3.6.27 \(\int \frac {(a+b \log (c (d+\frac {e}{x^{2/3}})^n))^3}{x^3} \, dx\) [527]

Optimal. Leaf size=449 \[ -\frac {9 b^3 d n^3 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^3}+\frac {b^3 n^3 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {9 a b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {9 b^3 d^2 n^3}{e^2 x^{2/3}}-\frac {9 b^3 d^2 n^2 \left (d+\frac {e}{x^{2/3}}\right ) \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{e^3}+\frac {9 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}+\frac {9 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {9 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {3 d^2 \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}+\frac {3 d \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}-\frac {\left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3} \]

[Out]

-9/8*b^3*d*n^3*(d+e/x^(2/3))^2/e^3+1/9*b^3*n^3*(d+e/x^(2/3))^3/e^3-9*a*b^2*d^2*n^2/e^2/x^(2/3)+9*b^3*d^2*n^3/e
^2/x^(2/3)-9*b^3*d^2*n^2*(d+e/x^(2/3))*ln(c*(d+e/x^(2/3))^n)/e^3+9/4*b^2*d*n^2*(d+e/x^(2/3))^2*(a+b*ln(c*(d+e/
x^(2/3))^n))/e^3-1/3*b^2*n^2*(d+e/x^(2/3))^3*(a+b*ln(c*(d+e/x^(2/3))^n))/e^3+9/2*b*d^2*n*(d+e/x^(2/3))*(a+b*ln
(c*(d+e/x^(2/3))^n))^2/e^3-9/4*b*d*n*(d+e/x^(2/3))^2*(a+b*ln(c*(d+e/x^(2/3))^n))^2/e^3+1/2*b*n*(d+e/x^(2/3))^3
*(a+b*ln(c*(d+e/x^(2/3))^n))^2/e^3-3/2*d^2*(d+e/x^(2/3))*(a+b*ln(c*(d+e/x^(2/3))^n))^3/e^3+3/2*d*(d+e/x^(2/3))
^2*(a+b*ln(c*(d+e/x^(2/3))^n))^3/e^3-1/2*(d+e/x^(2/3))^3*(a+b*ln(c*(d+e/x^(2/3))^n))^3/e^3

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Rubi [A]
time = 0.30, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 16, 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^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}+\frac {9 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^3}-\frac {9 a b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {9 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {3 d^2 \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {9 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 e^3}-\frac {\left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}+\frac {3 d \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}-\frac {9 b^3 d^2 n^2 \left (d+\frac {e}{x^{2/3}}\right ) \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{e^3}+\frac {9 b^3 d^2 n^3}{e^2 x^{2/3}}+\frac {b^3 n^3 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {9 b^3 d n^3 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*b^3*d*n^3*(d + e/x^(2/3))^2)/(8*e^3) + (b^3*n^3*(d + e/x^(2/3))^3)/(9*e^3) - (9*a*b^2*d^2*n^2)/(e^2*x^(2/3
)) + (9*b^3*d^2*n^3)/(e^2*x^(2/3)) - (9*b^3*d^2*n^2*(d + e/x^(2/3))*Log[c*(d + e/x^(2/3))^n])/e^3 + (9*b^2*d*n
^2*(d + e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(4*e^3) - (b^2*n^2*(d + e/x^(2/3))^3*(a + b*Log[c*(d +
e/x^(2/3))^n]))/(3*e^3) + (9*b*d^2*n*(d + e/x^(2/3))*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(2*e^3) - (9*b*d*n*(d
+ e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(4*e^3) + (b*n*(d + e/x^(2/3))^3*(a + b*Log[c*(d + e/x^(2/
3))^n])^2)/(2*e^3) - (3*d^2*(d + e/x^(2/3))*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/(2*e^3) + (3*d*(d + e/x^(2/3))
^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/(2*e^3) - ((d + e/x^(2/3))^3*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/(2*e^3
)

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}{x^{2/3}}\right )^n\right )\right )^3}{x^3} \, dx &=-\left (\frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{x^{2/3}}\right )\right )\\ &=-\left (\frac {3}{2} \text {Subst}\left (\int \left (\frac {d^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}-\frac {2 d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}\right ) \, dx,x,\frac {1}{x^{2/3}}\right )\right )\\ &=-\frac {3 \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}{x^{2/3}}\right )}{2 e^2}+\frac {(3 d) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{x^{2/3}}\right )}{e^2}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{x^{2/3}}\right )}{2 e^2}\\ &=-\frac {3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {(3 d) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{x^{2/3}}\right )}{e^3}-\frac {\left (3 d^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 e^3}\\ &=-\frac {3 d^2 \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}+\frac {3 d \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}-\frac {\left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}+\frac {(3 b n) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 e^3}-\frac {(9 b d n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {\left (9 b d^2 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 e^3}\\ &=\frac {9 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {9 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {3 d^2 \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}+\frac {3 d \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}-\frac {\left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{x^{2/3}}\right )}{e^3}+\frac {\left (9 b^2 d n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 e^3}-\frac {\left (9 b^2 d^2 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{x^{2/3}}\right )}{e^3}\\ &=-\frac {9 b^3 d n^3 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^3}+\frac {b^3 n^3 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {9 a b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {9 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}+\frac {9 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {9 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {3 d^2 \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}+\frac {3 d \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}-\frac {\left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}-\frac {\left (9 b^3 d^2 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac {e}{x^{2/3}}\right )}{e^3}\\ &=-\frac {9 b^3 d n^3 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^3}+\frac {b^3 n^3 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {9 a b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {9 b^3 d^2 n^3}{e^2 x^{2/3}}-\frac {9 b^3 d^2 n^2 \left (d+\frac {e}{x^{2/3}}\right ) \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{e^3}+\frac {9 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}+\frac {9 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {9 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 e^3}-\frac {3 d^2 \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}+\frac {3 d \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}-\frac {\left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{2 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.74, size = 692, normalized size = 1.54 \begin {gather*} \frac {-36 a^3 e^3+36 a^2 b e^3 n-24 a b^2 e^3 n^2+8 b^3 e^3 n^3-54 a^2 b d e^2 n x^{2/3}+90 a b^2 d e^2 n^2 x^{2/3}-57 b^3 d e^2 n^3 x^{2/3}+108 a^2 b d^2 e n x^{4/3}-396 a b^2 d^2 e n^2 x^{4/3}+510 b^3 d^2 e n^3 x^{4/3}+72 b^3 d^3 n^3 x^2 \log ^3\left (d+\frac {e}{x^{2/3}}\right )-36 b^3 e^3 \log ^3\left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-108 a^2 b d^3 n x^2 \log \left (e+d x^{2/3}\right )+396 a b^2 d^3 n^2 x^2 \log \left (e+d x^{2/3}\right )-510 b^3 d^3 n^3 x^2 \log \left (e+d x^{2/3}\right )+12 b^2 d^3 n^2 x^2 \log \left (d+\frac {e}{x^{2/3}}\right ) \left (6 a-11 b n+6 b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \left (3 \log \left (e+d x^{2/3}\right )-2 \log (x)\right )+72 a^2 b d^3 n x^2 \log (x)-264 a b^2 d^3 n^2 x^2 \log (x)+340 b^3 d^3 n^3 x^2 \log (x)-18 b^2 d^3 n^2 x^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right ) \left (6 a-11 b n+6 b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+6 b n \log \left (e+d x^{2/3}\right )-4 b n \log (x)\right )+18 b^2 \log ^2\left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) \left (e \left (-6 a e^2+2 b e^2 n-3 b d e n x^{2/3}+6 b d^2 n x^{4/3}\right )-6 b d^3 n x^2 \log \left (e+d x^{2/3}\right )+4 b d^3 n x^2 \log (x)\right )-6 b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) \left (18 a^2 e^3-6 a b e n \left (2 e^2-3 d e x^{2/3}+6 d^2 x^{4/3}\right )+b^2 e n^2 \left (4 e^2-15 d e x^{2/3}+66 d^2 x^{4/3}\right )+6 b d^3 n (6 a-11 b n) x^2 \log \left (e+d x^{2/3}\right )+4 b d^3 n (-6 a+11 b n) x^2 \log (x)\right )}{72 e^3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-36*a^3*e^3 + 36*a^2*b*e^3*n - 24*a*b^2*e^3*n^2 + 8*b^3*e^3*n^3 - 54*a^2*b*d*e^2*n*x^(2/3) + 90*a*b^2*d*e^2*n
^2*x^(2/3) - 57*b^3*d*e^2*n^3*x^(2/3) + 108*a^2*b*d^2*e*n*x^(4/3) - 396*a*b^2*d^2*e*n^2*x^(4/3) + 510*b^3*d^2*
e*n^3*x^(4/3) + 72*b^3*d^3*n^3*x^2*Log[d + e/x^(2/3)]^3 - 36*b^3*e^3*Log[c*(d + e/x^(2/3))^n]^3 - 108*a^2*b*d^
3*n*x^2*Log[e + d*x^(2/3)] + 396*a*b^2*d^3*n^2*x^2*Log[e + d*x^(2/3)] - 510*b^3*d^3*n^3*x^2*Log[e + d*x^(2/3)]
+ 12*b^2*d^3*n^2*x^2*Log[d + e/x^(2/3)]*(6*a - 11*b*n + 6*b*Log[c*(d + e/x^(2/3))^n])*(3*Log[e + d*x^(2/3)] -
2*Log[x]) + 72*a^2*b*d^3*n*x^2*Log[x] - 264*a*b^2*d^3*n^2*x^2*Log[x] + 340*b^3*d^3*n^3*x^2*Log[x] - 18*b^2*d^
3*n^2*x^2*Log[d + e/x^(2/3)]^2*(6*a - 11*b*n + 6*b*Log[c*(d + e/x^(2/3))^n] + 6*b*n*Log[e + d*x^(2/3)] - 4*b*n
*Log[x]) + 18*b^2*Log[c*(d + e/x^(2/3))^n]^2*(e*(-6*a*e^2 + 2*b*e^2*n - 3*b*d*e*n*x^(2/3) + 6*b*d^2*n*x^(4/3))
- 6*b*d^3*n*x^2*Log[e + d*x^(2/3)] + 4*b*d^3*n*x^2*Log[x]) - 6*b*Log[c*(d + e/x^(2/3))^n]*(18*a^2*e^3 - 6*a*b
*e*n*(2*e^2 - 3*d*e*x^(2/3) + 6*d^2*x^(4/3)) + b^2*e*n^2*(4*e^2 - 15*d*e*x^(2/3) + 66*d^2*x^(4/3)) + 6*b*d^3*n
*(6*a - 11*b*n)*x^2*Log[e + d*x^(2/3)] + 4*b*d^3*n*(-6*a + 11*b*n)*x^2*Log[x]))/(72*e^3*x^2)

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )^{3}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 689, normalized size = 1.53 \begin {gather*} -\frac {1}{4} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}}\right ) - \frac {{\left (6 \, d^{2} x^{\frac {4}{3}} - 3 \, d x^{\frac {2}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x^{2}}\right )} a^{2} b n e - \frac {1}{12} \, {\left (6 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}}\right ) - \frac {{\left (6 \, d^{2} x^{\frac {4}{3}} - 3 \, d x^{\frac {2}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x^{2}}\right )} n e \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) - \frac {{\left (18 \, d^{3} x^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 8 \, d^{3} x^{2} \log \left (x\right )^{2} - 44 \, d^{3} x^{2} \log \left (x\right ) - 66 \, d^{2} x^{\frac {4}{3}} e + 15 \, d x^{\frac {2}{3}} e^{2} - 6 \, {\left (4 \, d^{3} x^{2} \log \left (x\right ) - 11 \, d^{3} x^{2}\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 4 \, e^{3}\right )} n^{2} e^{\left (-3\right )}}{x^{2}}\right )} a b^{2} - \frac {1}{216} \, {\left (54 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}}\right ) - \frac {{\left (6 \, d^{2} x^{\frac {4}{3}} - 3 \, d x^{\frac {2}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x^{2}}\right )} n e \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )^{2} + {\left (\frac {{\left (108 \, d^{3} x^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{3} - 32 \, d^{3} x^{2} \log \left (x\right )^{3} + 264 \, d^{3} x^{2} \log \left (x\right )^{2} - 1020 \, d^{3} x^{2} \log \left (x\right ) - 1530 \, d^{2} x^{\frac {4}{3}} e - 54 \, {\left (4 \, d^{3} x^{2} \log \left (x\right ) - 11 \, d^{3} x^{2}\right )} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 171 \, d x^{\frac {2}{3}} e^{2} + 18 \, {\left (8 \, d^{3} x^{2} \log \left (x\right )^{2} - 44 \, d^{3} x^{2} \log \left (x\right ) + 85 \, d^{3} x^{2}\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 24 \, e^{3}\right )} n^{2} e^{\left (-4\right )}}{x^{2}} - \frac {18 \, {\left (18 \, d^{3} x^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 8 \, d^{3} x^{2} \log \left (x\right )^{2} - 44 \, d^{3} x^{2} \log \left (x\right ) - 66 \, d^{2} x^{\frac {4}{3}} e + 15 \, d x^{\frac {2}{3}} e^{2} - 6 \, {\left (4 \, d^{3} x^{2} \log \left (x\right ) - 11 \, d^{3} x^{2}\right )} \log \left (d x^{\frac {2}{3}} + e\right ) - 4 \, e^{3}\right )} n e^{\left (-4\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{x^{2}}\right )} n e\right )} b^{3} - \frac {b^{3} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )^{3}}{2 \, x^{2}} - \frac {3 \, a b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {3 \, a^{2} b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{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^(2/3))^n))^3/x^3,x, algorithm="maxima")

[Out]

-1/4*(6*d^3*e^(-4)*log(d*x^(2/3) + e) - 6*d^3*e^(-4)*log(x^(2/3)) - (6*d^2*x^(4/3) - 3*d*x^(2/3)*e + 2*e^2)*e^
(-3)/x^2)*a^2*b*n*e - 1/12*(6*(6*d^3*e^(-4)*log(d*x^(2/3) + e) - 6*d^3*e^(-4)*log(x^(2/3)) - (6*d^2*x^(4/3) -
3*d*x^(2/3)*e + 2*e^2)*e^(-3)/x^2)*n*e*log(c*(d + e/x^(2/3))^n) - (18*d^3*x^2*log(d*x^(2/3) + e)^2 + 8*d^3*x^2
*log(x)^2 - 44*d^3*x^2*log(x) - 66*d^2*x^(4/3)*e + 15*d*x^(2/3)*e^2 - 6*(4*d^3*x^2*log(x) - 11*d^3*x^2)*log(d*
x^(2/3) + e) - 4*e^3)*n^2*e^(-3)/x^2)*a*b^2 - 1/216*(54*(6*d^3*e^(-4)*log(d*x^(2/3) + e) - 6*d^3*e^(-4)*log(x^
(2/3)) - (6*d^2*x^(4/3) - 3*d*x^(2/3)*e + 2*e^2)*e^(-3)/x^2)*n*e*log(c*(d + e/x^(2/3))^n)^2 + ((108*d^3*x^2*lo
g(d*x^(2/3) + e)^3 - 32*d^3*x^2*log(x)^3 + 264*d^3*x^2*log(x)^2 - 1020*d^3*x^2*log(x) - 1530*d^2*x^(4/3)*e - 5
4*(4*d^3*x^2*log(x) - 11*d^3*x^2)*log(d*x^(2/3) + e)^2 + 171*d*x^(2/3)*e^2 + 18*(8*d^3*x^2*log(x)^2 - 44*d^3*x
^2*log(x) + 85*d^3*x^2)*log(d*x^(2/3) + e) - 24*e^3)*n^2*e^(-4)/x^2 - 18*(18*d^3*x^2*log(d*x^(2/3) + e)^2 + 8*
d^3*x^2*log(x)^2 - 44*d^3*x^2*log(x) - 66*d^2*x^(4/3)*e + 15*d*x^(2/3)*e^2 - 6*(4*d^3*x^2*log(x) - 11*d^3*x^2)
*log(d*x^(2/3) + e) - 4*e^3)*n*e^(-4)*log(c*(d + e/x^(2/3))^n)/x^2)*n*e)*b^3 - 1/2*b^3*log(c*(d + e/x^(2/3))^n
)^3/x^2 - 3/2*a*b^2*log(c*(d + e/x^(2/3))^n)^2/x^2 - 3/2*a^2*b*log(c*(d + e/x^(2/3))^n)/x^2 - 1/2*a^3/x^2

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Fricas [A]
time = 0.40, size = 686, normalized size = 1.53 \begin {gather*} -\frac {{\left (36 \, b^{3} e^{3} \log \left (c\right )^{3} - 36 \, {\left (b^{3} n - 3 \, a b^{2}\right )} e^{3} \log \left (c\right )^{2} + 36 \, {\left (b^{3} d^{3} n^{3} x^{2} + b^{3} n^{3} e^{3}\right )} \log \left (\frac {d x + x^{\frac {1}{3}} e}{x}\right )^{3} + 12 \, {\left (2 \, b^{3} n^{2} - 6 \, a b^{2} n + 9 \, a^{2} b\right )} e^{3} \log \left (c\right ) - 18 \, {\left (6 \, b^{3} d^{2} n^{3} x^{\frac {4}{3}} e - 3 \, b^{3} d n^{3} x^{\frac {2}{3}} e^{2} + {\left (11 \, b^{3} d^{3} n^{3} - 6 \, a b^{2} d^{3} n^{2}\right )} x^{2} + 2 \, {\left (b^{3} n^{3} - 3 \, a b^{2} n^{2}\right )} e^{3} - 6 \, {\left (b^{3} d^{3} n^{2} x^{2} + b^{3} n^{2} e^{3}\right )} \log \left (c\right )\right )} \log \left (\frac {d x + x^{\frac {1}{3}} e}{x}\right )^{2} - 4 \, {\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n - 9 \, a^{3}\right )} e^{3} + 6 \, {\left ({\left (85 \, b^{3} d^{3} n^{3} - 66 \, a b^{2} d^{3} n^{2} + 18 \, a^{2} b d^{3} n\right )} x^{2} + 18 \, {\left (b^{3} d^{3} n x^{2} + b^{3} n e^{3}\right )} \log \left (c\right )^{2} + 2 \, {\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n\right )} e^{3} - 6 \, {\left ({\left (11 \, b^{3} d^{3} n^{2} - 6 \, a b^{2} d^{3} n\right )} x^{2} + 2 \, {\left (b^{3} n^{2} - 3 \, a b^{2} n\right )} e^{3}\right )} \log \left (c\right ) + 3 \, {\left (6 \, b^{3} d n^{2} e^{2} \log \left (c\right ) - {\left (5 \, b^{3} d n^{3} - 6 \, a b^{2} d n^{2}\right )} e^{2}\right )} x^{\frac {2}{3}} - 6 \, {\left (6 \, b^{3} d^{2} n^{2} x e \log \left (c\right ) - {\left (11 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2}\right )} x e\right )} x^{\frac {1}{3}}\right )} \log \left (\frac {d x + x^{\frac {1}{3}} e}{x}\right ) + 3 \, {\left (18 \, b^{3} d n e^{2} \log \left (c\right )^{2} - 6 \, {\left (5 \, b^{3} d n^{2} - 6 \, a b^{2} d n\right )} e^{2} \log \left (c\right ) + {\left (19 \, b^{3} d n^{3} - 30 \, a b^{2} d n^{2} + 18 \, a^{2} b d n\right )} e^{2}\right )} x^{\frac {2}{3}} - 6 \, {\left (18 \, b^{3} d^{2} n x e \log \left (c\right )^{2} - 6 \, {\left (11 \, b^{3} d^{2} n^{2} - 6 \, a b^{2} d^{2} n\right )} x e \log \left (c\right ) + {\left (85 \, b^{3} d^{2} n^{3} - 66 \, a b^{2} d^{2} n^{2} + 18 \, a^{2} b d^{2} n\right )} x e\right )} x^{\frac {1}{3}}\right )} e^{\left (-3\right )}}{72 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(36*b^3*e^3*log(c)^3 - 36*(b^3*n - 3*a*b^2)*e^3*log(c)^2 + 36*(b^3*d^3*n^3*x^2 + b^3*n^3*e^3)*log((d*x +
x^(1/3)*e)/x)^3 + 12*(2*b^3*n^2 - 6*a*b^2*n + 9*a^2*b)*e^3*log(c) - 18*(6*b^3*d^2*n^3*x^(4/3)*e - 3*b^3*d*n^3
*x^(2/3)*e^2 + (11*b^3*d^3*n^3 - 6*a*b^2*d^3*n^2)*x^2 + 2*(b^3*n^3 - 3*a*b^2*n^2)*e^3 - 6*(b^3*d^3*n^2*x^2 + b
^3*n^2*e^3)*log(c))*log((d*x + x^(1/3)*e)/x)^2 - 4*(2*b^3*n^3 - 6*a*b^2*n^2 + 9*a^2*b*n - 9*a^3)*e^3 + 6*((85*
b^3*d^3*n^3 - 66*a*b^2*d^3*n^2 + 18*a^2*b*d^3*n)*x^2 + 18*(b^3*d^3*n*x^2 + b^3*n*e^3)*log(c)^2 + 2*(2*b^3*n^3
- 6*a*b^2*n^2 + 9*a^2*b*n)*e^3 - 6*((11*b^3*d^3*n^2 - 6*a*b^2*d^3*n)*x^2 + 2*(b^3*n^2 - 3*a*b^2*n)*e^3)*log(c)
+ 3*(6*b^3*d*n^2*e^2*log(c) - (5*b^3*d*n^3 - 6*a*b^2*d*n^2)*e^2)*x^(2/3) - 6*(6*b^3*d^2*n^2*x*e*log(c) - (11*
b^3*d^2*n^3 - 6*a*b^2*d^2*n^2)*x*e)*x^(1/3))*log((d*x + x^(1/3)*e)/x) + 3*(18*b^3*d*n*e^2*log(c)^2 - 6*(5*b^3*
d*n^2 - 6*a*b^2*d*n)*e^2*log(c) + (19*b^3*d*n^3 - 30*a*b^2*d*n^2 + 18*a^2*b*d*n)*e^2)*x^(2/3) - 6*(18*b^3*d^2*
n*x*e*log(c)^2 - 6*(11*b^3*d^2*n^2 - 6*a*b^2*d^2*n)*x*e*log(c) + (85*b^3*d^2*n^3 - 66*a*b^2*d^2*n^2 + 18*a^2*b
*d^2*n)*x*e)*x^(1/3))*e^(-3)/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**(2/3))**n))**3/x**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3879 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.75, size = 578, normalized size = 1.29 \begin {gather*} \frac {\frac {d\,\left (\frac {3\,a^3}{2}-\frac {3\,a^2\,b\,n}{2}+a\,b^2\,n^2-\frac {b^3\,n^3}{3}\right )}{2\,e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{8\,e}}{x^{4/3}}-{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^3\,\left (\frac {b^3}{2\,x^2}+\frac {b^3\,d^3}{2\,e^3}\right )-{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^2\,\left (\frac {b^2\,\left (3\,a-b\,n\right )}{2\,x^2}-\frac {\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {9\,a\,b^2\,d}{2\,e}}{2\,x^{4/3}}+\frac {d\,\left (6\,a\,b^2\,d^2-11\,b^3\,d^2\,n\right )}{4\,e^3}+\frac {d\,\left (\frac {6\,b^2\,d\,\left (3\,a-b\,n\right )}{e}-\frac {18\,a\,b^2\,d}{e}\right )}{4\,e\,x^{2/3}}\right )-\frac {\frac {d\,\left (\frac {d\,\left (\frac {3\,a^3}{2}-\frac {3\,a^2\,b\,n}{2}+a\,b^2\,n^2-\frac {b^3\,n^3}{3}\right )}{e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{4\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (6\,a-11\,b\,n\right )}{2\,e^2}}{x^{2/3}}-\frac {\frac {a^3}{2}-\frac {a^2\,b\,n}{2}+\frac {a\,b^2\,n^2}{3}-\frac {b^3\,n^3}{9}}{x^2}-\frac {\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\,\left (\frac {\frac {d\,\left (2\,b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-6\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{e}+12\,b^3\,d^2\,n^2}{2\,e\,x^{2/3}}-\frac {2\,b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-6\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )}{4\,e\,x^{4/3}}+\frac {b\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{3\,x^2}\right )}{2\,e}-\frac {\ln \left (d+\frac {e}{x^{2/3}}\right )\,\left (18\,a^2\,b\,d^3\,n-66\,a\,b^2\,d^3\,n^2+85\,b^3\,d^3\,n^3\right )}{12\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d*((3*a^3)/2 - (b^3*n^3)/3 + a*b^2*n^2 - (3*a^2*b*n)/2))/(2*e) - (d*(6*a^3 + 5*b^3*n^3 - 6*a*b^2*n^2))/(8*e)
)/x^(4/3) - log(c*(d + e/x^(2/3))^n)^3*(b^3/(2*x^2) + (b^3*d^3)/(2*e^3)) - log(c*(d + e/x^(2/3))^n)^2*((b^2*(3
*a - b*n))/(2*x^2) - ((3*b^2*d*(3*a - b*n))/(2*e) - (9*a*b^2*d)/(2*e))/(2*x^(4/3)) + (d*(6*a*b^2*d^2 - 11*b^3*
d^2*n))/(4*e^3) + (d*((6*b^2*d*(3*a - b*n))/e - (18*a*b^2*d)/e))/(4*e*x^(2/3))) - ((d*((d*((3*a^3)/2 - (b^3*n^
3)/3 + a*b^2*n^2 - (3*a^2*b*n)/2))/e - (d*(6*a^3 + 5*b^3*n^3 - 6*a*b^2*n^2))/(4*e)))/e + (b^2*d^2*n^2*(6*a - 1
1*b*n))/(2*e^2))/x^(2/3) - (a^3/2 - (b^3*n^3)/9 + (a*b^2*n^2)/3 - (a^2*b*n)/2)/x^2 - (log(c*(d + e/x^(2/3))^n)
*(((d*(2*b*d*e*(9*a^2 + 2*b^2*n^2 - 6*a*b*n) - 6*b*d*e*(3*a^2 - b^2*n^2)))/e + 12*b^3*d^2*n^2)/(2*e*x^(2/3)) -
(2*b*d*e*(9*a^2 + 2*b^2*n^2 - 6*a*b*n) - 6*b*d*e*(3*a^2 - b^2*n^2))/(4*e*x^(4/3)) + (b*e*(9*a^2 + 2*b^2*n^2 -
6*a*b*n))/(3*x^2)))/(2*e) - (log(d + e/x^(2/3))*(85*b^3*d^3*n^3 - 66*a*b^2*d^3*n^2 + 18*a^2*b*d^3*n))/(12*e^3
)

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