3.5.81 \(\int x^3 (a+b \log (c (d+e x^{2/3})^n))^3 \, dx\) [481]

Optimal. Leaf size=913 \[ -\frac {45 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{16 e^6}+\frac {10 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{128 e^6}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{144 e^6}-\frac {9 a b^2 d^5 n^2 x^{2/3}}{e^5}+\frac {9 b^3 d^5 n^3 x^{2/3}}{e^5}-\frac {9 b^3 d^5 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^6}+\frac {45 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 e^6}-\frac {10 b^2 d^3 n^2 \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^6}+\frac {45 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{32 e^6}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{24 e^6}+\frac {9 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^6}-\frac {45 b d^4 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}{8 e^6}+\frac {5 b d^3 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 )^2}{e^6}-\frac {45 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{16 e^6}+\frac {9 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{10 e^6}-\frac {b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \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^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6} \]

[Out]

1/24*b^2*n^2*(d+e*x^(2/3))^6*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6-1/8*b*n*(d+e*x^(2/3))^6*(a+b*ln(c*(d+e*x^(2/3))^n
))^2/e^6+1/4*(d+e*x^(2/3))^6*(a+b*ln(c*(d+e*x^(2/3))^n))^3/e^6-10/3*b^2*d^3*n^2*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e
*x^(2/3))^n))/e^6+45/32*b^2*d^2*n^2*(d+e*x^(2/3))^4*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6-9/25*b^2*d*n^2*(d+e*x^(2/3
))^5*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6+9/2*b*d^5*n*(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^6-45/8*b*d^4*n*
(d+e*x^(2/3))^2*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^6+5*b*d^3*n*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^6-
45/16*b*d^2*n*(d+e*x^(2/3))^4*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^6+9/10*b*d*n*(d+e*x^(2/3))^5*(a+b*ln(c*(d+e*x^(2
/3))^n))^2/e^6-9*b^3*d^5*n^2*(d+e*x^(2/3))*ln(c*(d+e*x^(2/3))^n)/e^6+45/8*b^2*d^4*n^2*(d+e*x^(2/3))^2*(a+b*ln(
c*(d+e*x^(2/3))^n))/e^6-3/2*d^5*(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))^3/e^6+15/4*d^4*(d+e*x^(2/3))^2*(a+b*
ln(c*(d+e*x^(2/3))^n))^3/e^6-5*d^3*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e*x^(2/3))^n))^3/e^6+15/4*d^2*(d+e*x^(2/3))^4*
(a+b*ln(c*(d+e*x^(2/3))^n))^3/e^6-3/2*d*(d+e*x^(2/3))^5*(a+b*ln(c*(d+e*x^(2/3))^n))^3/e^6-9*a*b^2*d^5*n^2*x^(2
/3)/e^5-45/16*b^3*d^4*n^3*(d+e*x^(2/3))^2/e^6+10/9*b^3*d^3*n^3*(d+e*x^(2/3))^3/e^6-45/128*b^3*d^2*n^3*(d+e*x^(
2/3))^4/e^6+9/125*b^3*d*n^3*(d+e*x^(2/3))^5/e^6+9*b^3*d^5*n^3*x^(2/3)/e^5-1/144*b^3*n^3*(d+e*x^(2/3))^6/e^6

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 913, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} -\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{144 e^6}+\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^6}{4 e^6}-\frac {b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^6}{8 e^6}+\frac {b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^6}{24 e^6}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {3 d \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^5}{2 e^6}+\frac {9 b d n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^5}{10 e^6}-\frac {9 b^2 d n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^5}{25 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{128 e^6}+\frac {15 d^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^4}{4 e^6}-\frac {45 b d^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^4}{16 e^6}+\frac {45 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^4}{32 e^6}+\frac {10 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {5 d^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^3}{e^6}+\frac {5 b d^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^3}{e^6}-\frac {10 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^3}{3 e^6}-\frac {45 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{16 e^6}+\frac {15 d^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^2}{4 e^6}-\frac {45 b d^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^2}{8 e^6}+\frac {45 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^2}{8 e^6}-\frac {3 d^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )}{2 e^6}+\frac {9 b d^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )}{2 e^6}-\frac {9 b^3 d^5 n^2 \log \left (c \left (d+e x^{2/3}\right )^n\right ) \left (d+e x^{2/3}\right )}{e^6}+\frac {9 b^3 d^5 n^3 x^{2/3}}{e^5}-\frac {9 a b^2 d^5 n^2 x^{2/3}}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*b^3*d^4*n^3*(d + e*x^(2/3))^2)/(16*e^6) + (10*b^3*d^3*n^3*(d + e*x^(2/3))^3)/(9*e^6) - (45*b^3*d^2*n^3*(d
+ e*x^(2/3))^4)/(128*e^6) + (9*b^3*d*n^3*(d + e*x^(2/3))^5)/(125*e^6) - (b^3*n^3*(d + e*x^(2/3))^6)/(144*e^6)
- (9*a*b^2*d^5*n^2*x^(2/3))/e^5 + (9*b^3*d^5*n^3*x^(2/3))/e^5 - (9*b^3*d^5*n^2*(d + e*x^(2/3))*Log[c*(d + e*x
^(2/3))^n])/e^6 + (45*b^2*d^4*n^2*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(8*e^6) - (10*b^2*d^3*n^
2*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^6) + (45*b^2*d^2*n^2*(d + e*x^(2/3))^4*(a + b*Log[c
*(d + e*x^(2/3))^n]))/(32*e^6) - (9*b^2*d*n^2*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n]))/(25*e^6) + (
b^2*n^2*(d + e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n]))/(24*e^6) + (9*b*d^5*n*(d + e*x^(2/3))*(a + b*Log[c
*(d + e*x^(2/3))^n])^2)/(2*e^6) - (45*b*d^4*n*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(8*e^6) +
(5*b*d^3*n*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e^6 - (45*b*d^2*n*(d + e*x^(2/3))^4*(a + b*Lo
g[c*(d + e*x^(2/3))^n])^2)/(16*e^6) + (9*b*d*n*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(10*e^6)
- (b*n*(d + e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(8*e^6) - (3*d^5*(d + e*x^(2/3))*(a + b*Log[c*(d
+ e*x^(2/3))^n])^3)/(2*e^6) + (15*d^4*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(4*e^6) - (5*d^3*(
d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/e^6 + (15*d^2*(d + e*x^(2/3))^4*(a + b*Log[c*(d + e*x^(2/
3))^n])^3)/(4*e^6) - (3*d*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(2*e^6) + ((d + e*x^(2/3))^6*(
a + b*Log[c*(d + e*x^(2/3))^n])^3)/(4*e^6)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

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

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

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

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

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*} \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx &=\frac {3}{2} \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )\\ &=\frac {3}{2} \text {Subst}\left (\int \left (-\frac {d^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {(d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}\right ) \, dx,x,x^{2/3}\right )\\ &=\frac {3 \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^5}-\frac {(15 d) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^5}+\frac {\left (15 d^2\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{e^5}-\frac {\left (15 d^3\right ) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{e^5}+\frac {\left (15 d^4\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^5}-\frac {\left (3 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^5}\\ &=\frac {3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^6}-\frac {(15 d) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^6}+\frac {\left (15 d^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{e^6}-\frac {\left (15 d^3\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{e^6}+\frac {\left (15 d^4\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^6}-\frac {\left (3 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^6}\\ &=-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \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^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {(3 b n) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{4 e^6}+\frac {(9 b d n) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{2 e^6}-\frac {\left (45 b d^2 n\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{4 e^6}+\frac {\left (15 b d^3 n\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{e^6}-\frac {\left (45 b d^4 n\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{4 e^6}+\frac {\left (9 b d^5 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{2 e^6}\\ &=\frac {9 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^6}-\frac {45 b d^4 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}{8 e^6}+\frac {5 b d^3 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 )^2}{e^6}-\frac {45 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{16 e^6}+\frac {9 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{10 e^6}-\frac {b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \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^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{4 e^6}-\frac {\left (9 b^2 d n^2\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{5 e^6}+\frac {\left (45 b^2 d^2 n^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{8 e^6}-\frac {\left (10 b^2 d^3 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{e^6}+\frac {\left (45 b^2 d^4 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{4 e^6}-\frac {\left (9 b^2 d^5 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{e^6}\\ &=-\frac {45 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{16 e^6}+\frac {10 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{128 e^6}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{144 e^6}-\frac {9 a b^2 d^5 n^2 x^{2/3}}{e^5}+\frac {45 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 e^6}-\frac {10 b^2 d^3 n^2 \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^6}+\frac {45 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{32 e^6}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{24 e^6}+\frac {9 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^6}-\frac {45 b d^4 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}{8 e^6}+\frac {5 b d^3 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 )^2}{e^6}-\frac {45 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{16 e^6}+\frac {9 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{10 e^6}-\frac {b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \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^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {\left (9 b^3 d^5 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x^{2/3}\right )}{e^6}\\ &=-\frac {45 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{16 e^6}+\frac {10 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{128 e^6}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{144 e^6}-\frac {9 a b^2 d^5 n^2 x^{2/3}}{e^5}+\frac {9 b^3 d^5 n^3 x^{2/3}}{e^5}-\frac {9 b^3 d^5 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^6}+\frac {45 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 e^6}-\frac {10 b^2 d^3 n^2 \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^6}+\frac {45 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{32 e^6}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{24 e^6}+\frac {9 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^6}-\frac {45 b d^4 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}{8 e^6}+\frac {5 b d^3 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 )^2}{e^6}-\frac {45 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{16 e^6}+\frac {9 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{10 e^6}-\frac {b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \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^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 598, normalized size = 0.65 \begin {gather*} \frac {e x^{2/3} \left (36000 a^3 e^5 x^{10/3}+b^3 n^3 \left (809340 d^5-140070 d^4 e x^{2/3}+41180 d^3 e^2 x^{4/3}-13785 d^2 e^3 x^2+4368 d e^4 x^{8/3}-1000 e^5 x^{10/3}\right )-60 a b^2 n^2 \left (8820 d^5-2610 d^4 e x^{2/3}+1140 d^3 e^2 x^{4/3}-555 d^2 e^3 x^2+264 d e^4 x^{8/3}-100 e^5 x^{10/3}\right )+1800 a^2 b n \left (60 d^5-30 d^4 e x^{2/3}+20 d^3 e^2 x^{4/3}-15 d^2 e^3 x^2+12 d e^4 x^{8/3}-10 e^5 x^{10/3}\right )\right )-280140 b^3 d^6 n^3 \log \left (d+e x^{2/3}\right )-60 b \left (b^2 n^2 \left (8820 d^6+8820 d^5 e x^{2/3}-2610 d^4 e^2 x^{4/3}+1140 d^3 e^3 x^2-555 d^2 e^4 x^{8/3}+264 d e^5 x^{10/3}-100 e^6 x^4\right )-60 a b n \left (147 d^6+60 d^5 e x^{2/3}-30 d^4 e^2 x^{4/3}+20 d^3 e^3 x^2-15 d^2 e^4 x^{8/3}+12 d e^5 x^{10/3}-10 e^6 x^4\right )+1800 a^2 \left (d^6-e^6 x^4\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+1800 b^2 \left (b n \left (147 d^6+60 d^5 e x^{2/3}-30 d^4 e^2 x^{4/3}+20 d^3 e^3 x^2-15 d^2 e^4 x^{8/3}+12 d e^5 x^{10/3}-10 e^6 x^4\right )-60 a \left (d^6-e^6 x^4\right )\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^4\right ) \log ^3\left (c \left (d+e x^{2/3}\right )^n\right )}{144000 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x^(2/3)*(36000*a^3*e^5*x^(10/3) + b^3*n^3*(809340*d^5 - 140070*d^4*e*x^(2/3) + 41180*d^3*e^2*x^(4/3) - 1378
5*d^2*e^3*x^2 + 4368*d*e^4*x^(8/3) - 1000*e^5*x^(10/3)) - 60*a*b^2*n^2*(8820*d^5 - 2610*d^4*e*x^(2/3) + 1140*d
^3*e^2*x^(4/3) - 555*d^2*e^3*x^2 + 264*d*e^4*x^(8/3) - 100*e^5*x^(10/3)) + 1800*a^2*b*n*(60*d^5 - 30*d^4*e*x^(
2/3) + 20*d^3*e^2*x^(4/3) - 15*d^2*e^3*x^2 + 12*d*e^4*x^(8/3) - 10*e^5*x^(10/3))) - 280140*b^3*d^6*n^3*Log[d +
e*x^(2/3)] - 60*b*(b^2*n^2*(8820*d^6 + 8820*d^5*e*x^(2/3) - 2610*d^4*e^2*x^(4/3) + 1140*d^3*e^3*x^2 - 555*d^2
*e^4*x^(8/3) + 264*d*e^5*x^(10/3) - 100*e^6*x^4) - 60*a*b*n*(147*d^6 + 60*d^5*e*x^(2/3) - 30*d^4*e^2*x^(4/3) +
20*d^3*e^3*x^2 - 15*d^2*e^4*x^(8/3) + 12*d*e^5*x^(10/3) - 10*e^6*x^4) + 1800*a^2*(d^6 - e^6*x^4))*Log[c*(d +
e*x^(2/3))^n] + 1800*b^2*(b*n*(147*d^6 + 60*d^5*e*x^(2/3) - 30*d^4*e^2*x^(4/3) + 20*d^3*e^3*x^2 - 15*d^2*e^4*x
^(8/3) + 12*d*e^5*x^(10/3) - 10*e^6*x^4) - 60*a*(d^6 - e^6*x^4))*Log[c*(d + e*x^(2/3))^n]^2 - 36000*b^3*(d^6 -
e^6*x^4)*Log[c*(d + e*x^(2/3))^n]^3)/(144000*e^6)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 669, normalized size = 0.73 \begin {gather*} \frac {1}{4} \, b^{3} x^{4} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )^{3} + \frac {3}{4} \, a b^{2} x^{4} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )^{2} + \frac {3}{4} \, a^{2} b x^{4} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right ) + \frac {1}{4} \, a^{3} x^{4} - \frac {1}{80} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (30 \, d^{4} x^{\frac {4}{3}} e - 20 \, d^{3} x^{2} e^{2} - 60 \, d^{5} x^{\frac {2}{3}} + 15 \, d^{2} x^{\frac {8}{3}} e^{3} - 12 \, d x^{\frac {10}{3}} e^{4} + 10 \, x^{4} e^{5}\right )} e^{\left (-6\right )}\right )} a^{2} b n e + \frac {1}{2400} \, {\left ({\left (1800 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 8820 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right ) - 8820 \, d^{5} x^{\frac {2}{3}} e + 2610 \, d^{4} x^{\frac {4}{3}} e^{2} - 1140 \, d^{3} x^{2} e^{3} + 555 \, d^{2} x^{\frac {8}{3}} e^{4} - 264 \, d x^{\frac {10}{3}} e^{5} + 100 \, x^{4} e^{6}\right )} n^{2} e^{\left (-6\right )} - 60 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (30 \, d^{4} x^{\frac {4}{3}} e - 20 \, d^{3} x^{2} e^{2} - 60 \, d^{5} x^{\frac {2}{3}} + 15 \, d^{2} x^{\frac {8}{3}} e^{3} - 12 \, d x^{\frac {10}{3}} e^{4} + 10 \, x^{4} e^{5}\right )} e^{\left (-6\right )}\right )} n e \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )\right )} a b^{2} - \frac {1}{144000} \, {\left (1800 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (30 \, d^{4} x^{\frac {4}{3}} e - 20 \, d^{3} x^{2} e^{2} - 60 \, d^{5} x^{\frac {2}{3}} + 15 \, d^{2} x^{\frac {8}{3}} e^{3} - 12 \, d x^{\frac {10}{3}} e^{4} + 10 \, x^{4} e^{5}\right )} e^{\left (-6\right )}\right )} n e \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )^{2} + {\left ({\left (36000 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right )^{3} + 264600 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 809340 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right ) - 809340 \, d^{5} x^{\frac {2}{3}} e + 140070 \, d^{4} x^{\frac {4}{3}} e^{2} - 41180 \, d^{3} x^{2} e^{3} + 13785 \, d^{2} x^{\frac {8}{3}} e^{4} - 4368 \, d x^{\frac {10}{3}} e^{5} + 1000 \, x^{4} e^{6}\right )} n^{2} e^{\left (-7\right )} - 60 \, {\left (1800 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 8820 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right ) - 8820 \, d^{5} x^{\frac {2}{3}} e + 2610 \, d^{4} x^{\frac {4}{3}} e^{2} - 1140 \, d^{3} x^{2} e^{3} + 555 \, d^{2} x^{\frac {8}{3}} e^{4} - 264 \, d x^{\frac {10}{3}} e^{5} + 100 \, x^{4} e^{6}\right )} n e^{\left (-7\right )} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )\right )} n e\right )} b^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^3*x^4*log((x^(2/3)*e + d)^n*c)^3 + 3/4*a*b^2*x^4*log((x^(2/3)*e + d)^n*c)^2 + 3/4*a^2*b*x^4*log((x^(2/3)
*e + d)^n*c) + 1/4*a^3*x^4 - 1/80*(60*d^6*e^(-7)*log(x^(2/3)*e + d) + (30*d^4*x^(4/3)*e - 20*d^3*x^2*e^2 - 60*
d^5*x^(2/3) + 15*d^2*x^(8/3)*e^3 - 12*d*x^(10/3)*e^4 + 10*x^4*e^5)*e^(-6))*a^2*b*n*e + 1/2400*((1800*d^6*log(x
^(2/3)*e + d)^2 + 8820*d^6*log(x^(2/3)*e + d) - 8820*d^5*x^(2/3)*e + 2610*d^4*x^(4/3)*e^2 - 1140*d^3*x^2*e^3 +
555*d^2*x^(8/3)*e^4 - 264*d*x^(10/3)*e^5 + 100*x^4*e^6)*n^2*e^(-6) - 60*(60*d^6*e^(-7)*log(x^(2/3)*e + d) + (
30*d^4*x^(4/3)*e - 20*d^3*x^2*e^2 - 60*d^5*x^(2/3) + 15*d^2*x^(8/3)*e^3 - 12*d*x^(10/3)*e^4 + 10*x^4*e^5)*e^(-
6))*n*e*log((x^(2/3)*e + d)^n*c))*a*b^2 - 1/144000*(1800*(60*d^6*e^(-7)*log(x^(2/3)*e + d) + (30*d^4*x^(4/3)*e
- 20*d^3*x^2*e^2 - 60*d^5*x^(2/3) + 15*d^2*x^(8/3)*e^3 - 12*d*x^(10/3)*e^4 + 10*x^4*e^5)*e^(-6))*n*e*log((x^(
2/3)*e + d)^n*c)^2 + ((36000*d^6*log(x^(2/3)*e + d)^3 + 264600*d^6*log(x^(2/3)*e + d)^2 + 809340*d^6*log(x^(2/
3)*e + d) - 809340*d^5*x^(2/3)*e + 140070*d^4*x^(4/3)*e^2 - 41180*d^3*x^2*e^3 + 13785*d^2*x^(8/3)*e^4 - 4368*d
*x^(10/3)*e^5 + 1000*x^4*e^6)*n^2*e^(-7) - 60*(1800*d^6*log(x^(2/3)*e + d)^2 + 8820*d^6*log(x^(2/3)*e + d) - 8
820*d^5*x^(2/3)*e + 2610*d^4*x^(4/3)*e^2 - 1140*d^3*x^2*e^3 + 555*d^2*x^(8/3)*e^4 - 264*d*x^(10/3)*e^5 + 100*x
^4*e^6)*n*e^(-7)*log((x^(2/3)*e + d)^n*c))*n*e)*b^3

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Fricas [A]
time = 0.45, size = 1142, normalized size = 1.25 \begin {gather*} \frac {1}{144000} \, {\left (36000 \, b^{3} x^{4} e^{6} \log \left (c\right )^{3} - 1000 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2} + 18 \, a^{2} b n - 36 \, a^{3}\right )} x^{4} e^{6} + 20 \, {\left (2059 \, b^{3} d^{3} n^{3} - 3420 \, a b^{2} d^{3} n^{2} + 1800 \, a^{2} b d^{3} n\right )} x^{2} e^{3} - 36000 \, {\left (b^{3} d^{6} n^{3} - b^{3} n^{3} x^{4} e^{6}\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{3} + 1800 \, {\left (147 \, b^{3} d^{6} n^{3} + 20 \, b^{3} d^{3} n^{3} x^{2} e^{3} - 60 \, a b^{2} d^{6} n^{2} - 10 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2}\right )} x^{4} e^{6} - 60 \, {\left (b^{3} d^{6} n^{2} - b^{3} n^{2} x^{4} e^{6}\right )} \log \left (c\right ) + 15 \, {\left (4 \, b^{3} d^{5} n^{3} e - b^{3} d^{2} n^{3} x^{2} e^{4}\right )} x^{\frac {2}{3}} - 6 \, {\left (5 \, b^{3} d^{4} n^{3} x e^{2} - 2 \, b^{3} d n^{3} x^{3} e^{5}\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 18000 \, {\left (2 \, b^{3} d^{3} n x^{2} e^{3} - {\left (b^{3} n - 6 \, a b^{2}\right )} x^{4} e^{6}\right )} \log \left (c\right )^{2} - 60 \, {\left (13489 \, b^{3} d^{6} n^{3} - 8820 \, a b^{2} d^{6} n^{2} + 1800 \, a^{2} b d^{6} n - 100 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2} + 18 \, a^{2} b n\right )} x^{4} e^{6} + 60 \, {\left (19 \, b^{3} d^{3} n^{3} - 20 \, a b^{2} d^{3} n^{2}\right )} x^{2} e^{3} + 1800 \, {\left (b^{3} d^{6} n - b^{3} n x^{4} e^{6}\right )} \log \left (c\right )^{2} - 60 \, {\left (147 \, b^{3} d^{6} n^{2} + 20 \, b^{3} d^{3} n^{2} x^{2} e^{3} - 60 \, a b^{2} d^{6} n - 10 \, {\left (b^{3} n^{2} - 6 \, a b^{2} n\right )} x^{4} e^{6}\right )} \log \left (c\right ) - 15 \, {\left ({\left (37 \, b^{3} d^{2} n^{3} - 60 \, a b^{2} d^{2} n^{2}\right )} x^{2} e^{4} - 12 \, {\left (49 \, b^{3} d^{5} n^{3} - 20 \, a b^{2} d^{5} n^{2}\right )} e + 60 \, {\left (4 \, b^{3} d^{5} n^{2} e - b^{3} d^{2} n^{2} x^{2} e^{4}\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} + 6 \, {\left (4 \, {\left (11 \, b^{3} d n^{3} - 30 \, a b^{2} d n^{2}\right )} x^{3} e^{5} - 15 \, {\left (29 \, b^{3} d^{4} n^{3} - 20 \, a b^{2} d^{4} n^{2}\right )} x e^{2} + 60 \, {\left (5 \, b^{3} d^{4} n^{2} x e^{2} - 2 \, b^{3} d n^{2} x^{3} e^{5}\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 1200 \, {\left (5 \, {\left (b^{3} n^{2} - 6 \, a b^{2} n + 18 \, a^{2} b\right )} x^{4} e^{6} - 3 \, {\left (19 \, b^{3} d^{3} n^{2} - 20 \, a b^{2} d^{3} n\right )} x^{2} e^{3}\right )} \log \left (c\right ) - 15 \, {\left ({\left (919 \, b^{3} d^{2} n^{3} - 2220 \, a b^{2} d^{2} n^{2} + 1800 \, a^{2} b d^{2} n\right )} x^{2} e^{4} - 1800 \, {\left (4 \, b^{3} d^{5} n e - b^{3} d^{2} n x^{2} e^{4}\right )} \log \left (c\right )^{2} - 4 \, {\left (13489 \, b^{3} d^{5} n^{3} - 8820 \, a b^{2} d^{5} n^{2} + 1800 \, a^{2} b d^{5} n\right )} e - 60 \, {\left ({\left (37 \, b^{3} d^{2} n^{2} - 60 \, a b^{2} d^{2} n\right )} x^{2} e^{4} - 12 \, {\left (49 \, b^{3} d^{5} n^{2} - 20 \, a b^{2} d^{5} n\right )} e\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} + 6 \, {\left (8 \, {\left (91 \, b^{3} d n^{3} - 330 \, a b^{2} d n^{2} + 450 \, a^{2} b d n\right )} x^{3} e^{5} - 5 \, {\left (4669 \, b^{3} d^{4} n^{3} - 5220 \, a b^{2} d^{4} n^{2} + 1800 \, a^{2} b d^{4} n\right )} x e^{2} - 1800 \, {\left (5 \, b^{3} d^{4} n x e^{2} - 2 \, b^{3} d n x^{3} e^{5}\right )} \log \left (c\right )^{2} - 60 \, {\left (4 \, {\left (11 \, b^{3} d n^{2} - 30 \, a b^{2} d n\right )} x^{3} e^{5} - 15 \, {\left (29 \, b^{3} d^{4} n^{2} - 20 \, a b^{2} d^{4} n\right )} x e^{2}\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144000*(36000*b^3*x^4*e^6*log(c)^3 - 1000*(b^3*n^3 - 6*a*b^2*n^2 + 18*a^2*b*n - 36*a^3)*x^4*e^6 + 20*(2059*b
^3*d^3*n^3 - 3420*a*b^2*d^3*n^2 + 1800*a^2*b*d^3*n)*x^2*e^3 - 36000*(b^3*d^6*n^3 - b^3*n^3*x^4*e^6)*log(x^(2/3
)*e + d)^3 + 1800*(147*b^3*d^6*n^3 + 20*b^3*d^3*n^3*x^2*e^3 - 60*a*b^2*d^6*n^2 - 10*(b^3*n^3 - 6*a*b^2*n^2)*x^
4*e^6 - 60*(b^3*d^6*n^2 - b^3*n^2*x^4*e^6)*log(c) + 15*(4*b^3*d^5*n^3*e - b^3*d^2*n^3*x^2*e^4)*x^(2/3) - 6*(5*
b^3*d^4*n^3*x*e^2 - 2*b^3*d*n^3*x^3*e^5)*x^(1/3))*log(x^(2/3)*e + d)^2 + 18000*(2*b^3*d^3*n*x^2*e^3 - (b^3*n -
6*a*b^2)*x^4*e^6)*log(c)^2 - 60*(13489*b^3*d^6*n^3 - 8820*a*b^2*d^6*n^2 + 1800*a^2*b*d^6*n - 100*(b^3*n^3 - 6
*a*b^2*n^2 + 18*a^2*b*n)*x^4*e^6 + 60*(19*b^3*d^3*n^3 - 20*a*b^2*d^3*n^2)*x^2*e^3 + 1800*(b^3*d^6*n - b^3*n*x^
4*e^6)*log(c)^2 - 60*(147*b^3*d^6*n^2 + 20*b^3*d^3*n^2*x^2*e^3 - 60*a*b^2*d^6*n - 10*(b^3*n^2 - 6*a*b^2*n)*x^4
*e^6)*log(c) - 15*((37*b^3*d^2*n^3 - 60*a*b^2*d^2*n^2)*x^2*e^4 - 12*(49*b^3*d^5*n^3 - 20*a*b^2*d^5*n^2)*e + 60
*(4*b^3*d^5*n^2*e - b^3*d^2*n^2*x^2*e^4)*log(c))*x^(2/3) + 6*(4*(11*b^3*d*n^3 - 30*a*b^2*d*n^2)*x^3*e^5 - 15*(
29*b^3*d^4*n^3 - 20*a*b^2*d^4*n^2)*x*e^2 + 60*(5*b^3*d^4*n^2*x*e^2 - 2*b^3*d*n^2*x^3*e^5)*log(c))*x^(1/3))*log
(x^(2/3)*e + d) + 1200*(5*(b^3*n^2 - 6*a*b^2*n + 18*a^2*b)*x^4*e^6 - 3*(19*b^3*d^3*n^2 - 20*a*b^2*d^3*n)*x^2*e
^3)*log(c) - 15*((919*b^3*d^2*n^3 - 2220*a*b^2*d^2*n^2 + 1800*a^2*b*d^2*n)*x^2*e^4 - 1800*(4*b^3*d^5*n*e - b^3
*d^2*n*x^2*e^4)*log(c)^2 - 4*(13489*b^3*d^5*n^3 - 8820*a*b^2*d^5*n^2 + 1800*a^2*b*d^5*n)*e - 60*((37*b^3*d^2*n
^2 - 60*a*b^2*d^2*n)*x^2*e^4 - 12*(49*b^3*d^5*n^2 - 20*a*b^2*d^5*n)*e)*log(c))*x^(2/3) + 6*(8*(91*b^3*d*n^3 -
330*a*b^2*d*n^2 + 450*a^2*b*d*n)*x^3*e^5 - 5*(4669*b^3*d^4*n^3 - 5220*a*b^2*d^4*n^2 + 1800*a^2*b*d^4*n)*x*e^2
- 1800*(5*b^3*d^4*n*x*e^2 - 2*b^3*d*n*x^3*e^5)*log(c)^2 - 60*(4*(11*b^3*d*n^2 - 30*a*b^2*d*n)*x^3*e^5 - 15*(29
*b^3*d^4*n^2 - 20*a*b^2*d^4*n)*x*e^2)*log(c))*x^(1/3))*e^(-6)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2174 vs. \(2 (803) = 1606\).
time = 4.30, size = 2174, normalized size = 2.38 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^3*x^4*log(c)^3 + 3/4*a*b^2*x^4*log(c)^2 + 3/4*a^2*b*x^4*log(c) + 1/144000*(36000*(x^(2/3)*e + d)^6*e^(-6
)*log(x^(2/3)*e + d)^3 - 216000*(x^(2/3)*e + d)^5*d*e^(-6)*log(x^(2/3)*e + d)^3 + 540000*(x^(2/3)*e + d)^4*d^2
*e^(-6)*log(x^(2/3)*e + d)^3 - 720000*(x^(2/3)*e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d)^3 + 540000*(x^(2/3)*e +
d)^2*d^4*e^(-6)*log(x^(2/3)*e + d)^3 - 18000*(x^(2/3)*e + d)^6*e^(-6)*log(x^(2/3)*e + d)^2 + 129600*(x^(2/3)*e
+ d)^5*d*e^(-6)*log(x^(2/3)*e + d)^2 - 405000*(x^(2/3)*e + d)^4*d^2*e^(-6)*log(x^(2/3)*e + d)^2 + 720000*(x^(
2/3)*e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d)^2 - 810000*(x^(2/3)*e + d)^2*d^4*e^(-6)*log(x^(2/3)*e + d)^2 + 600
0*(x^(2/3)*e + d)^6*e^(-6)*log(x^(2/3)*e + d) - 51840*(x^(2/3)*e + d)^5*d*e^(-6)*log(x^(2/3)*e + d) + 202500*(
x^(2/3)*e + d)^4*d^2*e^(-6)*log(x^(2/3)*e + d) - 480000*(x^(2/3)*e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d) + 8100
00*(x^(2/3)*e + d)^2*d^4*e^(-6)*log(x^(2/3)*e + d) - 1000*(x^(2/3)*e + d)^6*e^(-6) + 10368*(x^(2/3)*e + d)^5*d
*e^(-6) - 50625*(x^(2/3)*e + d)^4*d^2*e^(-6) + 160000*(x^(2/3)*e + d)^3*d^3*e^(-6) - 405000*(x^(2/3)*e + d)^2*
d^4*e^(-6) - 216000*((x^(2/3)*e + d)*log(x^(2/3)*e + d)^3 - 3*(x^(2/3)*e + d)*log(x^(2/3)*e + d)^2 + 6*(x^(2/3
)*e + d)*log(x^(2/3)*e + d) - 6*x^(2/3)*e - 6*d)*d^5*e^(-6))*b^3*n^3 + 1/4*a^3*x^4 + 1/2400*(1800*(x^(2/3)*e +
d)^6*e^(-6)*log(x^(2/3)*e + d)^2 - 10800*(x^(2/3)*e + d)^5*d*e^(-6)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e +
d)^4*d^2*e^(-6)*log(x^(2/3)*e + d)^2 - 36000*(x^(2/3)*e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/
3)*e + d)^2*d^4*e^(-6)*log(x^(2/3)*e + d)^2 - 600*(x^(2/3)*e + d)^6*e^(-6)*log(x^(2/3)*e + d) + 4320*(x^(2/3)*
e + d)^5*d*e^(-6)*log(x^(2/3)*e + d) - 13500*(x^(2/3)*e + d)^4*d^2*e^(-6)*log(x^(2/3)*e + d) + 24000*(x^(2/3)*
e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d) - 27000*(x^(2/3)*e + d)^2*d^4*e^(-6)*log(x^(2/3)*e + d) + 100*(x^(2/3)*
e + d)^6*e^(-6) - 864*(x^(2/3)*e + d)^5*d*e^(-6) + 3375*(x^(2/3)*e + d)^4*d^2*e^(-6) - 8000*(x^(2/3)*e + d)^3*
d^3*e^(-6) + 13500*(x^(2/3)*e + d)^2*d^4*e^(-6) - 10800*((x^(2/3)*e + d)*log(x^(2/3)*e + d)^2 - 2*(x^(2/3)*e +
d)*log(x^(2/3)*e + d) + 2*x^(2/3)*e + 2*d)*d^5*e^(-6))*b^3*n^2*log(c) + 1/80*(60*(x^(2/3)*e + d)^6*e^(-6)*log
(x^(2/3)*e + d) - 360*(x^(2/3)*e + d)^5*d*e^(-6)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^4*d^2*e^(-6)*log(x^(
2/3)*e + d) - 1200*(x^(2/3)*e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^2*d^4*e^(-6)*log(x^(2
/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-6) + 72*(x^(2/3)*e + d)^5*d*e^(-6) - 225*(x^(2/3)*e + d)^4*d^2*e^(-6) +
400*(x^(2/3)*e + d)^3*d^3*e^(-6) - 450*(x^(2/3)*e + d)^2*d^4*e^(-6) - 360*((x^(2/3)*e + d)*log(x^(2/3)*e + d)
- x^(2/3)*e - d)*d^5*e^(-6))*b^3*n*log(c)^2 + 1/2400*(1800*(x^(2/3)*e + d)^6*e^(-6)*log(x^(2/3)*e + d)^2 - 108
00*(x^(2/3)*e + d)^5*d*e^(-6)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e + d)^4*d^2*e^(-6)*log(x^(2/3)*e + d)^2 -
36000*(x^(2/3)*e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e + d)^2*d^4*e^(-6)*log(x^(2/3)*e +
d)^2 - 600*(x^(2/3)*e + d)^6*e^(-6)*log(x^(2/3)*e + d) + 4320*(x^(2/3)*e + d)^5*d*e^(-6)*log(x^(2/3)*e + d) -
13500*(x^(2/3)*e + d)^4*d^2*e^(-6)*log(x^(2/3)*e + d) + 24000*(x^(2/3)*e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d)
- 27000*(x^(2/3)*e + d)^2*d^4*e^(-6)*log(x^(2/3)*e + d) + 100*(x^(2/3)*e + d)^6*e^(-6) - 864*(x^(2/3)*e + d)^5
*d*e^(-6) + 3375*(x^(2/3)*e + d)^4*d^2*e^(-6) - 8000*(x^(2/3)*e + d)^3*d^3*e^(-6) + 13500*(x^(2/3)*e + d)^2*d^
4*e^(-6) - 10800*((x^(2/3)*e + d)*log(x^(2/3)*e + d)^2 - 2*(x^(2/3)*e + d)*log(x^(2/3)*e + d) + 2*x^(2/3)*e +
2*d)*d^5*e^(-6))*a*b^2*n^2 + 1/40*(60*(x^(2/3)*e + d)^6*e^(-6)*log(x^(2/3)*e + d) - 360*(x^(2/3)*e + d)^5*d*e^
(-6)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^4*d^2*e^(-6)*log(x^(2/3)*e + d) - 1200*(x^(2/3)*e + d)^3*d^3*e^(
-6)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^2*d^4*e^(-6)*log(x^(2/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-6) + 7
2*(x^(2/3)*e + d)^5*d*e^(-6) - 225*(x^(2/3)*e + d)^4*d^2*e^(-6) + 400*(x^(2/3)*e + d)^3*d^3*e^(-6) - 450*(x^(2
/3)*e + d)^2*d^4*e^(-6) - 360*((x^(2/3)*e + d)*log(x^(2/3)*e + d) - x^(2/3)*e - d)*d^5*e^(-6))*a*b^2*n*log(c)
+ 1/80*(60*(x^(2/3)*e + d)^6*e^(-6)*log(x^(2/3)*e + d) - 360*(x^(2/3)*e + d)^5*d*e^(-6)*log(x^(2/3)*e + d) + 9
00*(x^(2/3)*e + d)^4*d^2*e^(-6)*log(x^(2/3)*e + d) - 1200*(x^(2/3)*e + d)^3*d^3*e^(-6)*log(x^(2/3)*e + d) + 90
0*(x^(2/3)*e + d)^2*d^4*e^(-6)*log(x^(2/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-6) + 72*(x^(2/3)*e + d)^5*d*e^(-6
) - 225*(x^(2/3)*e + d)^4*d^2*e^(-6) + 400*(x^(2/3)*e + d)^3*d^3*e^(-6) - 450*(x^(2/3)*e + d)^2*d^4*e^(-6) - 3
60*((x^(2/3)*e + d)*log(x^(2/3)*e + d) - x^(2/3)*e - d)*d^5*e^(-6))*a^2*b*n

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Mupad [B]
time = 8.10, size = 992, normalized size = 1.09 \begin {gather*} \frac {a^3\,x^4}{4}+\frac {b^3\,x^4\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^3}{4}-\frac {b^3\,n^3\,x^4}{144}+\frac {3\,a\,b^2\,x^4\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{4}-\frac {b^3\,n\,x^4\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{8}+\frac {b^3\,n^2\,x^4\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{24}+\frac {a\,b^2\,n^2\,x^4}{24}-\frac {b^3\,d^6\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^3}{4\,e^6}+\frac {3\,a^2\,b\,x^4\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{4}-\frac {a^2\,b\,n\,x^4}{8}-\frac {a\,b^2\,n\,x^4\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{4}-\frac {13489\,b^3\,d^6\,n^3\,\ln \left (d+e\,x^{2/3}\right )}{2400\,e^6}+\frac {2059\,b^3\,d^3\,n^3\,x^2}{7200\,e^3}-\frac {919\,b^3\,d^2\,n^3\,x^{8/3}}{9600\,e^2}-\frac {4669\,b^3\,d^4\,n^3\,x^{4/3}}{4800\,e^4}+\frac {13489\,b^3\,d^5\,n^3\,x^{2/3}}{2400\,e^5}-\frac {3\,a\,b^2\,d^6\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{4\,e^6}+\frac {147\,b^3\,d^6\,n\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{80\,e^6}+\frac {91\,b^3\,d\,n^3\,x^{10/3}}{3000\,e}-\frac {3\,a^2\,b\,d^6\,n\,\ln \left (d+e\,x^{2/3}\right )}{4\,e^6}+\frac {3\,b^3\,d\,n\,x^{10/3}\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{20\,e}-\frac {11\,b^3\,d\,n^2\,x^{10/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{100\,e}+\frac {a^2\,b\,d^3\,n\,x^2}{4\,e^3}-\frac {3\,a^2\,b\,d^2\,n\,x^{8/3}}{16\,e^2}-\frac {3\,a^2\,b\,d^4\,n\,x^{4/3}}{8\,e^4}+\frac {3\,a^2\,b\,d^5\,n\,x^{2/3}}{4\,e^5}-\frac {11\,a\,b^2\,d\,n^2\,x^{10/3}}{100\,e}+\frac {147\,a\,b^2\,d^6\,n^2\,\ln \left (d+e\,x^{2/3}\right )}{40\,e^6}+\frac {b^3\,d^3\,n\,x^2\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{4\,e^3}-\frac {19\,b^3\,d^3\,n^2\,x^2\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{40\,e^3}-\frac {3\,b^3\,d^2\,n\,x^{8/3}\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{16\,e^2}+\frac {37\,b^3\,d^2\,n^2\,x^{8/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{160\,e^2}-\frac {3\,b^3\,d^4\,n\,x^{4/3}\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{8\,e^4}+\frac {87\,b^3\,d^4\,n^2\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{80\,e^4}+\frac {3\,b^3\,d^5\,n\,x^{2/3}\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{4\,e^5}-\frac {147\,b^3\,d^5\,n^2\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{40\,e^5}-\frac {19\,a\,b^2\,d^3\,n^2\,x^2}{40\,e^3}+\frac {37\,a\,b^2\,d^2\,n^2\,x^{8/3}}{160\,e^2}+\frac {87\,a\,b^2\,d^4\,n^2\,x^{4/3}}{80\,e^4}-\frac {147\,a\,b^2\,d^5\,n^2\,x^{2/3}}{40\,e^5}+\frac {3\,a^2\,b\,d\,n\,x^{10/3}}{20\,e}+\frac {3\,a\,b^2\,d\,n\,x^{10/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{10\,e}+\frac {a\,b^2\,d^3\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{2\,e^3}-\frac {3\,a\,b^2\,d^2\,n\,x^{8/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{8\,e^2}-\frac {3\,a\,b^2\,d^4\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{4\,e^4}+\frac {3\,a\,b^2\,d^5\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{2\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*x^4)/4 + (b^3*x^4*log(c*(d + e*x^(2/3))^n)^3)/4 - (b^3*n^3*x^4)/144 + (3*a*b^2*x^4*log(c*(d + e*x^(2/3))^
n)^2)/4 - (b^3*n*x^4*log(c*(d + e*x^(2/3))^n)^2)/8 + (b^3*n^2*x^4*log(c*(d + e*x^(2/3))^n))/24 + (a*b^2*n^2*x^
4)/24 - (b^3*d^6*log(c*(d + e*x^(2/3))^n)^3)/(4*e^6) + (3*a^2*b*x^4*log(c*(d + e*x^(2/3))^n))/4 - (a^2*b*n*x^4
)/8 - (a*b^2*n*x^4*log(c*(d + e*x^(2/3))^n))/4 - (13489*b^3*d^6*n^3*log(d + e*x^(2/3)))/(2400*e^6) + (2059*b^3
*d^3*n^3*x^2)/(7200*e^3) - (919*b^3*d^2*n^3*x^(8/3))/(9600*e^2) - (4669*b^3*d^4*n^3*x^(4/3))/(4800*e^4) + (134
89*b^3*d^5*n^3*x^(2/3))/(2400*e^5) - (3*a*b^2*d^6*log(c*(d + e*x^(2/3))^n)^2)/(4*e^6) + (147*b^3*d^6*n*log(c*(
d + e*x^(2/3))^n)^2)/(80*e^6) + (91*b^3*d*n^3*x^(10/3))/(3000*e) - (3*a^2*b*d^6*n*log(d + e*x^(2/3)))/(4*e^6)
+ (3*b^3*d*n*x^(10/3)*log(c*(d + e*x^(2/3))^n)^2)/(20*e) - (11*b^3*d*n^2*x^(10/3)*log(c*(d + e*x^(2/3))^n))/(1
00*e) + (a^2*b*d^3*n*x^2)/(4*e^3) - (3*a^2*b*d^2*n*x^(8/3))/(16*e^2) - (3*a^2*b*d^4*n*x^(4/3))/(8*e^4) + (3*a^
2*b*d^5*n*x^(2/3))/(4*e^5) - (11*a*b^2*d*n^2*x^(10/3))/(100*e) + (147*a*b^2*d^6*n^2*log(d + e*x^(2/3)))/(40*e^
6) + (b^3*d^3*n*x^2*log(c*(d + e*x^(2/3))^n)^2)/(4*e^3) - (19*b^3*d^3*n^2*x^2*log(c*(d + e*x^(2/3))^n))/(40*e^
3) - (3*b^3*d^2*n*x^(8/3)*log(c*(d + e*x^(2/3))^n)^2)/(16*e^2) + (37*b^3*d^2*n^2*x^(8/3)*log(c*(d + e*x^(2/3))
^n))/(160*e^2) - (3*b^3*d^4*n*x^(4/3)*log(c*(d + e*x^(2/3))^n)^2)/(8*e^4) + (87*b^3*d^4*n^2*x^(4/3)*log(c*(d +
e*x^(2/3))^n))/(80*e^4) + (3*b^3*d^5*n*x^(2/3)*log(c*(d + e*x^(2/3))^n)^2)/(4*e^5) - (147*b^3*d^5*n^2*x^(2/3)
*log(c*(d + e*x^(2/3))^n))/(40*e^5) - (19*a*b^2*d^3*n^2*x^2)/(40*e^3) + (37*a*b^2*d^2*n^2*x^(8/3))/(160*e^2) +
(87*a*b^2*d^4*n^2*x^(4/3))/(80*e^4) - (147*a*b^2*d^5*n^2*x^(2/3))/(40*e^5) + (3*a^2*b*d*n*x^(10/3))/(20*e) +
(3*a*b^2*d*n*x^(10/3)*log(c*(d + e*x^(2/3))^n))/(10*e) + (a*b^2*d^3*n*x^2*log(c*(d + e*x^(2/3))^n))/(2*e^3) -
(3*a*b^2*d^2*n*x^(8/3)*log(c*(d + e*x^(2/3))^n))/(8*e^2) - (3*a*b^2*d^4*n*x^(4/3)*log(c*(d + e*x^(2/3))^n))/(4
*e^4) + (3*a*b^2*d^5*n*x^(2/3)*log(c*(d + e*x^(2/3))^n))/(2*e^5)

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