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

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

[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*x^(1/3))/e
^2 - (18*b^3*d^2*n^3*x^(1/3))/e^2 + (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

________________________________________________________________________________________

Rubi [A]  time = 0.443928, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2451, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac{9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac{18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}+\frac{\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac{18 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac{18 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}-\frac{2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac{9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^(1/3))^n])^3,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*x^(1/3))/e
^2 - (18*b^3*d^2*n^3*x^(1/3))/e^2 + (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 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p,
 q}, x] && FractionQ[n]

Rule 2401

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

Rule 2389

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

Rule 2296

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

Rule 2295

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

Rule 2390

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

Rule 2305

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

Rule 2304

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

Rubi steps

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

Mathematica [A]  time = 0.224464, size = 362, normalized size = 0.83 \[ \frac{6 b \left (d+e \sqrt [3]{x}\right ) \left (18 a^2 \left (d^2-d e \sqrt [3]{x}+e^2 x^{2/3}\right )-6 a b n \left (11 d^2-5 d e \sqrt [3]{x}+2 e^2 x^{2/3}\right )+b^2 n^2 \left (85 d^2-19 d e \sqrt [3]{x}+4 e^2 x^{2/3}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-18 a^2 b n \left (6 d^2 e \sqrt [3]{x}+11 d^3-3 d e^2 x^{2/3}+2 e^3 x\right )+36 a^3 \left (d^3+e^3 x\right )+18 b^2 \left (6 a \left (d^3+e^3 x\right )-b n \left (6 d^2 e \sqrt [3]{x}+11 d^3-3 d e^2 x^{2/3}+2 e^3 x\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )-6 a b^2 n^2 \left (-66 d^2 e \sqrt [3]{x}+23 d^3+15 d e^2 x^{2/3}-4 e^3 x\right )+36 b^3 \left (d^3+e^3 x\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+b^3 e n^3 \sqrt [3]{x} \left (-510 d^2+57 d e \sqrt [3]{x}-8 e^2 x^{2/3}\right )}{36 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*e*n^3*(-510*d^2 + 57*d*e*x^(1/3) - 8*e^2*x^(2/3))*x^(1/3) - 6*a*b^2*n^2*(23*d^3 - 66*d^2*e*x^(1/3) + 15*d
*e^2*x^(2/3) - 4*e^3*x) + 36*a^3*(d^3 + e^3*x) - 18*a^2*b*n*(11*d^3 + 6*d^2*e*x^(1/3) - 3*d*e^2*x^(2/3) + 2*e^
3*x) + 6*b*(18*a^2*(d^2 - d*e*x^(1/3) + e^2*x^(2/3)) - 6*a*b*n*(11*d^2 - 5*d*e*x^(1/3) + 2*e^2*x^(2/3)) + b^2*
n^2*(85*d^2 - 19*d*e*x^(1/3) + 4*e^2*x^(2/3)))*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n] + 18*b^2*(6*a*(d^3 + e
^3*x) - b*n*(11*d^3 + 6*d^2*e*x^(1/3) - 3*d*e^2*x^(2/3) + 2*e^3*x))*Log[c*(d + e*x^(1/3))^n]^2 + 36*b^3*(d^3 +
 e^3*x)*Log[c*(d + e*x^(1/3))^n]^3)/(36*e^3)

________________________________________________________________________________________

Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08376, size = 614, normalized size = 1.4 \begin{align*} \frac{1}{2} \,{\left (e n{\left (\frac{6 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{4}} - \frac{2 \, e^{2} x - 3 \, d e x^{\frac{2}{3}} + 6 \, d^{2} x^{\frac{1}{3}}}{e^{3}}\right )} + 6 \, x \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )\right )} a^{2} b + \frac{1}{6} \,{\left (6 \, e n{\left (\frac{6 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{4}} - \frac{2 \, e^{2} x - 3 \, d e x^{\frac{2}{3}} + 6 \, d^{2} x^{\frac{1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + 18 \, x \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{2} - \frac{{\left (18 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac{2}{3}} - 66 \, d^{2} e x^{\frac{1}{3}}\right )} n^{2}}{e^{3}}\right )} a b^{2} + \frac{1}{36} \,{\left (18 \, e n{\left (\frac{6 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{4}} - \frac{2 \, e^{2} x - 3 \, d e x^{\frac{2}{3}} + 6 \, d^{2} x^{\frac{1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{2} + 36 \, x \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{3} + e n{\left (\frac{{\left (36 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{3} + 198 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 8 \, e^{3} x + 510 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right ) + 57 \, d e^{2} x^{\frac{2}{3}} - 510 \, d^{2} e x^{\frac{1}{3}}\right )} n^{2}}{e^{4}} - \frac{6 \,{\left (18 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac{2}{3}} - 66 \, d^{2} e x^{\frac{1}{3}}\right )} n \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )}{e^{4}}\right )}\right )} b^{3} + a^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(e*n*(6*d^3*log(e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2*x^(1/3))/e^3) + 6*x*log((e*x^(1/3) +
 d)^n*c))*a^2*b + 1/6*(6*e*n*(6*d^3*log(e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2*x^(1/3))/e^3)*lo
g((e*x^(1/3) + d)^n*c) + 18*x*log((e*x^(1/3) + d)^n*c)^2 - (18*d^3*log(e*x^(1/3) + d)^2 - 4*e^3*x + 66*d^3*log
(e*x^(1/3) + d) + 15*d*e^2*x^(2/3) - 66*d^2*e*x^(1/3))*n^2/e^3)*a*b^2 + 1/36*(18*e*n*(6*d^3*log(e*x^(1/3) + d)
/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2*x^(1/3))/e^3)*log((e*x^(1/3) + d)^n*c)^2 + 36*x*log((e*x^(1/3) + d)^n*
c)^3 + e*n*((36*d^3*log(e*x^(1/3) + d)^3 + 198*d^3*log(e*x^(1/3) + d)^2 - 8*e^3*x + 510*d^3*log(e*x^(1/3) + d)
 + 57*d*e^2*x^(2/3) - 510*d^2*e*x^(1/3))*n^2/e^4 - 6*(18*d^3*log(e*x^(1/3) + d)^2 - 4*e^3*x + 66*d^3*log(e*x^(
1/3) + d) + 15*d*e^2*x^(2/3) - 66*d^2*e*x^(1/3))*n*log((e*x^(1/3) + d)^n*c)/e^4))*b^3 + a^3*x

________________________________________________________________________________________

Fricas [A]  time = 2.09555, size = 1544, normalized size = 3.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.33871, size = 1492, normalized size = 3.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(36*b^3*x*e*log(c)^3 + (36*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d)^3 - 108*(x^(1/3)*e + d)^2*d*e^(-2)
*log(x^(1/3)*e + d)^3 + 108*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d)^3 - 36*(x^(1/3)*e + d)^3*e^(-2)*log(
x^(1/3)*e + d)^2 + 162*(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e + d)^2 - 324*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^
(1/3)*e + d)^2 + 24*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d) - 162*(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e
 + d) + 648*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d) - 8*(x^(1/3)*e + d)^3*e^(-2) + 81*(x^(1/3)*e + d)^2*
d*e^(-2) - 648*(x^(1/3)*e + d)*d^2*e^(-2))*b^3*n^3 + 6*(18*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d)^2 - 54*
(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e + d)^2 + 54*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d)^2 - 12*(x^(
1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d) + 54*(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e + d) - 108*(x^(1/3)*e +
d)*d^2*e^(-2)*log(x^(1/3)*e + d) + 4*(x^(1/3)*e + d)^3*e^(-2) - 27*(x^(1/3)*e + d)^2*d*e^(-2) + 108*(x^(1/3)*e
 + d)*d^2*e^(-2))*b^3*n^2*log(c) + 18*(6*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d) - 18*(x^(1/3)*e + d)^2*d*
e^(-2)*log(x^(1/3)*e + d) + 18*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d) - 2*(x^(1/3)*e + d)^3*e^(-2) + 9*
(x^(1/3)*e + d)^2*d*e^(-2) - 18*(x^(1/3)*e + d)*d^2*e^(-2))*b^3*n*log(c)^2 + 108*a*b^2*x*e*log(c)^2 + 6*(18*(x
^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d)^2 - 54*(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e + d)^2 + 54*(x^(1/3)
*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d)^2 - 12*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d) + 54*(x^(1/3)*e + d)^
2*d*e^(-2)*log(x^(1/3)*e + d) - 108*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d) + 4*(x^(1/3)*e + d)^3*e^(-2)
 - 27*(x^(1/3)*e + d)^2*d*e^(-2) + 108*(x^(1/3)*e + d)*d^2*e^(-2))*a*b^2*n^2 + 36*(6*(x^(1/3)*e + d)^3*e^(-2)*
log(x^(1/3)*e + d) - 18*(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e + d) + 18*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1
/3)*e + d) - 2*(x^(1/3)*e + d)^3*e^(-2) + 9*(x^(1/3)*e + d)^2*d*e^(-2) - 18*(x^(1/3)*e + d)*d^2*e^(-2))*a*b^2*
n*log(c) + 108*a^2*b*x*e*log(c) + 18*(6*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d) - 18*(x^(1/3)*e + d)^2*d*e
^(-2)*log(x^(1/3)*e + d) + 18*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d) - 2*(x^(1/3)*e + d)^3*e^(-2) + 9*(
x^(1/3)*e + d)^2*d*e^(-2) - 18*(x^(1/3)*e + d)*d^2*e^(-2))*a^2*b*n + 36*a^3*x*e)*e^(-1)