3.6.6 \(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n))^3}{x^2} \, dx\) [506]

Optimal. Leaf size=438 \[ -\frac {9 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^3}+\frac {2 b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac {18 a b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {18 b^3 d^2 n^3}{e^2 \sqrt [3]{x}}-\frac {18 b^3 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{e^3}+\frac {9 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^3}-\frac {2 b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}+\frac {9 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}-\frac {9 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 e^3}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}-\frac {3 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}+\frac {3 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3} \]

[Out]

-9/4*b^3*d*n^3*(d+e/x^(1/3))^2/e^3+2/9*b^3*n^3*(d+e/x^(1/3))^3/e^3-18*a*b^2*d^2*n^2/e^2/x^(1/3)+18*b^3*d^2*n^3
/e^2/x^(1/3)-18*b^3*d^2*n^2*(d+e/x^(1/3))*ln(c*(d+e/x^(1/3))^n)/e^3+9/2*b^2*d*n^2*(d+e/x^(1/3))^2*(a+b*ln(c*(d
+e/x^(1/3))^n))/e^3-2/3*b^2*n^2*(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1/3))^n))/e^3+9*b*d^2*n*(d+e/x^(1/3))*(a+b*l
n(c*(d+e/x^(1/3))^n))^2/e^3-9/2*b*d*n*(d+e/x^(1/3))^2*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^3+b*n*(d+e/x^(1/3))^3*(a
+b*ln(c*(d+e/x^(1/3))^n))^2/e^3-3*d^2*(d+e/x^(1/3))*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^3+3*d*(d+e/x^(1/3))^2*(a+b
*ln(c*(d+e/x^(1/3))^n))^3/e^3-(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^3

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Rubi [A]
time = 0.30, antiderivative size = 438, 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 {2 b^2 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^3}+\frac {9 b^2 d 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 )}{2 e^3}-\frac {18 a b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {9 b d^2 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^3}-\frac {3 d^2 \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^3}+\frac {b 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^3}-\frac {9 b d 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}{2 e^3}-\frac {\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^3}+\frac {3 d \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}{e^3}-\frac {18 b^3 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{e^3}+\frac {18 b^3 d^2 n^3}{e^2 \sqrt [3]{x}}+\frac {2 b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac {9 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^2,x]

[Out]

(-9*b^3*d*n^3*(d + e/x^(1/3))^2)/(4*e^3) + (2*b^3*n^3*(d + e/x^(1/3))^3)/(9*e^3) - (18*a*b^2*d^2*n^2)/(e^2*x^(
1/3)) + (18*b^3*d^2*n^3)/(e^2*x^(1/3)) - (18*b^3*d^2*n^2*(d + e/x^(1/3))*Log[c*(d + e/x^(1/3))^n])/e^3 + (9*b^
2*d*n^2*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(2*e^3) - (2*b^2*n^2*(d + e/x^(1/3))^3*(a + b*Log[
c*(d + e/x^(1/3))^n]))/(3*e^3) + (9*b*d^2*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^3 - (9*b*d*n
*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(2*e^3) + (b*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^
(1/3))^n])^2)/e^3 - (3*d^2*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^3 + (3*d*(d + e/x^(1/3))^2*(a
+ b*Log[c*(d + e/x^(1/3))^n])^3)/e^3 - ((d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/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}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx &=-\left (3 \text {Subst}\left (\int x^2 \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^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}{\sqrt [3]{x}}\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}{\sqrt [3]{x}}\right )}{e^2}+\frac {(6 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}{\sqrt [3]{x}}\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}{\sqrt [3]{x}}\right )}{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}{\sqrt [3]{x}}\right )}{e^3}+\frac {(6 d) \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^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}{\sqrt [3]{x}}\right )}{e^3}\\ &=-\frac {3 d^2 \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^3}+\frac {3 d \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}{e^3}-\frac {\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^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}{\sqrt [3]{x}}\right )}{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}{\sqrt [3]{x}}\right )}{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}{\sqrt [3]{x}}\right )}{e^3}\\ &=\frac {9 b d^2 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^3}-\frac {9 b d 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}{2 e^3}+\frac {b 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^3}-\frac {3 d^2 \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^3}+\frac {3 d \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}{e^3}-\frac {\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^3}-\frac {\left (2 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}{\sqrt [3]{x}}\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}{\sqrt [3]{x}}\right )}{e^3}-\frac {\left (18 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}{\sqrt [3]{x}}\right )}{e^3}\\ &=-\frac {9 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^3}+\frac {2 b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac {18 a b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {9 b^2 d 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 )}{2 e^3}-\frac {2 b^2 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^3}+\frac {9 b d^2 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^3}-\frac {9 b d 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}{2 e^3}+\frac {b 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^3}-\frac {3 d^2 \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^3}+\frac {3 d \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}{e^3}-\frac {\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^3}-\frac {\left (18 b^3 d^2 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}\\ &=-\frac {9 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^3}+\frac {2 b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac {18 a b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {18 b^3 d^2 n^3}{e^2 \sqrt [3]{x}}-\frac {18 b^3 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{e^3}+\frac {9 b^2 d 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 )}{2 e^3}-\frac {2 b^2 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^3}+\frac {9 b d^2 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^3}-\frac {9 b d 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}{2 e^3}+\frac {b 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^3}-\frac {3 d^2 \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^3}+\frac {3 d \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}{e^3}-\frac {\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^3}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 666, normalized size = 1.52 \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 \sqrt [3]{x}+90 a b^2 d e^2 n^2 \sqrt [3]{x}-57 b^3 d e^2 n^3 \sqrt [3]{x}+108 a^2 b d^2 e n x^{2/3}-396 a b^2 d^2 e n^2 x^{2/3}+510 b^3 d^2 e n^3 x^{2/3}+72 b^3 d^3 n^3 x \log ^3\left (d+\frac {e}{\sqrt [3]{x}}\right )-36 b^3 e^3 \log ^3\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-108 a^2 b d^3 n x \log \left (e+d \sqrt [3]{x}\right )+396 a b^2 d^3 n^2 x \log \left (e+d \sqrt [3]{x}\right )-510 b^3 d^3 n^3 x \log \left (e+d \sqrt [3]{x}\right )+12 b^2 d^3 n^2 x \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (6 a-11 b n+6 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 )+36 a^2 b d^3 n x \log (x)-132 a b^2 d^3 n^2 x \log (x)+170 b^3 d^3 n^3 x \log (x)-18 b^2 d^3 n^2 x \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (6 a-11 b n+6 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+6 b n \log \left (e+d \sqrt [3]{x}\right )-2 b n \log (x)\right )+18 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (e \left (-6 a e^2+2 b e^2 n-3 b d e n \sqrt [3]{x}+6 b d^2 n x^{2/3}\right )-6 b d^3 n x \log \left (e+d \sqrt [3]{x}\right )+2 b d^3 n x \log (x)\right )-6 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (18 a^2 e^3-6 a b e n \left (2 e^2-3 d e \sqrt [3]{x}+6 d^2 x^{2/3}\right )+b^2 e n^2 \left (4 e^2-15 d e \sqrt [3]{x}+66 d^2 x^{2/3}\right )+6 b d^3 n (6 a-11 b n) x \log \left (e+d \sqrt [3]{x}\right )+2 b d^3 n (-6 a+11 b n) x \log (x)\right )}{36 e^3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^2,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^(1/3) + 90*a*b^2*d*e^2*n
^2*x^(1/3) - 57*b^3*d*e^2*n^3*x^(1/3) + 108*a^2*b*d^2*e*n*x^(2/3) - 396*a*b^2*d^2*e*n^2*x^(2/3) + 510*b^3*d^2*
e*n^3*x^(2/3) + 72*b^3*d^3*n^3*x*Log[d + e/x^(1/3)]^3 - 36*b^3*e^3*Log[c*(d + e/x^(1/3))^n]^3 - 108*a^2*b*d^3*
n*x*Log[e + d*x^(1/3)] + 396*a*b^2*d^3*n^2*x*Log[e + d*x^(1/3)] - 510*b^3*d^3*n^3*x*Log[e + d*x^(1/3)] + 12*b^
2*d^3*n^2*x*Log[d + e/x^(1/3)]*(6*a - 11*b*n + 6*b*Log[c*(d + e/x^(1/3))^n])*(3*Log[e + d*x^(1/3)] - Log[x]) +
36*a^2*b*d^3*n*x*Log[x] - 132*a*b^2*d^3*n^2*x*Log[x] + 170*b^3*d^3*n^3*x*Log[x] - 18*b^2*d^3*n^2*x*Log[d + e/
x^(1/3)]^2*(6*a - 11*b*n + 6*b*Log[c*(d + e/x^(1/3))^n] + 6*b*n*Log[e + d*x^(1/3)] - 2*b*n*Log[x]) + 18*b^2*Lo
g[c*(d + e/x^(1/3))^n]^2*(e*(-6*a*e^2 + 2*b*e^2*n - 3*b*d*e*n*x^(1/3) + 6*b*d^2*n*x^(2/3)) - 6*b*d^3*n*x*Log[e
+ d*x^(1/3)] + 2*b*d^3*n*x*Log[x]) - 6*b*Log[c*(d + e/x^(1/3))^n]*(18*a^2*e^3 - 6*a*b*e*n*(2*e^2 - 3*d*e*x^(1
/3) + 6*d^2*x^(2/3)) + b^2*e*n^2*(4*e^2 - 15*d*e*x^(1/3) + 66*d^2*x^(2/3)) + 6*b*d^3*n*(6*a - 11*b*n)*x*Log[e
+ d*x^(1/3)] + 2*b*d^3*n*(-6*a + 11*b*n)*x*Log[x]))/(36*e^3*x)

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )^{3}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(1/3))^n))^3/x^2,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/3))^n))^3/x^2,x)

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Maxima [A]
time = 0.32, size = 645, normalized size = 1.47 \begin {gather*} -\frac {1}{2} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 2 \, d^{3} e^{\left (-4\right )} \log \left (x\right ) - \frac {{\left (6 \, d^{2} x^{\frac {2}{3}} - 3 \, d x^{\frac {1}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x}\right )} a^{2} b n e - \frac {b^{3} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{3}}{x} - \frac {1}{6} \, {\left (6 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 2 \, d^{3} e^{\left (-4\right )} \log \left (x\right ) - \frac {{\left (6 \, d^{2} x^{\frac {2}{3}} - 3 \, d x^{\frac {1}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x}\right )} n e \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) - \frac {{\left (18 \, d^{3} x \log \left (d x^{\frac {1}{3}} + e\right )^{2} + 2 \, d^{3} x \log \left (x\right )^{2} - 22 \, d^{3} x \log \left (x\right ) - 66 \, d^{2} x^{\frac {2}{3}} e + 15 \, d x^{\frac {1}{3}} e^{2} - 6 \, {\left (2 \, d^{3} x \log \left (x\right ) - 11 \, d^{3} x\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 4 \, e^{3}\right )} n^{2} e^{\left (-3\right )}}{x}\right )} a b^{2} - \frac {1}{108} \, {\left (54 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 2 \, d^{3} e^{\left (-4\right )} \log \left (x\right ) - \frac {{\left (6 \, d^{2} x^{\frac {2}{3}} - 3 \, d x^{\frac {1}{3}} e + 2 \, e^{2}\right )} e^{\left (-3\right )}}{x}\right )} n e \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{2} + {\left (\frac {{\left (108 \, d^{3} x \log \left (d x^{\frac {1}{3}} + e\right )^{3} - 4 \, d^{3} x \log \left (x\right )^{3} + 66 \, d^{3} x \log \left (x\right )^{2} - 510 \, d^{3} x \log \left (x\right ) - 1530 \, d^{2} x^{\frac {2}{3}} e - 54 \, {\left (2 \, d^{3} x \log \left (x\right ) - 11 \, d^{3} x\right )} \log \left (d x^{\frac {1}{3}} + e\right )^{2} + 171 \, d x^{\frac {1}{3}} e^{2} + 18 \, {\left (2 \, d^{3} x \log \left (x\right )^{2} - 22 \, d^{3} x \log \left (x\right ) + 85 \, d^{3} x\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 24 \, e^{3}\right )} n^{2} e^{\left (-4\right )}}{x} - \frac {18 \, {\left (18 \, d^{3} x \log \left (d x^{\frac {1}{3}} + e\right )^{2} + 2 \, d^{3} x \log \left (x\right )^{2} - 22 \, d^{3} x \log \left (x\right ) - 66 \, d^{2} x^{\frac {2}{3}} e + 15 \, d x^{\frac {1}{3}} e^{2} - 6 \, {\left (2 \, d^{3} x \log \left (x\right ) - 11 \, d^{3} x\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 4 \, e^{3}\right )} n e^{\left (-4\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{x}\right )} n e\right )} b^{3} - \frac {3 \, a b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{2}}{x} - \frac {3 \, a^{2} b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{x} - \frac {a^{3}}{x} \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^2,x, algorithm="maxima")

[Out]

-1/2*(6*d^3*e^(-4)*log(d*x^(1/3) + e) - 2*d^3*e^(-4)*log(x) - (6*d^2*x^(2/3) - 3*d*x^(1/3)*e + 2*e^2)*e^(-3)/x
)*a^2*b*n*e - b^3*log(c*(d + e/x^(1/3))^n)^3/x - 1/6*(6*(6*d^3*e^(-4)*log(d*x^(1/3) + e) - 2*d^3*e^(-4)*log(x)
- (6*d^2*x^(2/3) - 3*d*x^(1/3)*e + 2*e^2)*e^(-3)/x)*n*e*log(c*(d + e/x^(1/3))^n) - (18*d^3*x*log(d*x^(1/3) +
e)^2 + 2*d^3*x*log(x)^2 - 22*d^3*x*log(x) - 66*d^2*x^(2/3)*e + 15*d*x^(1/3)*e^2 - 6*(2*d^3*x*log(x) - 11*d^3*x
)*log(d*x^(1/3) + e) - 4*e^3)*n^2*e^(-3)/x)*a*b^2 - 1/108*(54*(6*d^3*e^(-4)*log(d*x^(1/3) + e) - 2*d^3*e^(-4)*
log(x) - (6*d^2*x^(2/3) - 3*d*x^(1/3)*e + 2*e^2)*e^(-3)/x)*n*e*log(c*(d + e/x^(1/3))^n)^2 + ((108*d^3*x*log(d*
x^(1/3) + e)^3 - 4*d^3*x*log(x)^3 + 66*d^3*x*log(x)^2 - 510*d^3*x*log(x) - 1530*d^2*x^(2/3)*e - 54*(2*d^3*x*lo
g(x) - 11*d^3*x)*log(d*x^(1/3) + e)^2 + 171*d*x^(1/3)*e^2 + 18*(2*d^3*x*log(x)^2 - 22*d^3*x*log(x) + 85*d^3*x)
*log(d*x^(1/3) + e) - 24*e^3)*n^2*e^(-4)/x - 18*(18*d^3*x*log(d*x^(1/3) + e)^2 + 2*d^3*x*log(x)^2 - 22*d^3*x*l
og(x) - 66*d^2*x^(2/3)*e + 15*d*x^(1/3)*e^2 - 6*(2*d^3*x*log(x) - 11*d^3*x)*log(d*x^(1/3) + e) - 4*e^3)*n*e^(-
4)*log(c*(d + e/x^(1/3))^n)/x)*n*e)*b^3 - 3*a*b^2*log(c*(d + e/x^(1/3))^n)^2/x - 3*a^2*b*log(c*(d + e/x^(1/3))
^n)/x - a^3/x

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Fricas [A]
time = 0.42, size = 750, normalized size = 1.71 \begin {gather*} \frac {{\left (36 \, {\left (b^{3} x - b^{3}\right )} e^{3} \log \left (c\right )^{3} + 36 \, {\left (b^{3} n - 3 \, a b^{2} - {\left (b^{3} n - 3 \, a b^{2}\right )} x\right )} e^{3} \log \left (c\right )^{2} - 36 \, {\left (b^{3} d^{3} n^{3} x + b^{3} n^{3} e^{3}\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right )^{3} - 12 \, {\left (2 \, b^{3} n^{2} - 6 \, a b^{2} n + 9 \, a^{2} b - {\left (2 \, b^{3} n^{2} - 6 \, a b^{2} n + 9 \, a^{2} b\right )} x\right )} e^{3} \log \left (c\right ) + 18 \, {\left (6 \, b^{3} d^{2} n^{3} x^{\frac {2}{3}} e - 3 \, b^{3} d n^{3} x^{\frac {1}{3}} e^{2} + {\left (11 \, b^{3} d^{3} n^{3} - 6 \, a b^{2} d^{3} n^{2}\right )} x + 2 \, {\left (b^{3} n^{3} - 3 \, a b^{2} n^{2}\right )} e^{3} - 6 \, {\left (b^{3} d^{3} n^{2} x + b^{3} n^{2} e^{3}\right )} \log \left (c\right )\right )} \log \left (\frac {d x + x^{\frac {2}{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} - {\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n - 9 \, a^{3}\right )} x\right )} e^{3} - 6 \, {\left (18 \, {\left (b^{3} d^{3} n x + b^{3} n e^{3}\right )} \log \left (c\right )^{2} + {\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 \, {\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 \, {\left (b^{3} n^{2} - 3 \, a b^{2} n\right )} e^{3}\right )} \log \left (c\right ) - 6 \, {\left (6 \, b^{3} d^{2} n^{2} e \log \left (c\right ) - {\left (11 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2}\right )} e\right )} x^{\frac {2}{3}} + 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 {1}{3}}\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right ) + 6 \, {\left (18 \, b^{3} d^{2} n e \log \left (c\right )^{2} - 6 \, {\left (11 \, b^{3} d^{2} n^{2} - 6 \, a b^{2} d^{2} n\right )} 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 )} e\right )} x^{\frac {2}{3}} - 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 {1}{3}}\right )} e^{\left (-3\right )}}{36 \, x} \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{3}}{x^{2}}\, dx \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**2,x)

[Out]

Integral((a + b*log(c*(d + e/x**(1/3))**n))**3/x**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1758 vs. \(2 (391) = 782\).
time = 3.97, size = 1758, normalized size = 4.01 \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^2,x, algorithm="giac")

[Out]

-1/36*(108*(d*x^(1/3) + e)*b^3*d^2*n^3*log((d*x^(1/3) + e)/x^(1/3))^3/x^(1/3) - 108*(d*x^(1/3) + e)^2*b^3*d*n^
3*log((d*x^(1/3) + e)/x^(1/3))^3/x^(2/3) + 36*(d*x^(1/3) + e)^3*b^3*n^3*log((d*x^(1/3) + e)/x^(1/3))^3/x - 324
*(d*x^(1/3) + e)*b^3*d^2*n^3*log((d*x^(1/3) + e)/x^(1/3))^2/x^(1/3) + 324*(d*x^(1/3) + e)*b^3*d^2*n^2*log(c)*l
og((d*x^(1/3) + e)/x^(1/3))^2/x^(1/3) + 162*(d*x^(1/3) + e)^2*b^3*d*n^3*log((d*x^(1/3) + e)/x^(1/3))^2/x^(2/3)
- 324*(d*x^(1/3) + e)^2*b^3*d*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))^2/x^(2/3) - 36*(d*x^(1/3) + e)^3*b^3*n^
3*log((d*x^(1/3) + e)/x^(1/3))^2/x + 108*(d*x^(1/3) + e)^3*b^3*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))^2/x + 6
48*(d*x^(1/3) + e)*b^3*d^2*n^3*log((d*x^(1/3) + e)/x^(1/3))/x^(1/3) - 648*(d*x^(1/3) + e)*b^3*d^2*n^2*log(c)*l
og((d*x^(1/3) + e)/x^(1/3))/x^(1/3) + 324*(d*x^(1/3) + e)*b^3*d^2*n*log(c)^2*log((d*x^(1/3) + e)/x^(1/3))/x^(1
/3) + 324*(d*x^(1/3) + e)*a*b^2*d^2*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x^(1/3) - 162*(d*x^(1/3) + e)^2*b^3*d*n
^3*log((d*x^(1/3) + e)/x^(1/3))/x^(2/3) + 324*(d*x^(1/3) + e)^2*b^3*d*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))/
x^(2/3) - 324*(d*x^(1/3) + e)^2*b^3*d*n*log(c)^2*log((d*x^(1/3) + e)/x^(1/3))/x^(2/3) - 324*(d*x^(1/3) + e)^2*
a*b^2*d*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x^(2/3) + 24*(d*x^(1/3) + e)^3*b^3*n^3*log((d*x^(1/3) + e)/x^(1/3))
/x - 72*(d*x^(1/3) + e)^3*b^3*n^2*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x + 108*(d*x^(1/3) + e)^3*b^3*n*log(c)^2
*log((d*x^(1/3) + e)/x^(1/3))/x + 108*(d*x^(1/3) + e)^3*a*b^2*n^2*log((d*x^(1/3) + e)/x^(1/3))^2/x - 648*(d*x^
(1/3) + e)*b^3*d^2*n^3/x^(1/3) + 648*(d*x^(1/3) + e)*b^3*d^2*n^2*log(c)/x^(1/3) - 324*(d*x^(1/3) + e)*b^3*d^2*
n*log(c)^2/x^(1/3) + 108*(d*x^(1/3) + e)*b^3*d^2*log(c)^3/x^(1/3) - 648*(d*x^(1/3) + e)*a*b^2*d^2*n^2*log((d*x
^(1/3) + e)/x^(1/3))/x^(1/3) + 648*(d*x^(1/3) + e)*a*b^2*d^2*n*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x^(1/3) + 8
1*(d*x^(1/3) + e)^2*b^3*d*n^3/x^(2/3) - 162*(d*x^(1/3) + e)^2*b^3*d*n^2*log(c)/x^(2/3) + 162*(d*x^(1/3) + e)^2
*b^3*d*n*log(c)^2/x^(2/3) - 108*(d*x^(1/3) + e)^2*b^3*d*log(c)^3/x^(2/3) + 324*(d*x^(1/3) + e)^2*a*b^2*d*n^2*l
og((d*x^(1/3) + e)/x^(1/3))/x^(2/3) - 648*(d*x^(1/3) + e)^2*a*b^2*d*n*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x^(2
/3) - 8*(d*x^(1/3) + e)^3*b^3*n^3/x + 24*(d*x^(1/3) + e)^3*b^3*n^2*log(c)/x - 36*(d*x^(1/3) + e)^3*b^3*n*log(c
)^2/x + 36*(d*x^(1/3) + e)^3*b^3*log(c)^3/x - 72*(d*x^(1/3) + e)^3*a*b^2*n^2*log((d*x^(1/3) + e)/x^(1/3))/x +
216*(d*x^(1/3) + e)^3*a*b^2*n*log(c)*log((d*x^(1/3) + e)/x^(1/3))/x + 648*(d*x^(1/3) + e)*a*b^2*d^2*n^2/x^(1/3
) - 648*(d*x^(1/3) + e)*a*b^2*d^2*n*log(c)/x^(1/3) + 324*(d*x^(1/3) + e)*a*b^2*d^2*log(c)^2/x^(1/3) + 324*(d*x
^(1/3) + e)*a^2*b*d^2*n*log((d*x^(1/3) + e)/x^(1/3))/x^(1/3) - 162*(d*x^(1/3) + e)^2*a*b^2*d*n^2/x^(2/3) + 324
*(d*x^(1/3) + e)^2*a*b^2*d*n*log(c)/x^(2/3) - 324*(d*x^(1/3) + e)^2*a*b^2*d*log(c)^2/x^(2/3) - 324*(d*x^(1/3)
+ e)^2*a^2*b*d*n*log((d*x^(1/3) + e)/x^(1/3))/x^(2/3) + 24*(d*x^(1/3) + e)^3*a*b^2*n^2/x - 72*(d*x^(1/3) + e)^
3*a*b^2*n*log(c)/x + 108*(d*x^(1/3) + e)^3*a*b^2*log(c)^2/x + 108*(d*x^(1/3) + e)^3*a^2*b*n*log((d*x^(1/3) + e
)/x^(1/3))/x - 324*(d*x^(1/3) + e)*a^2*b*d^2*n/x^(1/3) + 324*(d*x^(1/3) + e)*a^2*b*d^2*log(c)/x^(1/3) + 162*(d
*x^(1/3) + e)^2*a^2*b*d*n/x^(2/3) - 324*(d*x^(1/3) + e)^2*a^2*b*d*log(c)/x^(2/3) - 36*(d*x^(1/3) + e)^3*a^2*b*
n/x + 108*(d*x^(1/3) + e)^3*a^2*b*log(c)/x + 108*(d*x^(1/3) + e)*a^3*d^2/x^(1/3) - 108*(d*x^(1/3) + e)^2*a^3*d
/x^(2/3) + 36*(d*x^(1/3) + e)^3*a^3/x)*e^(-3)

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Mupad [B]
time = 0.74, size = 570, normalized size = 1.30 \begin {gather*} \frac {\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,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 )}{4\,e}}{x^{2/3}}-{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^3\,\left (\frac {b^3}{x}+\frac {b^3\,d^3}{e^3}\right )-{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2\,\left (\frac {b^2\,\left (3\,a-b\,n\right )}{x}-\frac {\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {9\,a\,b^2\,d}{2\,e}}{x^{2/3}}+\frac {d\,\left (6\,a\,b^2\,d^2-11\,b^3\,d^2\,n\right )}{2\,e^3}+\frac {d\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{e}-\frac {9\,a\,b^2\,d}{e}\right )}{e\,x^{1/3}}\right )-\frac {a^3-a^2\,b\,n+\frac {2\,a\,b^2\,n^2}{3}-\frac {2\,b^3\,n^3}{9}}{x}-\frac {\frac {d\,\left (\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,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 )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (6\,a-11\,b\,n\right )}{e^2}}{x^{1/3}}-\frac {\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\,\left (\frac {\frac {d\,\left (b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{e}+6\,b^3\,d^2\,n^2}{e\,x^{1/3}}-\frac {b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )}{2\,e\,x^{2/3}}+\frac {b\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{3\,x}\right )}{e}-\frac {\ln \left (d+\frac {e}{x^{1/3}}\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 )}{6\,e^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^2,x)

[Out]

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

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