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

Optimal. Leaf size=913 \[ -\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} \]

[Out]

1/4*(d+e*x^(2/3))^6*(a+b*ln(c*(d+e*x^(2/3))^n))^3/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^6-10/3*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^6-45/16*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^5-3/2*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^6-1/144*b^3*n^3*(d+e*x^(2/3))^6/e^6+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^6-9*a*b^2*d^5*n^2*x^(2/3)/e^5

________________________________________________________________________________________

Rubi [A]  time = 1.03, 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 = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ -\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} \]

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 2295

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

Rule 2296

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 2304

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 2305

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 2389

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 2390

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 2401

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 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])

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} \operatorname {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} \operatorname {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 \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 1.07, size = 598, normalized size = 0.65 \[ \frac {-60 b \left (1800 a^2 \left (d^6-e^6 x^4\right )-60 a b n \left (147 d^6+60 d^5 e x^{2/3}-30 d^4 e^2 x^{4/3}+20 d^3 e^3 x^2-15 d^2 e^4 x^{8/3}+12 d e^5 x^{10/3}-10 e^6 x^4\right )+b^2 n^2 \left (8820 d^6+8820 d^5 e x^{2/3}-2610 d^4 e^2 x^{4/3}+1140 d^3 e^3 x^2-555 d^2 e^4 x^{8/3}+264 d e^5 x^{10/3}-100 e^6 x^4\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+e x^{2/3} \left (36000 a^3 e^5 x^{10/3}+1800 a^2 b n \left (60 d^5-30 d^4 e x^{2/3}+20 d^3 e^2 x^{4/3}-15 d^2 e^3 x^2+12 d e^4 x^{8/3}-10 e^5 x^{10/3}\right )-60 a b^2 n^2 \left (8820 d^5-2610 d^4 e x^{2/3}+1140 d^3 e^2 x^{4/3}-555 d^2 e^3 x^2+264 d e^4 x^{8/3}-100 e^5 x^{10/3}\right )+b^3 n^3 \left (809340 d^5-140070 d^4 e x^{2/3}+41180 d^3 e^2 x^{4/3}-13785 d^2 e^3 x^2+4368 d e^4 x^{8/3}-1000 e^5 x^{10/3}\right )\right )+1800 b^2 \left (b n \left (147 d^6+60 d^5 e x^{2/3}-30 d^4 e^2 x^{4/3}+20 d^3 e^3 x^2-15 d^2 e^4 x^{8/3}+12 d e^5 x^{10/3}-10 e^6 x^4\right )-60 a \left (d^6-e^6 x^4\right )\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^4\right ) \log ^3\left (c \left (d+e x^{2/3}\right )^n\right )-280140 b^3 d^6 n^3 \log \left (d+e x^{2/3}\right )}{144000 e^6} \]

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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fricas [A]  time = 0.63, size = 1241, normalized size = 1.36 \[ \text {result too large to display} \]

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*e^6*x^4*log(c)^3 - 1000*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n - 36*a^3*e^6)*x^4
+ 36000*(b^3*e^6*n^3*x^4 - b^3*d^6*n^3)*log(e*x^(2/3) + d)^3 + 20*(2059*b^3*d^3*e^3*n^3 - 3420*a*b^2*d^3*e^3*n
^2 + 1800*a^2*b*d^3*e^3*n)*x^2 + 1800*(20*b^3*d^3*e^3*n^3*x^2 + 147*b^3*d^6*n^3 - 60*a*b^2*d^6*n^2 - 10*(b^3*e
^6*n^3 - 6*a*b^2*e^6*n^2)*x^4 + 60*(b^3*e^6*n^2*x^4 - b^3*d^6*n^2)*log(c) - 15*(b^3*d^2*e^4*n^3*x^2 - 4*b^3*d^
5*e*n^3)*x^(2/3) + 6*(2*b^3*d*e^5*n^3*x^3 - 5*b^3*d^4*e^2*n^3*x)*x^(1/3))*log(e*x^(2/3) + d)^2 + 18000*(2*b^3*
d^3*e^3*n*x^2 - (b^3*e^6*n - 6*a*b^2*e^6)*x^4)*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*e^6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n)*x^4 + 60*(19*b^3*d^3*e^3*n^3 - 20*a*b^2*d^3*e
^3*n^2)*x^2 - 1800*(b^3*e^6*n*x^4 - b^3*d^6*n)*log(c)^2 - 60*(20*b^3*d^3*e^3*n^2*x^2 + 147*b^3*d^6*n^2 - 60*a*
b^2*d^6*n - 10*(b^3*e^6*n^2 - 6*a*b^2*e^6*n)*x^4)*log(c) + 15*(588*b^3*d^5*e*n^3 - 240*a*b^2*d^5*e*n^2 - (37*b
^3*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2)*x^2 + 60*(b^3*d^2*e^4*n^2*x^2 - 4*b^3*d^5*e*n^2)*log(c))*x^(2/3) + 6*(4
*(11*b^3*d*e^5*n^3 - 30*a*b^2*d*e^5*n^2)*x^3 - 15*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4*e^2*n^2)*x - 60*(2*b^3*d*
e^5*n^2*x^3 - 5*b^3*d^4*e^2*n^2*x)*log(c))*x^(1/3))*log(e*x^(2/3) + d) + 1200*(5*(b^3*e^6*n^2 - 6*a*b^2*e^6*n
+ 18*a^2*b*e^6)*x^4 - 3*(19*b^3*d^3*e^3*n^2 - 20*a*b^2*d^3*e^3*n)*x^2)*log(c) + 15*(53956*b^3*d^5*e*n^3 - 3528
0*a*b^2*d^5*e*n^2 + 7200*a^2*b*d^5*e*n - (919*b^3*d^2*e^4*n^3 - 2220*a*b^2*d^2*e^4*n^2 + 1800*a^2*b*d^2*e^4*n)
*x^2 - 1800*(b^3*d^2*e^4*n*x^2 - 4*b^3*d^5*e*n)*log(c)^2 - 60*(588*b^3*d^5*e*n^2 - 240*a*b^2*d^5*e*n - (37*b^3
*d^2*e^4*n^2 - 60*a*b^2*d^2*e^4*n)*x^2)*log(c))*x^(2/3) + 6*(8*(91*b^3*d*e^5*n^3 - 330*a*b^2*d*e^5*n^2 + 450*a
^2*b*d*e^5*n)*x^3 + 1800*(2*b^3*d*e^5*n*x^3 - 5*b^3*d^4*e^2*n*x)*log(c)^2 - 5*(4669*b^3*d^4*e^2*n^3 - 5220*a*b
^2*d^4*e^2*n^2 + 1800*a^2*b*d^4*e^2*n)*x - 60*(4*(11*b^3*d*e^5*n^2 - 30*a*b^2*d*e^5*n)*x^3 - 15*(29*b^3*d^4*e^
2*n^2 - 20*a*b^2*d^4*e^2*n)*x)*log(c))*x^(1/3))/e^6

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giac [B]  time = 1.35, size = 2224, normalized size = 2.44 \[ \text {result too large to display} \]

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 + 1/144000*(36000*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e + d)^3
- 216000*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d)^3 + 540000*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e +
d)^3 - 720000*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d)^3 + 540000*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(
2/3)*e + d)^3 - 216000*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d)^3 - 18000*(x^(2/3)*e + d)^6*e^(-5)*log(x^
(2/3)*e + d)^2 + 129600*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d)^2 - 405000*(x^(2/3)*e + d)^4*d^2*e^(-5)*
log(x^(2/3)*e + d)^2 + 720000*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d)^2 - 810000*(x^(2/3)*e + d)^2*d^4
*e^(-5)*log(x^(2/3)*e + d)^2 + 648000*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d)^2 + 6000*(x^(2/3)*e + d)^6
*e^(-5)*log(x^(2/3)*e + d) - 51840*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d) + 202500*(x^(2/3)*e + d)^4*d^
2*e^(-5)*log(x^(2/3)*e + d) - 480000*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d) + 810000*(x^(2/3)*e + d)^
2*d^4*e^(-5)*log(x^(2/3)*e + d) - 1296000*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d) - 1000*(x^(2/3)*e + d)
^6*e^(-5) + 10368*(x^(2/3)*e + d)^5*d*e^(-5) - 50625*(x^(2/3)*e + d)^4*d^2*e^(-5) + 160000*(x^(2/3)*e + d)^3*d
^3*e^(-5) - 405000*(x^(2/3)*e + d)^2*d^4*e^(-5) + 1296000*(x^(2/3)*e + d)*d^5*e^(-5))*b^3*n^3*e^(-1) + 3/4*a^2
*b*x^4*log(c) + 1/2400*(1800*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e + d)^2 - 10800*(x^(2/3)*e + d)^5*d*e^(-5)*
log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e + d)^2 - 36000*(x^(2/3)*e + d)^3*d^3*e
^(-5)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*e + d)^2 - 10800*(x^(2/3)*e + d)*d
^5*e^(-5)*log(x^(2/3)*e + d)^2 - 600*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e + d) + 4320*(x^(2/3)*e + d)^5*d*e^
(-5)*log(x^(2/3)*e + d) - 13500*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e + d) + 24000*(x^(2/3)*e + d)^3*d^3*
e^(-5)*log(x^(2/3)*e + d) - 27000*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*e + d) + 21600*(x^(2/3)*e + d)*d^5*
e^(-5)*log(x^(2/3)*e + d) + 100*(x^(2/3)*e + d)^6*e^(-5) - 864*(x^(2/3)*e + d)^5*d*e^(-5) + 3375*(x^(2/3)*e +
d)^4*d^2*e^(-5) - 8000*(x^(2/3)*e + d)^3*d^3*e^(-5) + 13500*(x^(2/3)*e + d)^2*d^4*e^(-5) - 21600*(x^(2/3)*e +
d)*d^5*e^(-5))*b^3*n^2*e^(-1)*log(c) + 1/80*(60*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e + d) - 360*(x^(2/3)*e +
d)^5*d*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e + d) - 1200*(x^(2/3)*e + d)
^3*d^3*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*e + d) - 360*(x^(2/3)*e + d)*d
^5*e^(-5)*log(x^(2/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-5) + 72*(x^(2/3)*e + d)^5*d*e^(-5) - 225*(x^(2/3)*e +
d)^4*d^2*e^(-5) + 400*(x^(2/3)*e + d)^3*d^3*e^(-5) - 450*(x^(2/3)*e + d)^2*d^4*e^(-5) + 360*(x^(2/3)*e + d)*d^
5*e^(-5))*b^3*n*e^(-1)*log(c)^2 + 1/4*a^3*x^4 + 1/2400*(1800*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e + d)^2 - 1
0800*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e + d)^2
- 36000*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*e
+ d)^2 - 10800*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d)^2 - 600*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e +
d) + 4320*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d) - 13500*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e + d
) + 24000*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d) - 27000*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*e +
d) + 21600*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d) + 100*(x^(2/3)*e + d)^6*e^(-5) - 864*(x^(2/3)*e + d)
^5*d*e^(-5) + 3375*(x^(2/3)*e + d)^4*d^2*e^(-5) - 8000*(x^(2/3)*e + d)^3*d^3*e^(-5) + 13500*(x^(2/3)*e + d)^2*
d^4*e^(-5) - 21600*(x^(2/3)*e + d)*d^5*e^(-5))*a*b^2*n^2*e^(-1) + 1/40*(60*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3
)*e + d) - 360*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e
+ d) - 1200*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*e +
d) - 360*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-5) + 72*(x^(2/3)*e + d)^5*d
*e^(-5) - 225*(x^(2/3)*e + d)^4*d^2*e^(-5) + 400*(x^(2/3)*e + d)^3*d^3*e^(-5) - 450*(x^(2/3)*e + d)^2*d^4*e^(-
5) + 360*(x^(2/3)*e + d)*d^5*e^(-5))*a*b^2*n*e^(-1)*log(c) + 1/80*(60*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e +
d) - 360*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e + d)
- 1200*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*e + d) -
360*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-5) + 72*(x^(2/3)*e + d)^5*d*e^(-
5) - 225*(x^(2/3)*e + d)^4*d^2*e^(-5) + 400*(x^(2/3)*e + d)^3*d^3*e^(-5) - 450*(x^(2/3)*e + d)^2*d^4*e^(-5) +
360*(x^(2/3)*e + d)*d^5*e^(-5))*a^2*b*n*e^(-1)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{3} x^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 680, normalized size = 0.74 \[ \frac {1}{4} \, b^{3} x^{4} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{3} + \frac {3}{4} \, a b^{2} x^{4} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + \frac {3}{4} \, a^{2} b x^{4} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + \frac {1}{4} \, a^{3} x^{4} - \frac {1}{80} \, a^{2} b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac {10}{3}} + 15 \, d^{2} e^{3} x^{\frac {8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac {4}{3}} - 60 \, d^{5} x^{\frac {2}{3}}}{e^{6}}\right )} - \frac {1}{2400} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac {10}{3}} + 15 \, d^{2} e^{3} x^{\frac {8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac {4}{3}} - 60 \, d^{5} x^{\frac {2}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{4} - 264 \, d e^{5} x^{\frac {10}{3}} + 555 \, d^{2} e^{4} x^{\frac {8}{3}} - 1140 \, d^{3} e^{3} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 2610 \, d^{4} e^{2} x^{\frac {4}{3}} + 8820 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right ) - 8820 \, d^{5} e x^{\frac {2}{3}}\right )} n^{2}}{e^{6}}\right )} a b^{2} - \frac {1}{144000} \, {\left (1800 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac {10}{3}} + 15 \, d^{2} e^{3} x^{\frac {8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac {4}{3}} - 60 \, d^{5} x^{\frac {2}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + e n {\left (\frac {{\left (1000 \, e^{6} x^{4} - 4368 \, d e^{5} x^{\frac {10}{3}} + 36000 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )^{3} + 13785 \, d^{2} e^{4} x^{\frac {8}{3}} - 41180 \, d^{3} e^{3} x^{2} + 264600 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 140070 \, d^{4} e^{2} x^{\frac {4}{3}} + 809340 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right ) - 809340 \, d^{5} e x^{\frac {2}{3}}\right )} n^{2}}{e^{7}} - \frac {60 \, {\left (100 \, e^{6} x^{4} - 264 \, d e^{5} x^{\frac {10}{3}} + 555 \, d^{2} e^{4} x^{\frac {8}{3}} - 1140 \, d^{3} e^{3} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 2610 \, d^{4} e^{2} x^{\frac {4}{3}} + 8820 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right ) - 8820 \, d^{5} e x^{\frac {2}{3}}\right )} n \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )}{e^{7}}\right )}\right )} b^{3} \]

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

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mupad [B]  time = 8.10, size = 992, normalized size = 1.09 \[ \text {result too large to display} \]

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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