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]

9/4*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*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))*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*ln
(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 = 0.44, antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 2295

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

Rule 2296

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

Rule 2304

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

Rule 2305

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

Rule 2389

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

Rule 2390

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

Rule 2401

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

Rule 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]

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*}

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Mathematica [A]  time = 0.23, size = 362, normalized size = 0.83 \[ \frac {36 a^3 \left (d^3+e^3 x\right )+6 b \left (d+e \sqrt [3]{x}\right ) \left (18 a^2 \left (d^2-d e \sqrt [3]{x}+e^2 x^{2/3}\right )-6 a b n \left (11 d^2-5 d e \sqrt [3]{x}+2 e^2 x^{2/3}\right )+b^2 n^2 \left (85 d^2-19 d e \sqrt [3]{x}+4 e^2 x^{2/3}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-18 a^2 b n \left (11 d^3+6 d^2 e \sqrt [3]{x}-3 d e^2 x^{2/3}+2 e^3 x\right )+18 b^2 \left (6 a \left (d^3+e^3 x\right )-b n \left (11 d^3+6 d^2 e \sqrt [3]{x}-3 d e^2 x^{2/3}+2 e^3 x\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )-6 a b^2 n^2 \left (23 d^3-66 d^2 e \sqrt [3]{x}+15 d e^2 x^{2/3}-4 e^3 x\right )+36 b^3 \left (d^3+e^3 x\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+b^3 e n^3 \sqrt [3]{x} \left (-510 d^2+57 d e \sqrt [3]{x}-8 e^2 x^{2/3}\right )}{36 e^3} \]

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)

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fricas [A]  time = 0.48, size = 690, normalized size = 1.58 \[ \frac {36 \, b^{3} e^{3} x \log \relax (c)^{3} + 36 \, {\left (b^{3} e^{3} n^{3} x + b^{3} d^{3} n^{3}\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{3} - 36 \, {\left (b^{3} e^{3} n - 3 \, a b^{2} e^{3}\right )} x \log \relax (c)^{2} + 18 \, {\left (3 \, b^{3} d e^{2} n^{3} x^{\frac {2}{3}} - 6 \, b^{3} d^{2} e n^{3} x^{\frac {1}{3}} - 11 \, b^{3} d^{3} n^{3} + 6 \, a b^{2} d^{3} n^{2} - 2 \, {\left (b^{3} e^{3} n^{3} - 3 \, a b^{2} e^{3} n^{2}\right )} x + 6 \, {\left (b^{3} e^{3} n^{2} x + b^{3} d^{3} n^{2}\right )} \log \relax (c)\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{2} + 12 \, {\left (2 \, b^{3} e^{3} n^{2} - 6 \, a b^{2} e^{3} n + 9 \, a^{2} b e^{3}\right )} x \log \relax (c) - 4 \, {\left (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}\right )} x + 6 \, {\left (85 \, b^{3} d^{3} n^{3} - 66 \, a b^{2} d^{3} n^{2} + 18 \, a^{2} b d^{3} n + 18 \, {\left (b^{3} e^{3} n x + b^{3} d^{3} n\right )} \log \relax (c)^{2} + 2 \, {\left (2 \, b^{3} e^{3} n^{3} - 6 \, a b^{2} e^{3} n^{2} + 9 \, a^{2} b e^{3} n\right )} x - 6 \, {\left (11 \, b^{3} d^{3} n^{2} - 6 \, a b^{2} d^{3} n + 2 \, {\left (b^{3} e^{3} n^{2} - 3 \, a b^{2} e^{3} n\right )} x\right )} \log \relax (c) - 3 \, {\left (5 \, b^{3} d e^{2} n^{3} - 6 \, b^{3} d e^{2} n^{2} \log \relax (c) - 6 \, a b^{2} d e^{2} n^{2}\right )} x^{\frac {2}{3}} + 6 \, {\left (11 \, b^{3} d^{2} e n^{3} - 6 \, b^{3} d^{2} e n^{2} \log \relax (c) - 6 \, a b^{2} d^{2} e n^{2}\right )} x^{\frac {1}{3}}\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 3 \, {\left (19 \, b^{3} d e^{2} n^{3} + 18 \, b^{3} d e^{2} n \log \relax (c)^{2} - 30 \, a b^{2} d e^{2} n^{2} + 18 \, a^{2} b d e^{2} n - 6 \, {\left (5 \, b^{3} d e^{2} n^{2} - 6 \, a b^{2} d e^{2} n\right )} \log \relax (c)\right )} x^{\frac {2}{3}} - 6 \, {\left (85 \, b^{3} d^{2} e n^{3} + 18 \, b^{3} d^{2} e n \log \relax (c)^{2} - 66 \, a b^{2} d^{2} e n^{2} + 18 \, a^{2} b d^{2} e n - 6 \, {\left (11 \, b^{3} d^{2} e n^{2} - 6 \, a b^{2} d^{2} e n\right )} \log \relax (c)\right )} x^{\frac {1}{3}}}{36 \, e^{3}} \]

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

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giac [B]  time = 0.28, size = 1105, normalized size = 2.52 \[ \text {result too large to display} \]

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)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \,x^{\frac {1}{3}}+d \right )^{n}\right )+a \right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 455, normalized size = 1.04 \[ \frac {1}{2} \, {\left (e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} + 6 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )\right )} a^{2} b + \frac {1}{6} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + 18 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} - \frac {{\left (18 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac {2}{3}} - 66 \, d^{2} e x^{\frac {1}{3}}\right )} n^{2}}{e^{3}}\right )} a b^{2} + \frac {1}{36} \, {\left (18 \, e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} + 36 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{3} + e n {\left (\frac {{\left (36 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{3} + 198 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 8 \, e^{3} x + 510 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right ) + 57 \, d e^{2} x^{\frac {2}{3}} - 510 \, d^{2} e x^{\frac {1}{3}}\right )} n^{2}}{e^{4}} - \frac {6 \, {\left (18 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac {2}{3}} - 66 \, d^{2} e x^{\frac {1}{3}}\right )} n \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )}{e^{4}}\right )}\right )} b^{3} + a^{3} x \]

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

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mupad [B]  time = 0.69, size = 558, normalized size = 1.27 \[ x\,\left (a^3-a^2\,b\,n+\frac {2\,a\,b^2\,n^2}{3}-\frac {2\,b^3\,n^3}{9}\right )-x^{2/3}\,\left (\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,b^3\,n^3}{3}\right )}{2\,e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{4\,e}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,d^3}{e^3}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2\,\left (\frac {d\,\left (6\,a\,b^2\,d^2-11\,b^3\,d^2\,n\right )}{2\,e^3}-x^{2/3}\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {9\,a\,b^2\,d}{2\,e}\right )+b^2\,x\,\left (3\,a-b\,n\right )+\frac {d\,x^{1/3}\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{e}-\frac {9\,a\,b^2\,d}{e}\right )}{e}\right )+x^{1/3}\,\left (\frac {d\,\left (\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,b^3\,n^3}{3}\right )}{e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (6\,a-11\,b\,n\right )}{e^2}\right )+\frac {\ln \left (d+e\,x^{1/3}\right )\,\left (18\,a^2\,b\,d^3\,n-66\,a\,b^2\,d^3\,n^2+85\,b^3\,d^3\,n^3\right )}{6\,e^3}+\frac {\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\,\left (\frac {x^{1/3}\,\left (\frac {d\,\left (b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{e}+6\,b^3\,d^2\,n^2\right )}{e}-\frac {x^{2/3}\,\left (b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{2\,e}+\frac {b\,e\,x\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{3}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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