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

Optimal. Leaf size=275 \[ \frac{b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}-\frac{3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac{3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac{b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac{b^2 d^3 n^2 \log ^2\left (d+e x^{2/3}\right )}{2 e^3}-\frac{3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac{b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3} \]

[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

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Rubi [A]  time = 0.305088, antiderivative size = 217, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac{1}{6} b n \left (\frac{18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac{6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}-\frac{9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac{2 \left (d+e x^{2/3}\right )^3}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac{b^2 d^3 n^2 \log ^2\left (d+e x^{2/3}\right )}{2 e^3}-\frac{3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac{b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3} \]

Antiderivative was successfully verified.

[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) - (b*n*((18*d^2*(d + e*x^(2/3)))/e^3 - (9*d*(d + e*x^(2/3))^2)/e^3
+ (2*(d + e*x^(2/3))^3)/e^3 - (6*d^3*Log[d + e*x^(2/3)])/e^3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/6 + (x^2*(a +
b*Log[c*(d + e*x^(2/3))^n])^2)/2

Rule 2454

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])

Rule 2398

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 2411

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 43

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 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 2301

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]

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx &=\frac{3}{2} \operatorname{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) \operatorname{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) \operatorname{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{1}{6} b n \left (\frac{18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac{9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac{2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\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 ) \operatorname{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{1}{6} b n \left (\frac{18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac{9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac{2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\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 ) \operatorname{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{1}{6} b n \left (\frac{18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac{9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac{2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\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 ) \operatorname{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{1}{6} b n \left (\frac{18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac{9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac{2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\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 ) \operatorname{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{1}{6} b n \left (\frac{18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac{9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac{2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\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]  time = 0.156801, size = 239, normalized size = 0.87 \[ \frac{18 a^2 d^3+18 a^2 e^3 x^2+6 b \left (6 a \left (d^3+e^3 x^2\right )-b n \left (6 d^2 e x^{2/3}+6 d^3-3 d e^2 x^{4/3}+2 e^3 x^2\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )-36 a b d^2 e n x^{2/3}+18 a b d e^2 n x^{4/3}-12 a b e^3 n x^2+18 b^2 \left (d^3+e^3 x^2\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+66 b^2 d^2 e n^2 x^{2/3}-30 b^2 d^3 n^2 \log \left (d+e x^{2/3}\right )-15 b^2 d e^2 n^2 x^{4/3}+4 b^2 e^3 n^2 x^2}{36 e^3} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.341, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 1.04515, size = 312, normalized size = 1.13 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 2.10064, size = 689, normalized size = 2.51 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.70717, size = 427, normalized size = 1.55 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} + \frac{1}{36} \,{\left (18 \, x^{2} \log \left (x^{\frac{2}{3}} e + d\right )^{2} +{\left (18 \, d^{3} \log \left (x^{\frac{2}{3}} e + d\right )^{2} - 12 \,{\left (x^{\frac{2}{3}} e + d\right )}^{3} \log \left (x^{\frac{2}{3}} e + d\right ) + 54 \,{\left (x^{\frac{2}{3}} e + d\right )}^{2} d \log \left (x^{\frac{2}{3}} e + d\right ) - 108 \,{\left (x^{\frac{2}{3}} e + d\right )} d^{2} \log \left (x^{\frac{2}{3}} e + d\right ) + 4 \,{\left (x^{\frac{2}{3}} e + d\right )}^{3} - 27 \,{\left (x^{\frac{2}{3}} e + d\right )}^{2} d + 108 \,{\left (x^{\frac{2}{3}} e + d\right )} d^{2}\right )} e^{\left (-3\right )}\right )} b^{2} n^{2} + \frac{1}{6} \,{\left (6 \, x^{2} \log \left (x^{\frac{2}{3}} e + d\right ) +{\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x^{\frac{2}{3}} e + d \right |}\right ) +{\left (3 \, d x^{\frac{4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac{2}{3}}\right )} e^{\left (-3\right )}\right )} e\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 (x^{\frac{2}{3}} e + d\right ) +{\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x^{\frac{2}{3}} e + d \right |}\right ) +{\left (3 \, d x^{\frac{4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac{2}{3}}\right )} e^{\left (-3\right )}\right )} e\right )} a b n + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(x^(2/3)*e + d)^2 + (18*d^3*log(x^(2/3)*e + d)^2 - 12*(x^(2/3)*e + d)^3
*log(x^(2/3)*e + d) + 54*(x^(2/3)*e + d)^2*d*log(x^(2/3)*e + d) - 108*(x^(2/3)*e + d)*d^2*log(x^(2/3)*e + d) +
4*(x^(2/3)*e + d)^3 - 27*(x^(2/3)*e + d)^2*d + 108*(x^(2/3)*e + d)*d^2)*e^(-3))*b^2*n^2 + 1/6*(6*x^2*log(x^(2
/3)*e + d) + (6*d^3*e^(-4)*log(abs(x^(2/3)*e + d)) + (3*d*x^(4/3)*e - 2*x^2*e^2 - 6*d^2*x^(2/3))*e^(-3))*e)*b^
2*n*log(c) + a*b*x^2*log(c) + 1/6*(6*x^2*log(x^(2/3)*e + d) + (6*d^3*e^(-4)*log(abs(x^(2/3)*e + d)) + (3*d*x^(
4/3)*e - 2*x^2*e^2 - 6*d^2*x^(2/3))*e^(-3))*e)*a*b*n + 1/2*a^2*x^2