3.6.7 \(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n))^3}{x^3} \, dx\) [507]

Optimal. Leaf size=907 \[ \frac {45 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{8 e^6}-\frac {20 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}+\frac {45 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{64 e^6}-\frac {18 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{125 e^6}+\frac {b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{72 e^6}+\frac {18 a b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {18 b^3 d^5 n^3}{e^5 \sqrt [3]{x}}+\frac {18 b^3 d^5 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{e^6}-\frac {45 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}-\frac {45 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{16 e^6}+\frac {18 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 e^6}-\frac {9 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{8 e^6}-\frac {9 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}+\frac {3 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {10 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {3 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6} \]

[Out]

-1/12*b^2*n^2*(d+e/x^(1/3))^6*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6+1/4*b*n*(d+e/x^(1/3))^6*(a+b*ln(c*(d+e/x^(1/3))^
n))^2/e^6+45/8*b^3*d^4*n^3*(d+e/x^(1/3))^2/e^6-20/9*b^3*d^3*n^3*(d+e/x^(1/3))^3/e^6+45/64*b^3*d^2*n^3*(d+e/x^(
1/3))^4/e^6-18/125*b^3*d*n^3*(d+e/x^(1/3))^5/e^6-18*b^3*d^5*n^3/e^5/x^(1/3)-1/2*(d+e/x^(1/3))^6*(a+b*ln(c*(d+e
/x^(1/3))^n))^3/e^6-9*b*d^5*n*(d+e/x^(1/3))*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^6+45/4*b*d^4*n*(d+e/x^(1/3))^2*(a+
b*ln(c*(d+e/x^(1/3))^n))^2/e^6-10*b*d^3*n*(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^6+45/8*b*d^2*n*(d+e/
x^(1/3))^4*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^6-9/5*b*d*n*(d+e/x^(1/3))^5*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^6+18*b^
3*d^5*n^2*(d+e/x^(1/3))*ln(c*(d+e/x^(1/3))^n)/e^6-45/4*b^2*d^4*n^2*(d+e/x^(1/3))^2*(a+b*ln(c*(d+e/x^(1/3))^n))
/e^6+20/3*b^2*d^3*n^2*(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6-45/16*b^2*d^2*n^2*(d+e/x^(1/3))^4*(a+b*l
n(c*(d+e/x^(1/3))^n))/e^6+18/25*b^2*d*n^2*(d+e/x^(1/3))^5*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6+18*a*b^2*d^5*n^2/e^5
/x^(1/3)+3*d^5*(d+e/x^(1/3))*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^6-15/2*d^4*(d+e/x^(1/3))^2*(a+b*ln(c*(d+e/x^(1/3)
)^n))^3/e^6+10*d^3*(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^6-15/2*d^2*(d+e/x^(1/3))^4*(a+b*ln(c*(d+e/x
^(1/3))^n))^3/e^6+3*d*(d+e/x^(1/3))^5*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^6+1/72*b^3*n^3*(d+e/x^(1/3))^6/e^6

________________________________________________________________________________________

Rubi [A]
time = 0.64, antiderivative size = 907, 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+\frac {e}{\sqrt [3]{x}}\right )^6}{72 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{2 e^6}+\frac {b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{4 e^6}-\frac {b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{12 e^6}-\frac {18 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{125 e^6}+\frac {3 d \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{e^6}-\frac {9 b d n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{5 e^6}+\frac {18 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}+\frac {45 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{64 e^6}-\frac {15 d^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{2 e^6}+\frac {45 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{8 e^6}-\frac {45 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}-\frac {20 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}+\frac {10 d^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{e^6}-\frac {10 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{e^6}+\frac {20 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{3 e^6}+\frac {45 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{8 e^6}-\frac {15 d^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{2 e^6}+\frac {45 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}-\frac {45 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {3 d^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {9 b d^5 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {18 b^3 d^5 n^2 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {18 b^3 d^5 n^3}{e^5 \sqrt [3]{x}}+\frac {18 a b^2 d^5 n^2}{e^5 \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(45*b^3*d^4*n^3*(d + e/x^(1/3))^2)/(8*e^6) - (20*b^3*d^3*n^3*(d + e/x^(1/3))^3)/(9*e^6) + (45*b^3*d^2*n^3*(d +
e/x^(1/3))^4)/(64*e^6) - (18*b^3*d*n^3*(d + e/x^(1/3))^5)/(125*e^6) + (b^3*n^3*(d + e/x^(1/3))^6)/(72*e^6) +
(18*a*b^2*d^5*n^2)/(e^5*x^(1/3)) - (18*b^3*d^5*n^3)/(e^5*x^(1/3)) + (18*b^3*d^5*n^2*(d + e/x^(1/3))*Log[c*(d +
e/x^(1/3))^n])/e^6 - (45*b^2*d^4*n^2*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(4*e^6) + (20*b^2*d^
3*n^2*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*e^6) - (45*b^2*d^2*n^2*(d + e/x^(1/3))^4*(a + b*L
og[c*(d + e/x^(1/3))^n]))/(16*e^6) + (18*b^2*d*n^2*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n]))/(25*e^6
) - (b^2*n^2*(d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n]))/(12*e^6) - (9*b*d^5*n*(d + e/x^(1/3))*(a + b*
Log[c*(d + e/x^(1/3))^n])^2)/e^6 + (45*b*d^4*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(4*e^6) -
(10*b*d^3*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^6 + (45*b*d^2*n*(d + e/x^(1/3))^4*(a + b*
Log[c*(d + e/x^(1/3))^n])^2)/(8*e^6) - (9*b*d*n*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(5*e^6)
+ (b*n*(d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(4*e^6) + (3*d^5*(d + e/x^(1/3))*(a + b*Log[c*(d
+ e/x^(1/3))^n])^3)/e^6 - (15*d^4*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/(2*e^6) + (10*d^3*(d +
e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^6 - (15*d^2*(d + e/x^(1/3))^4*(a + b*Log[c*(d + e/x^(1/3))
^n])^3)/(2*e^6) + (3*d*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^6 - ((d + e/x^(1/3))^6*(a + b*L
og[c*(d + e/x^(1/3))^n])^3)/(2*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 \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx &=-\left (3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \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,\frac {1}{\sqrt [3]{x}}\right )\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,\frac {1}{\sqrt [3]{x}}\right )}{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,\frac {1}{\sqrt [3]{x}}\right )}{e^5}-\frac {\left (30 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,\frac {1}{\sqrt [3]{x}}\right )}{e^5}+\frac {\left (30 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,\frac {1}{\sqrt [3]{x}}\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,\frac {1}{\sqrt [3]{x}}\right )}{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,\frac {1}{\sqrt [3]{x}}\right )}{e^5}\\ &=-\frac {3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{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+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {\left (30 d^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {\left (30 d^3\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\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+\frac {e}{\sqrt [3]{x}}\right )}{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+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\\ &=\frac {3 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {10 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {3 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 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+\frac {e}{\sqrt [3]{x}}\right )}{2 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+\frac {e}{\sqrt [3]{x}}\right )}{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+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac {\left (30 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+\frac {e}{\sqrt [3]{x}}\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+\frac {e}{\sqrt [3]{x}}\right )}{2 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+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\\ &=-\frac {9 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{8 e^6}-\frac {9 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}+\frac {3 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {10 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {3 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 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+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}+\frac {\left (18 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+\frac {e}{\sqrt [3]{x}}\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+\frac {e}{\sqrt [3]{x}}\right )}{4 e^6}+\frac {\left (20 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+\frac {e}{\sqrt [3]{x}}\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+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}+\frac {\left (18 b^2 d^5 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\\ &=\frac {45 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{8 e^6}-\frac {20 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}+\frac {45 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{64 e^6}-\frac {18 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{125 e^6}+\frac {b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{72 e^6}+\frac {18 a b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {45 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}-\frac {45 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{16 e^6}+\frac {18 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 e^6}-\frac {9 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{8 e^6}-\frac {9 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}+\frac {3 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {10 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {3 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (18 b^3 d^5 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\\ &=\frac {45 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{8 e^6}-\frac {20 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}+\frac {45 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{64 e^6}-\frac {18 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{125 e^6}+\frac {b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{72 e^6}+\frac {18 a b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {18 b^3 d^5 n^3}{e^5 \sqrt [3]{x}}+\frac {18 b^3 d^5 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{e^6}-\frac {45 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}-\frac {45 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{16 e^6}+\frac {18 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 e^6}-\frac {9 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{8 e^6}-\frac {9 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}+\frac {3 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {10 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {3 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}\\ \end {align*}

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Mathematica [A]
time = 1.08, size = 962, normalized size = 1.06 \begin {gather*} \frac {-36000 a^3 e^6+18000 a^2 b e^6 n-6000 a b^2 e^6 n^2+1000 b^3 e^6 n^3-21600 a^2 b d e^5 n \sqrt [3]{x}+15840 a b^2 d e^5 n^2 \sqrt [3]{x}-4368 b^3 d e^5 n^3 \sqrt [3]{x}+27000 a^2 b d^2 e^4 n x^{2/3}-33300 a b^2 d^2 e^4 n^2 x^{2/3}+13785 b^3 d^2 e^4 n^3 x^{2/3}-36000 a^2 b d^3 e^3 n x+68400 a b^2 d^3 e^3 n^2 x-41180 b^3 d^3 e^3 n^3 x+54000 a^2 b d^4 e^2 n x^{4/3}-156600 a b^2 d^4 e^2 n^2 x^{4/3}+140070 b^3 d^4 e^2 n^3 x^{4/3}-108000 a^2 b d^5 e n x^{5/3}+529200 a b^2 d^5 e n^2 x^{5/3}-809340 b^3 d^5 e n^3 x^{5/3}-72000 b^3 d^6 n^3 x^2 \log ^3\left (d+\frac {e}{\sqrt [3]{x}}\right )-36000 b^3 e^6 \log ^3\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+108000 a^2 b d^6 n x^2 \log \left (e+d \sqrt [3]{x}\right )-529200 a b^2 d^6 n^2 x^2 \log \left (e+d \sqrt [3]{x}\right )+809340 b^3 d^6 n^3 x^2 \log \left (e+d \sqrt [3]{x}\right )+3600 b^2 d^6 n^2 x^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (-20 a+49 b n-20 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (3 \log \left (e+d \sqrt [3]{x}\right )-\log (x)\right )-36000 a^2 b d^6 n x^2 \log (x)+176400 a b^2 d^6 n^2 x^2 \log (x)-269780 b^3 d^6 n^3 x^2 \log (x)+1800 b^2 d^6 n^2 x^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (60 a-147 b n+60 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+60 b n \log \left (e+d \sqrt [3]{x}\right )-20 b n \log (x)\right )+1800 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (e \left (-60 a e^5+10 b e^5 n-12 b d e^4 n \sqrt [3]{x}+15 b d^2 e^3 n x^{2/3}-20 b d^3 e^2 n x+30 b d^4 e n x^{4/3}-60 b d^5 n x^{5/3}\right )+60 b d^6 n x^2 \log \left (e+d \sqrt [3]{x}\right )-20 b d^6 n x^2 \log (x)\right )-60 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (1800 a^2 e^6+b^2 e n^2 \left (100 e^5-264 d e^4 \sqrt [3]{x}+555 d^2 e^3 x^{2/3}-1140 d^3 e^2 x+2610 d^4 e x^{4/3}-8820 d^5 x^{5/3}\right )-60 a b e n \left (10 e^5-12 d e^4 \sqrt [3]{x}+15 d^2 e^3 x^{2/3}-20 d^3 e^2 x+30 d^4 e x^{4/3}-60 d^5 x^{5/3}\right )+180 b d^6 n (-20 a+49 b n) x^2 \log \left (e+d \sqrt [3]{x}\right )+60 b d^6 n (20 a-49 b n) x^2 \log (x)\right )}{72000 e^6 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-36000*a^3*e^6 + 18000*a^2*b*e^6*n - 6000*a*b^2*e^6*n^2 + 1000*b^3*e^6*n^3 - 21600*a^2*b*d*e^5*n*x^(1/3) + 15
840*a*b^2*d*e^5*n^2*x^(1/3) - 4368*b^3*d*e^5*n^3*x^(1/3) + 27000*a^2*b*d^2*e^4*n*x^(2/3) - 33300*a*b^2*d^2*e^4
*n^2*x^(2/3) + 13785*b^3*d^2*e^4*n^3*x^(2/3) - 36000*a^2*b*d^3*e^3*n*x + 68400*a*b^2*d^3*e^3*n^2*x - 41180*b^3
*d^3*e^3*n^3*x + 54000*a^2*b*d^4*e^2*n*x^(4/3) - 156600*a*b^2*d^4*e^2*n^2*x^(4/3) + 140070*b^3*d^4*e^2*n^3*x^(
4/3) - 108000*a^2*b*d^5*e*n*x^(5/3) + 529200*a*b^2*d^5*e*n^2*x^(5/3) - 809340*b^3*d^5*e*n^3*x^(5/3) - 72000*b^
3*d^6*n^3*x^2*Log[d + e/x^(1/3)]^3 - 36000*b^3*e^6*Log[c*(d + e/x^(1/3))^n]^3 + 108000*a^2*b*d^6*n*x^2*Log[e +
d*x^(1/3)] - 529200*a*b^2*d^6*n^2*x^2*Log[e + d*x^(1/3)] + 809340*b^3*d^6*n^3*x^2*Log[e + d*x^(1/3)] + 3600*b
^2*d^6*n^2*x^2*Log[d + e/x^(1/3)]*(-20*a + 49*b*n - 20*b*Log[c*(d + e/x^(1/3))^n])*(3*Log[e + d*x^(1/3)] - Log
[x]) - 36000*a^2*b*d^6*n*x^2*Log[x] + 176400*a*b^2*d^6*n^2*x^2*Log[x] - 269780*b^3*d^6*n^3*x^2*Log[x] + 1800*b
^2*d^6*n^2*x^2*Log[d + e/x^(1/3)]^2*(60*a - 147*b*n + 60*b*Log[c*(d + e/x^(1/3))^n] + 60*b*n*Log[e + d*x^(1/3)
] - 20*b*n*Log[x]) + 1800*b^2*Log[c*(d + e/x^(1/3))^n]^2*(e*(-60*a*e^5 + 10*b*e^5*n - 12*b*d*e^4*n*x^(1/3) + 1
5*b*d^2*e^3*n*x^(2/3) - 20*b*d^3*e^2*n*x + 30*b*d^4*e*n*x^(4/3) - 60*b*d^5*n*x^(5/3)) + 60*b*d^6*n*x^2*Log[e +
d*x^(1/3)] - 20*b*d^6*n*x^2*Log[x]) - 60*b*Log[c*(d + e/x^(1/3))^n]*(1800*a^2*e^6 + b^2*e*n^2*(100*e^5 - 264*
d*e^4*x^(1/3) + 555*d^2*e^3*x^(2/3) - 1140*d^3*e^2*x + 2610*d^4*e*x^(4/3) - 8820*d^5*x^(5/3)) - 60*a*b*e*n*(10
*e^5 - 12*d*e^4*x^(1/3) + 15*d^2*e^3*x^(2/3) - 20*d^3*e^2*x + 30*d^4*e*x^(4/3) - 60*d^5*x^(5/3)) + 180*b*d^6*n
*(-20*a + 49*b*n)*x^2*Log[e + d*x^(1/3)] + 60*b*d^6*n*(20*a - 49*b*n)*x^2*Log[x]))/(72000*e^6*x^2)

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )^{3}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.34, size = 851, normalized size = 0.94 \begin {gather*} \frac {1}{40} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 20 \, d^{6} e^{\left (-7\right )} \log \left (x\right ) - \frac {{\left (60 \, d^{5} x^{\frac {5}{3}} - 30 \, d^{4} x^{\frac {4}{3}} e + 20 \, d^{3} x e^{2} - 15 \, d^{2} x^{\frac {2}{3}} e^{3} + 12 \, d x^{\frac {1}{3}} e^{4} - 10 \, e^{5}\right )} e^{\left (-6\right )}}{x^{2}}\right )} a^{2} b n e + \frac {1}{1200} \, {\left (60 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 20 \, d^{6} e^{\left (-7\right )} \log \left (x\right ) - \frac {{\left (60 \, d^{5} x^{\frac {5}{3}} - 30 \, d^{4} x^{\frac {4}{3}} e + 20 \, d^{3} x e^{2} - 15 \, d^{2} x^{\frac {2}{3}} e^{3} + 12 \, d x^{\frac {1}{3}} e^{4} - 10 \, e^{5}\right )} e^{\left (-6\right )}}{x^{2}}\right )} n e \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) - \frac {{\left (1800 \, d^{6} x^{2} \log \left (d x^{\frac {1}{3}} + e\right )^{2} + 200 \, d^{6} x^{2} \log \left (x\right )^{2} - 2940 \, d^{6} x^{2} \log \left (x\right ) - 8820 \, d^{5} x^{\frac {5}{3}} e + 2610 \, d^{4} x^{\frac {4}{3}} e^{2} - 1140 \, d^{3} x e^{3} + 555 \, d^{2} x^{\frac {2}{3}} e^{4} - 264 \, d x^{\frac {1}{3}} e^{5} - 60 \, {\left (20 \, d^{6} x^{2} \log \left (x\right ) - 147 \, d^{6} x^{2}\right )} \log \left (d x^{\frac {1}{3}} + e\right ) + 100 \, e^{6}\right )} n^{2} e^{\left (-6\right )}}{x^{2}}\right )} a b^{2} + \frac {1}{216000} \, {\left (5400 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 20 \, d^{6} e^{\left (-7\right )} \log \left (x\right ) - \frac {{\left (60 \, d^{5} x^{\frac {5}{3}} - 30 \, d^{4} x^{\frac {4}{3}} e + 20 \, d^{3} x e^{2} - 15 \, d^{2} x^{\frac {2}{3}} e^{3} + 12 \, d x^{\frac {1}{3}} e^{4} - 10 \, e^{5}\right )} e^{\left (-6\right )}}{x^{2}}\right )} n e \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{2} + {\left (\frac {{\left (108000 \, d^{6} x^{2} \log \left (d x^{\frac {1}{3}} + e\right )^{3} - 4000 \, d^{6} x^{2} \log \left (x\right )^{3} + 88200 \, d^{6} x^{2} \log \left (x\right )^{2} - 809340 \, d^{6} x^{2} \log \left (x\right ) - 2428020 \, d^{5} x^{\frac {5}{3}} e + 420210 \, d^{4} x^{\frac {4}{3}} e^{2} - 123540 \, d^{3} x e^{3} + 41355 \, d^{2} x^{\frac {2}{3}} e^{4} - 5400 \, {\left (20 \, d^{6} x^{2} \log \left (x\right ) - 147 \, d^{6} x^{2}\right )} \log \left (d x^{\frac {1}{3}} + e\right )^{2} - 13104 \, d x^{\frac {1}{3}} e^{5} + 180 \, {\left (200 \, d^{6} x^{2} \log \left (x\right )^{2} - 2940 \, d^{6} x^{2} \log \left (x\right ) + 13489 \, d^{6} x^{2}\right )} \log \left (d x^{\frac {1}{3}} + e\right ) + 3000 \, e^{6}\right )} n^{2} e^{\left (-7\right )}}{x^{2}} - \frac {180 \, {\left (1800 \, d^{6} x^{2} \log \left (d x^{\frac {1}{3}} + e\right )^{2} + 200 \, d^{6} x^{2} \log \left (x\right )^{2} - 2940 \, d^{6} x^{2} \log \left (x\right ) - 8820 \, d^{5} x^{\frac {5}{3}} e + 2610 \, d^{4} x^{\frac {4}{3}} e^{2} - 1140 \, d^{3} x e^{3} + 555 \, d^{2} x^{\frac {2}{3}} e^{4} - 264 \, d x^{\frac {1}{3}} e^{5} - 60 \, {\left (20 \, d^{6} x^{2} \log \left (x\right ) - 147 \, d^{6} x^{2}\right )} \log \left (d x^{\frac {1}{3}} + e\right ) + 100 \, e^{6}\right )} n e^{\left (-7\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{x^{2}}\right )} n e\right )} b^{3} - \frac {b^{3} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{3}}{2 \, x^{2}} - \frac {3 \, a b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {3 \, a^{2} b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{2 \, x^{2}} - \frac {a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(60*d^6*e^(-7)*log(d*x^(1/3) + e) - 20*d^6*e^(-7)*log(x) - (60*d^5*x^(5/3) - 30*d^4*x^(4/3)*e + 20*d^3*x*
e^2 - 15*d^2*x^(2/3)*e^3 + 12*d*x^(1/3)*e^4 - 10*e^5)*e^(-6)/x^2)*a^2*b*n*e + 1/1200*(60*(60*d^6*e^(-7)*log(d*
x^(1/3) + e) - 20*d^6*e^(-7)*log(x) - (60*d^5*x^(5/3) - 30*d^4*x^(4/3)*e + 20*d^3*x*e^2 - 15*d^2*x^(2/3)*e^3 +
12*d*x^(1/3)*e^4 - 10*e^5)*e^(-6)/x^2)*n*e*log(c*(d + e/x^(1/3))^n) - (1800*d^6*x^2*log(d*x^(1/3) + e)^2 + 20
0*d^6*x^2*log(x)^2 - 2940*d^6*x^2*log(x) - 8820*d^5*x^(5/3)*e + 2610*d^4*x^(4/3)*e^2 - 1140*d^3*x*e^3 + 555*d^
2*x^(2/3)*e^4 - 264*d*x^(1/3)*e^5 - 60*(20*d^6*x^2*log(x) - 147*d^6*x^2)*log(d*x^(1/3) + e) + 100*e^6)*n^2*e^(
-6)/x^2)*a*b^2 + 1/216000*(5400*(60*d^6*e^(-7)*log(d*x^(1/3) + e) - 20*d^6*e^(-7)*log(x) - (60*d^5*x^(5/3) - 3
0*d^4*x^(4/3)*e + 20*d^3*x*e^2 - 15*d^2*x^(2/3)*e^3 + 12*d*x^(1/3)*e^4 - 10*e^5)*e^(-6)/x^2)*n*e*log(c*(d + e/
x^(1/3))^n)^2 + ((108000*d^6*x^2*log(d*x^(1/3) + e)^3 - 4000*d^6*x^2*log(x)^3 + 88200*d^6*x^2*log(x)^2 - 80934
0*d^6*x^2*log(x) - 2428020*d^5*x^(5/3)*e + 420210*d^4*x^(4/3)*e^2 - 123540*d^3*x*e^3 + 41355*d^2*x^(2/3)*e^4 -
5400*(20*d^6*x^2*log(x) - 147*d^6*x^2)*log(d*x^(1/3) + e)^2 - 13104*d*x^(1/3)*e^5 + 180*(200*d^6*x^2*log(x)^2
- 2940*d^6*x^2*log(x) + 13489*d^6*x^2)*log(d*x^(1/3) + e) + 3000*e^6)*n^2*e^(-7)/x^2 - 180*(1800*d^6*x^2*log(
d*x^(1/3) + e)^2 + 200*d^6*x^2*log(x)^2 - 2940*d^6*x^2*log(x) - 8820*d^5*x^(5/3)*e + 2610*d^4*x^(4/3)*e^2 - 11
40*d^3*x*e^3 + 555*d^2*x^(2/3)*e^4 - 264*d*x^(1/3)*e^5 - 60*(20*d^6*x^2*log(x) - 147*d^6*x^2)*log(d*x^(1/3) +
e) + 100*e^6)*n*e^(-7)*log(c*(d + e/x^(1/3))^n)/x^2)*n*e)*b^3 - 1/2*b^3*log(c*(d + e/x^(1/3))^n)^3/x^2 - 3/2*a
*b^2*log(c*(d + e/x^(1/3))^n)^2/x^2 - 3/2*a^2*b*log(c*(d + e/x^(1/3))^n)/x^2 - 1/2*a^3/x^2

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Fricas [A]
time = 0.44, size = 1290, normalized size = 1.42 \begin {gather*} \frac {{\left (36000 \, {\left (b^{3} x^{2} - b^{3}\right )} e^{6} \log \left (c\right )^{3} + 36000 \, {\left (b^{3} d^{6} n^{3} x^{2} - b^{3} n^{3} e^{6}\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right )^{3} + 18000 \, {\left ({\left (b^{3} n - 6 \, a b^{2} - {\left (b^{3} n - 6 \, a b^{2}\right )} x^{2}\right )} e^{6} + 2 \, {\left (b^{3} d^{3} n x^{2} - b^{3} d^{3} n x\right )} e^{3}\right )} \log \left (c\right )^{2} - 1800 \, {\left (20 \, b^{3} d^{3} n^{3} x e^{3} + 3 \, {\left (49 \, b^{3} d^{6} n^{3} - 20 \, a b^{2} d^{6} n^{2}\right )} x^{2} - 10 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2}\right )} e^{6} - 60 \, {\left (b^{3} d^{6} n^{2} x^{2} - b^{3} n^{2} e^{6}\right )} \log \left (c\right ) + 15 \, {\left (4 \, b^{3} d^{5} n^{3} x e - b^{3} d^{2} n^{3} 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} e^{5}\right )} x^{\frac {1}{3}}\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right )^{2} + 1000 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2} + 18 \, a^{2} b n - 36 \, a^{3} - {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2} + 18 \, a^{2} b n - 36 \, a^{3}\right )} x^{2}\right )} e^{6} + 20 \, {\left ({\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} - {\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\right )} e^{3} - 1200 \, {\left (5 \, {\left (b^{3} n^{2} - 6 \, a b^{2} n + 18 \, a^{2} b - {\left (b^{3} n^{2} - 6 \, a b^{2} n + 18 \, a^{2} b\right )} x^{2}\right )} e^{6} + 3 \, {\left ({\left (19 \, b^{3} d^{3} n^{2} - 20 \, a b^{2} d^{3} n\right )} x^{2} - {\left (19 \, b^{3} d^{3} n^{2} - 20 \, a b^{2} d^{3} n\right )} x\right )} e^{3}\right )} \log \left (c\right ) + 60 \, {\left ({\left (13489 \, b^{3} d^{6} n^{3} - 8820 \, a b^{2} d^{6} n^{2} + 1800 \, a^{2} b d^{6} n\right )} x^{2} + 60 \, {\left (19 \, b^{3} d^{3} n^{3} - 20 \, a b^{2} d^{3} n^{2}\right )} x e^{3} + 1800 \, {\left (b^{3} d^{6} n x^{2} - b^{3} n e^{6}\right )} \log \left (c\right )^{2} - 100 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2} + 18 \, a^{2} b n\right )} e^{6} - 60 \, {\left (20 \, b^{3} d^{3} n^{2} x e^{3} + 3 \, {\left (49 \, b^{3} d^{6} n^{2} - 20 \, a b^{2} d^{6} n\right )} x^{2} - 10 \, {\left (b^{3} n^{2} - 6 \, a b^{2} n\right )} e^{6}\right )} \log \left (c\right ) + 15 \, {\left (12 \, {\left (49 \, b^{3} d^{5} n^{3} - 20 \, a b^{2} d^{5} n^{2}\right )} x e - {\left (37 \, b^{3} d^{2} n^{3} - 60 \, a b^{2} d^{2} n^{2}\right )} e^{4} - 60 \, {\left (4 \, b^{3} d^{5} n^{2} x e - b^{3} d^{2} n^{2} e^{4}\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} - 6 \, {\left (15 \, {\left (29 \, b^{3} d^{4} n^{3} - 20 \, a b^{2} d^{4} n^{2}\right )} x e^{2} - 4 \, {\left (11 \, b^{3} d n^{3} - 30 \, a b^{2} d n^{2}\right )} e^{5} - 60 \, {\left (5 \, b^{3} d^{4} n^{2} x e^{2} - 2 \, b^{3} d n^{2} e^{5}\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right ) - 15 \, {\left (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 )} x e + 1800 \, {\left (4 \, b^{3} d^{5} n x e - b^{3} d^{2} n e^{4}\right )} \log \left (c\right )^{2} - {\left (919 \, b^{3} d^{2} n^{3} - 2220 \, a b^{2} d^{2} n^{2} + 1800 \, a^{2} b d^{2} n\right )} e^{4} - 60 \, {\left (12 \, {\left (49 \, b^{3} d^{5} n^{2} - 20 \, a b^{2} d^{5} n\right )} x e - {\left (37 \, b^{3} d^{2} n^{2} - 60 \, a b^{2} d^{2} n\right )} e^{4}\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} + 6 \, {\left (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 e^{5}\right )} \log \left (c\right )^{2} - 8 \, {\left (91 \, b^{3} d n^{3} - 330 \, a b^{2} d n^{2} + 450 \, a^{2} b d n\right )} e^{5} - 60 \, {\left (15 \, {\left (29 \, b^{3} d^{4} n^{2} - 20 \, a b^{2} d^{4} n\right )} x e^{2} - 4 \, {\left (11 \, b^{3} d n^{2} - 30 \, a b^{2} d n\right )} e^{5}\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}\right )} e^{\left (-6\right )}}{72000 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72000*(36000*(b^3*x^2 - b^3)*e^6*log(c)^3 + 36000*(b^3*d^6*n^3*x^2 - b^3*n^3*e^6)*log((d*x + x^(2/3)*e)/x)^3
+ 18000*((b^3*n - 6*a*b^2 - (b^3*n - 6*a*b^2)*x^2)*e^6 + 2*(b^3*d^3*n*x^2 - b^3*d^3*n*x)*e^3)*log(c)^2 - 1800
*(20*b^3*d^3*n^3*x*e^3 + 3*(49*b^3*d^6*n^3 - 20*a*b^2*d^6*n^2)*x^2 - 10*(b^3*n^3 - 6*a*b^2*n^2)*e^6 - 60*(b^3*
d^6*n^2*x^2 - b^3*n^2*e^6)*log(c) + 15*(4*b^3*d^5*n^3*x*e - b^3*d^2*n^3*e^4)*x^(2/3) - 6*(5*b^3*d^4*n^3*x*e^2
- 2*b^3*d*n^3*e^5)*x^(1/3))*log((d*x + x^(2/3)*e)/x)^2 + 1000*(b^3*n^3 - 6*a*b^2*n^2 + 18*a^2*b*n - 36*a^3 - (
b^3*n^3 - 6*a*b^2*n^2 + 18*a^2*b*n - 36*a^3)*x^2)*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 - (2059*b^3*d^3*n^3 - 3420*a*b^2*d^3*n^2 + 1800*a^2*b*d^3*n)*x)*e^3 - 1200*(5*(b^3*n^2 - 6*a*b^2*
n + 18*a^2*b - (b^3*n^2 - 6*a*b^2*n + 18*a^2*b)*x^2)*e^6 + 3*((19*b^3*d^3*n^2 - 20*a*b^2*d^3*n)*x^2 - (19*b^3*
d^3*n^2 - 20*a*b^2*d^3*n)*x)*e^3)*log(c) + 60*((13489*b^3*d^6*n^3 - 8820*a*b^2*d^6*n^2 + 1800*a^2*b*d^6*n)*x^2
+ 60*(19*b^3*d^3*n^3 - 20*a*b^2*d^3*n^2)*x*e^3 + 1800*(b^3*d^6*n*x^2 - b^3*n*e^6)*log(c)^2 - 100*(b^3*n^3 - 6
*a*b^2*n^2 + 18*a^2*b*n)*e^6 - 60*(20*b^3*d^3*n^2*x*e^3 + 3*(49*b^3*d^6*n^2 - 20*a*b^2*d^6*n)*x^2 - 10*(b^3*n^
2 - 6*a*b^2*n)*e^6)*log(c) + 15*(12*(49*b^3*d^5*n^3 - 20*a*b^2*d^5*n^2)*x*e - (37*b^3*d^2*n^3 - 60*a*b^2*d^2*n
^2)*e^4 - 60*(4*b^3*d^5*n^2*x*e - b^3*d^2*n^2*e^4)*log(c))*x^(2/3) - 6*(15*(29*b^3*d^4*n^3 - 20*a*b^2*d^4*n^2)
*x*e^2 - 4*(11*b^3*d*n^3 - 30*a*b^2*d*n^2)*e^5 - 60*(5*b^3*d^4*n^2*x*e^2 - 2*b^3*d*n^2*e^5)*log(c))*x^(1/3))*l
og((d*x + x^(2/3)*e)/x) - 15*(4*(13489*b^3*d^5*n^3 - 8820*a*b^2*d^5*n^2 + 1800*a^2*b*d^5*n)*x*e + 1800*(4*b^3*
d^5*n*x*e - b^3*d^2*n*e^4)*log(c)^2 - (919*b^3*d^2*n^3 - 2220*a*b^2*d^2*n^2 + 1800*a^2*b*d^2*n)*e^4 - 60*(12*(
49*b^3*d^5*n^2 - 20*a*b^2*d^5*n)*x*e - (37*b^3*d^2*n^2 - 60*a*b^2*d^2*n)*e^4)*log(c))*x^(2/3) + 6*(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*e^5)*log(c)^2 -
8*(91*b^3*d*n^3 - 330*a*b^2*d*n^2 + 450*a^2*b*d*n)*e^5 - 60*(15*(29*b^3*d^4*n^2 - 20*a*b^2*d^4*n)*x*e^2 - 4*(1
1*b^3*d*n^2 - 30*a*b^2*d*n)*e^5)*log(c))*x^(1/3))*e^(-6)/x^2

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3651 vs. \(2 (803) = 1606\).
time = 3.53, size = 3651, normalized size = 4.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72000*(216000*(d*x^(1/3) + e)*b^3*d^5*n^3*log((d*x^(1/3) + e)/x^(1/3))^3/x^(1/3) - 540000*(d*x^(1/3) + e)^2*
b^3*d^4*n^3*log((d*x^(1/3) + e)/x^(1/3))^3/x^(2/3) + 720000*(d*x^(1/3) + e)^3*b^3*d^3*n^3*log((d*x^(1/3) + e)/
x^(1/3))^3/x - 648000*(d*x^(1/3) + e)*b^3*d^5*n^3*log((d*x^(1/3) + e)/x^(1/3))^2/x^(1/3) + 648000*(d*x^(1/3) +
e)*b^3*d^5*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))^2/x^(1/3) - 540000*(d*x^(1/3) + e)^4*b^3*d^2*n^3*log((d*x^
(1/3) + e)/x^(1/3))^3/x^(4/3) + 810000*(d*x^(1/3) + e)^2*b^3*d^4*n^3*log((d*x^(1/3) + e)/x^(1/3))^2/x^(2/3) -
1620000*(d*x^(1/3) + e)^2*b^3*d^4*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))^2/x^(2/3) + 216000*(d*x^(1/3) + e)^5
*b^3*d*n^3*log((d*x^(1/3) + e)/x^(1/3))^3/x^(5/3) - 720000*(d*x^(1/3) + e)^3*b^3*d^3*n^3*log((d*x^(1/3) + e)/x
^(1/3))^2/x + 2160000*(d*x^(1/3) + e)^3*b^3*d^3*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))^2/x - 36000*(d*x^(1/3)
+ e)^6*b^3*n^3*log((d*x^(1/3) + e)/x^(1/3))^3/x^2 + 1296000*(d*x^(1/3) + e)*b^3*d^5*n^3*log((d*x^(1/3) + e)/x
^(1/3))/x^(1/3) - 1296000*(d*x^(1/3) + e)*b^3*d^5*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x^(1/3) + 648000*(d*
x^(1/3) + e)*b^3*d^5*n*log(c)^2*log((d*x^(1/3) + e)/x^(1/3))/x^(1/3) + 405000*(d*x^(1/3) + e)^4*b^3*d^2*n^3*lo
g((d*x^(1/3) + e)/x^(1/3))^2/x^(4/3) + 648000*(d*x^(1/3) + e)*a*b^2*d^5*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x^(
1/3) - 1620000*(d*x^(1/3) + e)^4*b^3*d^2*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))^2/x^(4/3) - 810000*(d*x^(1/3)
+ e)^2*b^3*d^4*n^3*log((d*x^(1/3) + e)/x^(1/3))/x^(2/3) + 1620000*(d*x^(1/3) + e)^2*b^3*d^4*n^2*log(c)*log((d
*x^(1/3) + e)/x^(1/3))/x^(2/3) - 1620000*(d*x^(1/3) + e)^2*b^3*d^4*n*log(c)^2*log((d*x^(1/3) + e)/x^(1/3))/x^(
2/3) - 129600*(d*x^(1/3) + e)^5*b^3*d*n^3*log((d*x^(1/3) + e)/x^(1/3))^2/x^(5/3) - 1620000*(d*x^(1/3) + e)^2*a
*b^2*d^4*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x^(2/3) + 648000*(d*x^(1/3) + e)^5*b^3*d*n^2*log(c)*log((d*x^(1/3)
+ e)/x^(1/3))^2/x^(5/3) + 480000*(d*x^(1/3) + e)^3*b^3*d^3*n^3*log((d*x^(1/3) + e)/x^(1/3))/x - 1440000*(d*x^
(1/3) + e)^3*b^3*d^3*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x + 2160000*(d*x^(1/3) + e)^3*b^3*d^3*n*log(c)^2*
log((d*x^(1/3) + e)/x^(1/3))/x + 18000*(d*x^(1/3) + e)^6*b^3*n^3*log((d*x^(1/3) + e)/x^(1/3))^2/x^2 + 2160000*
(d*x^(1/3) + e)^3*a*b^2*d^3*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x - 108000*(d*x^(1/3) + e)^6*b^3*n^2*log(c)*log
((d*x^(1/3) + e)/x^(1/3))^2/x^2 - 1296000*(d*x^(1/3) + e)*b^3*d^5*n^3/x^(1/3) + 1296000*(d*x^(1/3) + e)*b^3*d^
5*n^2*log(c)/x^(1/3) - 648000*(d*x^(1/3) + e)*b^3*d^5*n*log(c)^2/x^(1/3) + 216000*(d*x^(1/3) + e)*b^3*d^5*log(
c)^3/x^(1/3) - 202500*(d*x^(1/3) + e)^4*b^3*d^2*n^3*log((d*x^(1/3) + e)/x^(1/3))/x^(4/3) - 1296000*(d*x^(1/3)
+ e)*a*b^2*d^5*n^2*log((d*x^(1/3) + e)/x^(1/3))/x^(1/3) + 810000*(d*x^(1/3) + e)^4*b^3*d^2*n^2*log(c)*log((d*x
^(1/3) + e)/x^(1/3))/x^(4/3) + 1296000*(d*x^(1/3) + e)*a*b^2*d^5*n*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x^(1/3)
- 1620000*(d*x^(1/3) + e)^4*b^3*d^2*n*log(c)^2*log((d*x^(1/3) + e)/x^(1/3))/x^(4/3) - 1620000*(d*x^(1/3) + e)
^4*a*b^2*d^2*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x^(4/3) + 405000*(d*x^(1/3) + e)^2*b^3*d^4*n^3/x^(2/3) - 81000
0*(d*x^(1/3) + e)^2*b^3*d^4*n^2*log(c)/x^(2/3) + 810000*(d*x^(1/3) + e)^2*b^3*d^4*n*log(c)^2/x^(2/3) - 540000*
(d*x^(1/3) + e)^2*b^3*d^4*log(c)^3/x^(2/3) + 51840*(d*x^(1/3) + e)^5*b^3*d*n^3*log((d*x^(1/3) + e)/x^(1/3))/x^
(5/3) + 1620000*(d*x^(1/3) + e)^2*a*b^2*d^4*n^2*log((d*x^(1/3) + e)/x^(1/3))/x^(2/3) - 259200*(d*x^(1/3) + e)^
5*b^3*d*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x^(5/3) - 3240000*(d*x^(1/3) + e)^2*a*b^2*d^4*n*log(c)*log((d*
x^(1/3) + e)/x^(1/3))/x^(2/3) + 648000*(d*x^(1/3) + e)^5*b^3*d*n*log(c)^2*log((d*x^(1/3) + e)/x^(1/3))/x^(5/3)
+ 648000*(d*x^(1/3) + e)^5*a*b^2*d*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x^(5/3) - 160000*(d*x^(1/3) + e)^3*b^3*
d^3*n^3/x + 480000*(d*x^(1/3) + e)^3*b^3*d^3*n^2*log(c)/x - 720000*(d*x^(1/3) + e)^3*b^3*d^3*n*log(c)^2/x + 72
0000*(d*x^(1/3) + e)^3*b^3*d^3*log(c)^3/x - 6000*(d*x^(1/3) + e)^6*b^3*n^3*log((d*x^(1/3) + e)/x^(1/3))/x^2 -
1440000*(d*x^(1/3) + e)^3*a*b^2*d^3*n^2*log((d*x^(1/3) + e)/x^(1/3))/x + 36000*(d*x^(1/3) + e)^6*b^3*n^2*log(c
)*log((d*x^(1/3) + e)/x^(1/3))/x^2 + 4320000*(d*x^(1/3) + e)^3*a*b^2*d^3*n*log(c)*log((d*x^(1/3) + e)/x^(1/3))
/x - 108000*(d*x^(1/3) + e)^6*b^3*n*log(c)^2*log((d*x^(1/3) + e)/x^(1/3))/x^2 - 108000*(d*x^(1/3) + e)^6*a*b^2
*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x^2 + 50625*(d*x^(1/3) + e)^4*b^3*d^2*n^3/x^(4/3) + 1296000*(d*x^(1/3) + e
)*a*b^2*d^5*n^2/x^(1/3) - 202500*(d*x^(1/3) + e)^4*b^3*d^2*n^2*log(c)/x^(4/3) - 1296000*(d*x^(1/3) + e)*a*b^2*
d^5*n*log(c)/x^(1/3) + 405000*(d*x^(1/3) + e)^4*b^3*d^2*n*log(c)^2/x^(4/3) + 648000*(d*x^(1/3) + e)*a*b^2*d^5*
log(c)^2/x^(1/3) - 540000*(d*x^(1/3) + e)^4*b^3*d^2*log(c)^3/x^(4/3) + 810000*(d*x^(1/3) + e)^4*a*b^2*d^2*n^2*
log((d*x^(1/3) + e)/x^(1/3))/x^(4/3) + 648000*(d*x^(1/3) + e)*a^2*b*d^5*n*log((d*x^(1/3) + e)/x^(1/3))/x^(1/3)
- 3240000*(d*x^(1/3) + e)^4*a*b^2*d^2*n*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x^(4/3) - 10368*(d*x^(1/3) + e)^5
*b^3*d*n^3/x^(5/3) - 810000*(d*x^(1/3) + e)^2*a...

________________________________________________________________________________________

Mupad [B]
time = 8.20, size = 992, normalized size = 1.09 \begin {gather*} \frac {b^3\,n^3}{72\,x^2}-\frac {b^3\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^3}{2\,x^2}-\frac {a^3}{2\,x^2}-\frac {3\,a\,b^2\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{2\,x^2}+\frac {b^3\,n\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{4\,x^2}-\frac {b^3\,n^2\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{12\,x^2}-\frac {a\,b^2\,n^2}{12\,x^2}+\frac {b^3\,d^6\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^3}{2\,e^6}-\frac {3\,a^2\,b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{2\,x^2}+\frac {a^2\,b\,n}{4\,x^2}+\frac {a\,b^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{2\,x^2}+\frac {13489\,b^3\,d^6\,n^3\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{1200\,e^6}-\frac {2059\,b^3\,d^3\,n^3}{3600\,e^3\,x}+\frac {919\,b^3\,d^2\,n^3}{4800\,e^2\,x^{4/3}}+\frac {4669\,b^3\,d^4\,n^3}{2400\,e^4\,x^{2/3}}-\frac {13489\,b^3\,d^5\,n^3}{1200\,e^5\,x^{1/3}}+\frac {3\,a\,b^2\,d^6\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{2\,e^6}-\frac {147\,b^3\,d^6\,n\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{40\,e^6}-\frac {91\,b^3\,d\,n^3}{1500\,e\,x^{5/3}}+\frac {3\,a^2\,b\,d^6\,n\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{2\,e^6}-\frac {3\,b^3\,d\,n\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{10\,e\,x^{5/3}}+\frac {11\,b^3\,d\,n^2\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{50\,e\,x^{5/3}}-\frac {a^2\,b\,d^3\,n}{2\,e^3\,x}+\frac {11\,a\,b^2\,d\,n^2}{50\,e\,x^{5/3}}+\frac {3\,a^2\,b\,d^2\,n}{8\,e^2\,x^{4/3}}+\frac {3\,a^2\,b\,d^4\,n}{4\,e^4\,x^{2/3}}-\frac {3\,a^2\,b\,d^5\,n}{2\,e^5\,x^{1/3}}-\frac {147\,a\,b^2\,d^6\,n^2\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{20\,e^6}-\frac {b^3\,d^3\,n\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{2\,e^3\,x}+\frac {19\,b^3\,d^3\,n^2\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{20\,e^3\,x}+\frac {3\,b^3\,d^2\,n\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{8\,e^2\,x^{4/3}}-\frac {37\,b^3\,d^2\,n^2\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{80\,e^2\,x^{4/3}}+\frac {3\,b^3\,d^4\,n\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{4\,e^4\,x^{2/3}}-\frac {87\,b^3\,d^4\,n^2\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{40\,e^4\,x^{2/3}}-\frac {3\,b^3\,d^5\,n\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{2\,e^5\,x^{1/3}}+\frac {147\,b^3\,d^5\,n^2\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{20\,e^5\,x^{1/3}}+\frac {19\,a\,b^2\,d^3\,n^2}{20\,e^3\,x}-\frac {37\,a\,b^2\,d^2\,n^2}{80\,e^2\,x^{4/3}}-\frac {87\,a\,b^2\,d^4\,n^2}{40\,e^4\,x^{2/3}}+\frac {147\,a\,b^2\,d^5\,n^2}{20\,e^5\,x^{1/3}}-\frac {3\,a^2\,b\,d\,n}{10\,e\,x^{5/3}}-\frac {3\,a\,b^2\,d\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{5\,e\,x^{5/3}}-\frac {a\,b^2\,d^3\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{e^3\,x}+\frac {3\,a\,b^2\,d^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{4\,e^2\,x^{4/3}}+\frac {3\,a\,b^2\,d^4\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{2\,e^4\,x^{2/3}}-\frac {3\,a\,b^2\,d^5\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{e^5\,x^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*n^3)/(72*x^2) - (b^3*log(c*(d + e/x^(1/3))^n)^3)/(2*x^2) - a^3/(2*x^2) - (3*a*b^2*log(c*(d + e/x^(1/3))^n
)^2)/(2*x^2) + (b^3*n*log(c*(d + e/x^(1/3))^n)^2)/(4*x^2) - (b^3*n^2*log(c*(d + e/x^(1/3))^n))/(12*x^2) - (a*b
^2*n^2)/(12*x^2) + (b^3*d^6*log(c*(d + e/x^(1/3))^n)^3)/(2*e^6) - (3*a^2*b*log(c*(d + e/x^(1/3))^n))/(2*x^2) +
(a^2*b*n)/(4*x^2) + (a*b^2*n*log(c*(d + e/x^(1/3))^n))/(2*x^2) + (13489*b^3*d^6*n^3*log(d + e/x^(1/3)))/(1200
*e^6) - (2059*b^3*d^3*n^3)/(3600*e^3*x) + (919*b^3*d^2*n^3)/(4800*e^2*x^(4/3)) + (4669*b^3*d^4*n^3)/(2400*e^4*
x^(2/3)) - (13489*b^3*d^5*n^3)/(1200*e^5*x^(1/3)) + (3*a*b^2*d^6*log(c*(d + e/x^(1/3))^n)^2)/(2*e^6) - (147*b^
3*d^6*n*log(c*(d + e/x^(1/3))^n)^2)/(40*e^6) - (91*b^3*d*n^3)/(1500*e*x^(5/3)) + (3*a^2*b*d^6*n*log(d + e/x^(1
/3)))/(2*e^6) - (3*b^3*d*n*log(c*(d + e/x^(1/3))^n)^2)/(10*e*x^(5/3)) + (11*b^3*d*n^2*log(c*(d + e/x^(1/3))^n)
)/(50*e*x^(5/3)) - (a^2*b*d^3*n)/(2*e^3*x) + (11*a*b^2*d*n^2)/(50*e*x^(5/3)) + (3*a^2*b*d^2*n)/(8*e^2*x^(4/3))
+ (3*a^2*b*d^4*n)/(4*e^4*x^(2/3)) - (3*a^2*b*d^5*n)/(2*e^5*x^(1/3)) - (147*a*b^2*d^6*n^2*log(d + e/x^(1/3)))/
(20*e^6) - (b^3*d^3*n*log(c*(d + e/x^(1/3))^n)^2)/(2*e^3*x) + (19*b^3*d^3*n^2*log(c*(d + e/x^(1/3))^n))/(20*e^
3*x) + (3*b^3*d^2*n*log(c*(d + e/x^(1/3))^n)^2)/(8*e^2*x^(4/3)) - (37*b^3*d^2*n^2*log(c*(d + e/x^(1/3))^n))/(8
0*e^2*x^(4/3)) + (3*b^3*d^4*n*log(c*(d + e/x^(1/3))^n)^2)/(4*e^4*x^(2/3)) - (87*b^3*d^4*n^2*log(c*(d + e/x^(1/
3))^n))/(40*e^4*x^(2/3)) - (3*b^3*d^5*n*log(c*(d + e/x^(1/3))^n)^2)/(2*e^5*x^(1/3)) + (147*b^3*d^5*n^2*log(c*(
d + e/x^(1/3))^n))/(20*e^5*x^(1/3)) + (19*a*b^2*d^3*n^2)/(20*e^3*x) - (37*a*b^2*d^2*n^2)/(80*e^2*x^(4/3)) - (8
7*a*b^2*d^4*n^2)/(40*e^4*x^(2/3)) + (147*a*b^2*d^5*n^2)/(20*e^5*x^(1/3)) - (3*a^2*b*d*n)/(10*e*x^(5/3)) - (3*a
*b^2*d*n*log(c*(d + e/x^(1/3))^n))/(5*e*x^(5/3)) - (a*b^2*d^3*n*log(c*(d + e/x^(1/3))^n))/(e^3*x) + (3*a*b^2*d
^2*n*log(c*(d + e/x^(1/3))^n))/(4*e^2*x^(4/3)) + (3*a*b^2*d^4*n*log(c*(d + e/x^(1/3))^n))/(2*e^4*x^(2/3)) - (3
*a*b^2*d^5*n*log(c*(d + e/x^(1/3))^n))/(e^5*x^(1/3))

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