3) + d)^2*d/e^4 - 108*(e*x^(2/3) + d)*d^2/e^4)*e)*b^2*n^2 + 1/6*(6*x^2*log(e*x^(2/3) + d) + e
*(6*d^3*log(abs(e*x^(2/3) + d))/e^4 - (2*e^2*x^2 - 3*d*e*x^(4/3) + 6*d^2*x^(2/3))/e^3))*b^2*n*log(c) + a*b*x^2
*log(c) + 1/6*(6*x^2*log(e*x^(2/3) + d) + e*(6*d^3*log(abs(e*x^(2/3) + d))/e^4 - (2*e^2*x^2 - 3*d*e*x^(4/3) +
6*d^2*x^(2/3))/e^3))*a*b*n + 1/2*a^2*x^2

Mupad [B] (verification not implemented)

Time = 1.73 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.09 \[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx={\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2\,\left (\frac {b^2\,x^2}{2}+\frac {b^2\,d^3}{2\,e^3}\right )-x^{4/3}\,\left (\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{4\,e}\right )+x^2\,\left (\frac {a^2}{2}-\frac {a\,b\,n}{3}+\frac {b^2\,n^2}{9}\right )+\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\,\left (\frac {b\,x^2\,\left (3\,a-b\,n\right )}{3}-x^{4/3}\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {3\,a\,b\,d}{2\,e}\right )+\frac {d\,x^{2/3}\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}\right )}{e}\right )+x^{2/3}\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{e^2}\right )-\frac {\ln \left (d+e\,x^{2/3}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{6\,e^3} \]

[In]

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

[Out]

log(c*(d + e*x^(2/3))^n)^2*((b^2*x^2)/2 + (b^2*d^3)/(2*e^3)) - x^(4/3)*((d*((3*a^2)/2 + (b^2*n^2)/3 - a*b*n))/
(2*e) - (d*(3*a^2 - b^2*n^2))/(4*e)) + x^2*(a^2/2 + (b^2*n^2)/9 - (a*b*n)/3) + log(c*(d + e*x^(2/3))^n)*((b*x^
2*(3*a - b*n))/3 - x^(4/3)*((b*d*(3*a - b*n))/(2*e) - (3*a*b*d)/(2*e)) + (d*x^(2/3)*((b*d*(3*a - b*n))/e - (3*
a*b*d)/e))/e) + x^(2/3)*((d*((d*((3*a^2)/2 + (b^2*n^2)/3 - a*b*n))/e - (d*(3*a^2 - b^2*n^2))/(2*e)))/e + (b^2*
d^2*n^2)/e^2) - (log(d + e*x^(2/3))*(11*b^2*d^3*n^2 - 6*a*b*d^3*n))/(6*e^3)