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

Optimal. Leaf size=680 \[ \frac{2 b d^9 n \log \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^9}-\frac{6 b d^8 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^9}+\frac{12 b d^7 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 )}{e^9}-\frac{56 b d^6 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 )}{3 e^9}+\frac{21 b d^5 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^9}-\frac{84 b d^4 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^9}+\frac{28 b d^3 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^9}-\frac{24 b d^2 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{7 e^9}+\frac{3 b d n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^9}-\frac{2 b n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{27 e^9}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{6 b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac{6 b^2 d^7 n^2 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{56 b^2 d^6 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}-\frac{21 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^4}{4 e^9}+\frac{84 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^9}-\frac{14 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}+\frac{24 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^7}{49 e^9}-\frac{b^2 d^9 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{3 e^9}-\frac{3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^8}{32 e^9}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^9}{243 e^9} \]

[Out]

(-6*b^2*d^7*n^2*(d + e*x^(1/3))^2)/e^9 + (56*b^2*d^6*n^2*(d + e*x^(1/3))^3)/(9*e^9) - (21*b^2*d^5*n^2*(d + e*x
^(1/3))^4)/(4*e^9) + (84*b^2*d^4*n^2*(d + e*x^(1/3))^5)/(25*e^9) - (14*b^2*d^3*n^2*(d + e*x^(1/3))^6)/(9*e^9)
+ (24*b^2*d^2*n^2*(d + e*x^(1/3))^7)/(49*e^9) - (3*b^2*d*n^2*(d + e*x^(1/3))^8)/(32*e^9) + (2*b^2*n^2*(d + e*x
^(1/3))^9)/(243*e^9) + (6*b^2*d^8*n^2*x^(1/3))/e^8 - (b^2*d^9*n^2*Log[d + e*x^(1/3)]^2)/(3*e^9) - (6*b*d^8*n*(
d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^9 + (12*b*d^7*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/
3))^n]))/e^9 - (56*b*d^6*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) + (21*b*d^5*n*(d + e*x^
(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^9 - (84*b*d^4*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]
))/(5*e^9) + (28*b*d^3*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) - (24*b*d^2*n*(d + e*x^(1
/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(7*e^9) + (3*b*d*n*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])
)/(4*e^9) - (2*b*n*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(27*e^9) + (2*b*d^9*n*Log[d + e*x^(1/3)
]*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) + (x^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/3

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Rubi [A]  time = 0.697383, antiderivative size = 491, normalized size of antiderivative = 0.72, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac{b n \left (\frac{22680 d^8 \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac{45360 d^7 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{70560 d^6 \left (d+e \sqrt [3]{x}\right )^3}{e^9}-\frac{79380 d^5 \left (d+e \sqrt [3]{x}\right )^4}{e^9}+\frac{63504 d^4 \left (d+e \sqrt [3]{x}\right )^5}{e^9}-\frac{35280 d^3 \left (d+e \sqrt [3]{x}\right )^6}{e^9}+\frac{12960 d^2 \left (d+e \sqrt [3]{x}\right )^7}{e^9}-\frac{2520 d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac{2835 d \left (d+e \sqrt [3]{x}\right )^8}{e^9}+\frac{280 \left (d+e \sqrt [3]{x}\right )^9}{e^9}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3780}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{6 b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac{6 b^2 d^7 n^2 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{56 b^2 d^6 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}-\frac{21 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^4}{4 e^9}+\frac{84 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^9}-\frac{14 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}+\frac{24 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^7}{49 e^9}-\frac{b^2 d^9 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{3 e^9}-\frac{3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^8}{32 e^9}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^9}{243 e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*b^2*d^7*n^2*(d + e*x^(1/3))^2)/e^9 + (56*b^2*d^6*n^2*(d + e*x^(1/3))^3)/(9*e^9) - (21*b^2*d^5*n^2*(d + e*x
^(1/3))^4)/(4*e^9) + (84*b^2*d^4*n^2*(d + e*x^(1/3))^5)/(25*e^9) - (14*b^2*d^3*n^2*(d + e*x^(1/3))^6)/(9*e^9)
+ (24*b^2*d^2*n^2*(d + e*x^(1/3))^7)/(49*e^9) - (3*b^2*d*n^2*(d + e*x^(1/3))^8)/(32*e^9) + (2*b^2*n^2*(d + e*x
^(1/3))^9)/(243*e^9) + (6*b^2*d^8*n^2*x^(1/3))/e^8 - (b^2*d^9*n^2*Log[d + e*x^(1/3)]^2)/(3*e^9) - (b*n*((22680
*d^8*(d + e*x^(1/3)))/e^9 - (45360*d^7*(d + e*x^(1/3))^2)/e^9 + (70560*d^6*(d + e*x^(1/3))^3)/e^9 - (79380*d^5
*(d + e*x^(1/3))^4)/e^9 + (63504*d^4*(d + e*x^(1/3))^5)/e^9 - (35280*d^3*(d + e*x^(1/3))^6)/e^9 + (12960*d^2*(
d + e*x^(1/3))^7)/e^9 - (2835*d*(d + e*x^(1/3))^8)/e^9 + (280*(d + e*x^(1/3))^9)/e^9 - (2520*d^9*Log[d + e*x^(
1/3)])/e^9)*(a + b*Log[c*(d + e*x^(1/3))^n]))/3780 + (x^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/3

Rule 2454

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])

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx &=3 \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac{1}{3} (2 b e n) \operatorname{Subst}\left (\int \frac{x^9 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac{1}{3} (2 b n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^9 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=-\frac{b n \left (\frac{22680 d^8 \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac{45360 d^7 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{70560 d^6 \left (d+e \sqrt [3]{x}\right )^3}{e^9}-\frac{79380 d^5 \left (d+e \sqrt [3]{x}\right )^4}{e^9}+\frac{63504 d^4 \left (d+e \sqrt [3]{x}\right )^5}{e^9}-\frac{35280 d^3 \left (d+e \sqrt [3]{x}\right )^6}{e^9}+\frac{12960 d^2 \left (d+e \sqrt [3]{x}\right )^7}{e^9}-\frac{2835 d \left (d+e \sqrt [3]{x}\right )^8}{e^9}+\frac{280 \left (d+e \sqrt [3]{x}\right )^9}{e^9}-\frac{2520 d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^9}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3780}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{1}{3} \left (2 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{22680 d^8 x-45360 d^7 x^2+70560 d^6 x^3-79380 d^5 x^4+63504 d^4 x^5-35280 d^3 x^6+12960 d^2 x^7-2835 d x^8+280 x^9-2520 d^9 \log (x)}{2520 e^9 x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=-\frac{b n \left (\frac{22680 d^8 \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac{45360 d^7 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{70560 d^6 \left (d+e \sqrt [3]{x}\right )^3}{e^9}-\frac{79380 d^5 \left (d+e \sqrt [3]{x}\right )^4}{e^9}+\frac{63504 d^4 \left (d+e \sqrt [3]{x}\right )^5}{e^9}-\frac{35280 d^3 \left (d+e \sqrt [3]{x}\right )^6}{e^9}+\frac{12960 d^2 \left (d+e \sqrt [3]{x}\right )^7}{e^9}-\frac{2835 d \left (d+e \sqrt [3]{x}\right )^8}{e^9}+\frac{280 \left (d+e \sqrt [3]{x}\right )^9}{e^9}-\frac{2520 d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^9}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3780}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{22680 d^8 x-45360 d^7 x^2+70560 d^6 x^3-79380 d^5 x^4+63504 d^4 x^5-35280 d^3 x^6+12960 d^2 x^7-2835 d x^8+280 x^9-2520 d^9 \log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{3780 e^9}\\ &=-\frac{b n \left (\frac{22680 d^8 \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac{45360 d^7 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{70560 d^6 \left (d+e \sqrt [3]{x}\right )^3}{e^9}-\frac{79380 d^5 \left (d+e \sqrt [3]{x}\right )^4}{e^9}+\frac{63504 d^4 \left (d+e \sqrt [3]{x}\right )^5}{e^9}-\frac{35280 d^3 \left (d+e \sqrt [3]{x}\right )^6}{e^9}+\frac{12960 d^2 \left (d+e \sqrt [3]{x}\right )^7}{e^9}-\frac{2835 d \left (d+e \sqrt [3]{x}\right )^8}{e^9}+\frac{280 \left (d+e \sqrt [3]{x}\right )^9}{e^9}-\frac{2520 d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^9}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3780}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (22680 d^8-45360 d^7 x+70560 d^6 x^2-79380 d^5 x^3+63504 d^4 x^4-35280 d^3 x^5+12960 d^2 x^6-2835 d x^7+280 x^8-\frac{2520 d^9 \log (x)}{x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{3780 e^9}\\ &=-\frac{6 b^2 d^7 n^2 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{56 b^2 d^6 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}-\frac{21 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^4}{4 e^9}+\frac{84 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^9}-\frac{14 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}+\frac{24 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^7}{49 e^9}-\frac{3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^8}{32 e^9}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^9}{243 e^9}+\frac{6 b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac{b n \left (\frac{22680 d^8 \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac{45360 d^7 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{70560 d^6 \left (d+e \sqrt [3]{x}\right )^3}{e^9}-\frac{79380 d^5 \left (d+e \sqrt [3]{x}\right )^4}{e^9}+\frac{63504 d^4 \left (d+e \sqrt [3]{x}\right )^5}{e^9}-\frac{35280 d^3 \left (d+e \sqrt [3]{x}\right )^6}{e^9}+\frac{12960 d^2 \left (d+e \sqrt [3]{x}\right )^7}{e^9}-\frac{2835 d \left (d+e \sqrt [3]{x}\right )^8}{e^9}+\frac{280 \left (d+e \sqrt [3]{x}\right )^9}{e^9}-\frac{2520 d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^9}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3780}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac{\left (2 b^2 d^9 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{3 e^9}\\ &=-\frac{6 b^2 d^7 n^2 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{56 b^2 d^6 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}-\frac{21 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^4}{4 e^9}+\frac{84 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^9}-\frac{14 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}+\frac{24 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^7}{49 e^9}-\frac{3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^8}{32 e^9}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^9}{243 e^9}+\frac{6 b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac{b^2 d^9 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{3 e^9}-\frac{b n \left (\frac{22680 d^8 \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac{45360 d^7 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac{70560 d^6 \left (d+e \sqrt [3]{x}\right )^3}{e^9}-\frac{79380 d^5 \left (d+e \sqrt [3]{x}\right )^4}{e^9}+\frac{63504 d^4 \left (d+e \sqrt [3]{x}\right )^5}{e^9}-\frac{35280 d^3 \left (d+e \sqrt [3]{x}\right )^6}{e^9}+\frac{12960 d^2 \left (d+e \sqrt [3]{x}\right )^7}{e^9}-\frac{2835 d \left (d+e \sqrt [3]{x}\right )^8}{e^9}+\frac{280 \left (d+e \sqrt [3]{x}\right )^9}{e^9}-\frac{2520 d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^9}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3780}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\\ \end{align*}

Mathematica [A]  time = 0.549727, size = 411, normalized size = 0.6 \[ \frac{e \sqrt [3]{x} \left (3175200 a^2 e^8 x^{8/3}-2520 a b n \left (840 d^6 e^2 x^{2/3}+504 d^4 e^4 x^{4/3}-420 d^3 e^5 x^{5/3}+360 d^2 e^6 x^2-630 d^5 e^3 x-1260 d^7 e \sqrt [3]{x}+2520 d^8-315 d e^7 x^{7/3}+280 e^8 x^{8/3}\right )+b^2 n^2 \left (2813160 d^6 e^2 x^{2/3}+947016 d^4 e^4 x^{4/3}-577500 d^3 e^5 x^{5/3}+343800 d^2 e^6 x^2-1580670 d^5 e^3 x-5807340 d^7 e \sqrt [3]{x}[phone] d^8-187425 d e^7 x^{7/3}+78400 e^8 x^{8/3}\right )\right )+2520 b \left (2520 a \left (d^9+e^9 x^3\right )-b n \left (-1260 d^7 e^2 x^{2/3}-630 d^5 e^4 x^{4/3}+504 d^4 e^5 x^{5/3}-420 d^3 e^6 x^2+360 d^2 e^7 x^{7/3}+840 d^6 e^3 x+2520 d^8 e \sqrt [3]{x}+7129 d^9-315 d e^8 x^{8/3}+280 e^9 x^3\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+3175200 b^2 \left (d^9+e^9 x^3\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{9525600 e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x^(1/3)*(3175200*a^2*e^8*x^(8/3) - 2520*a*b*n*(2520*d^8 - 1260*d^7*e*x^(1/3) + 840*d^6*e^2*x^(2/3) - 630*d^
5*e^3*x + 504*d^4*e^4*x^(4/3) - 420*d^3*e^5*x^(5/3) + 360*d^2*e^6*x^2 - 315*d*e^7*x^(7/3) + 280*e^8*x^(8/3)) +
b^2*n^2*(17965080*d^8 - 5807340*d^7*e*x^(1/3) + 2813160*d^6*e^2*x^(2/3) - 1580670*d^5*e^3*x + 947016*d^4*e^4*
x^(4/3) - 577500*d^3*e^5*x^(5/3) + 343800*d^2*e^6*x^2 - 187425*d*e^7*x^(7/3) + 78400*e^8*x^(8/3))) + 2520*b*(2
520*a*(d^9 + e^9*x^3) - b*n*(7129*d^9 + 2520*d^8*e*x^(1/3) - 1260*d^7*e^2*x^(2/3) + 840*d^6*e^3*x - 630*d^5*e^
4*x^(4/3) + 504*d^4*e^5*x^(5/3) - 420*d^3*e^6*x^2 + 360*d^2*e^7*x^(7/3) - 315*d*e^8*x^(8/3) + 280*e^9*x^3))*Lo
g[c*(d + e*x^(1/3))^n] + 3175200*b^2*(d^9 + e^9*x^3)*Log[c*(d + e*x^(1/3))^n]^2)/(9525600*e^9)

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04935, size = 572, normalized size = 0.84 \begin{align*} \frac{1}{3} \, b^{2} x^{3} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{2} + \frac{2}{3} \, a b x^{3} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + \frac{1}{3} \, a^{2} x^{3} + \frac{1}{3780} \, a b e n{\left (\frac{2520 \, d^{9} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{10}} - \frac{280 \, e^{8} x^{3} - 315 \, d e^{7} x^{\frac{8}{3}} + 360 \, d^{2} e^{6} x^{\frac{7}{3}} - 420 \, d^{3} e^{5} x^{2} + 504 \, d^{4} e^{4} x^{\frac{5}{3}} - 630 \, d^{5} e^{3} x^{\frac{4}{3}} + 840 \, d^{6} e^{2} x - 1260 \, d^{7} e x^{\frac{2}{3}} + 2520 \, d^{8} x^{\frac{1}{3}}}{e^{9}}\right )} + \frac{1}{9525600} \,{\left (2520 \, e n{\left (\frac{2520 \, d^{9} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{10}} - \frac{280 \, e^{8} x^{3} - 315 \, d e^{7} x^{\frac{8}{3}} + 360 \, d^{2} e^{6} x^{\frac{7}{3}} - 420 \, d^{3} e^{5} x^{2} + 504 \, d^{4} e^{4} x^{\frac{5}{3}} - 630 \, d^{5} e^{3} x^{\frac{4}{3}} + 840 \, d^{6} e^{2} x - 1260 \, d^{7} e x^{\frac{2}{3}} + 2520 \, d^{8} x^{\frac{1}{3}}}{e^{9}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + \frac{{\left (78400 \, e^{9} x^{3} - 187425 \, d e^{8} x^{\frac{8}{3}} + 343800 \, d^{2} e^{7} x^{\frac{7}{3}} - 577500 \, d^{3} e^{6} x^{2} - 3175200 \, d^{9} \log \left (e x^{\frac{1}{3}} + d\right )^{2} + 947016 \, d^{4} e^{5} x^{\frac{5}{3}} - 1580670 \, d^{5} e^{4} x^{\frac{4}{3}} + 2813160 \, d^{6} e^{3} x - 17965080 \, d^{9} \log \left (e x^{\frac{1}{3}} + d\right ) - 5807340 \, d^{7} e^{2} x^{\frac{2}{3}} + 17965080 \, d^{8} e x^{\frac{1}{3}}\right )} n^{2}}{e^{9}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="maxima")

[Out]

1/3*b^2*x^3*log((e*x^(1/3) + d)^n*c)^2 + 2/3*a*b*x^3*log((e*x^(1/3) + d)^n*c) + 1/3*a^2*x^3 + 1/3780*a*b*e*n*(
2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8*x^3 - 315*d*e^7*x^(8/3) + 360*d^2*e^6*x^(7/3) - 420*d^3*e^5*x^2 +
504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) + 840*d^6*e^2*x - 1260*d^7*e*x^(2/3) + 2520*d^8*x^(1/3))/e^9) + 1/95
25600*(2520*e*n*(2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8*x^3 - 315*d*e^7*x^(8/3) + 360*d^2*e^6*x^(7/3) - 4
20*d^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) + 840*d^6*e^2*x - 1260*d^7*e*x^(2/3) + 2520*d^8*x^(
1/3))/e^9)*log((e*x^(1/3) + d)^n*c) + (78400*e^9*x^3 - 187425*d*e^8*x^(8/3) + 343800*d^2*e^7*x^(7/3) - 577500*
d^3*e^6*x^2 - 3175200*d^9*log(e*x^(1/3) + d)^2 + 947016*d^4*e^5*x^(5/3) - 1580670*d^5*e^4*x^(4/3) + 2813160*d^
6*e^3*x - 17965080*d^9*log(e*x^(1/3) + d) - 5807340*d^7*e^2*x^(2/3) + 17965080*d^8*e*x^(1/3))*n^2/e^9)*b^2

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Fricas [A]  time = 2.47932, size = 1574, normalized size = 2.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="fricas")

[Out]

1/9525600*(3175200*b^2*e^9*x^3*log(c)^2 + 39200*(2*b^2*e^9*n^2 - 18*a*b*e^9*n + 81*a^2*e^9)*x^3 - 2100*(275*b^
2*d^3*e^6*n^2 - 504*a*b*d^3*e^6*n)*x^2 + 3175200*(b^2*e^9*n^2*x^3 + b^2*d^9*n^2)*log(e*x^(1/3) + d)^2 + 840*(3
349*b^2*d^6*e^3*n^2 - 2520*a*b*d^6*e^3*n)*x + 2520*(420*b^2*d^3*e^6*n^2*x^2 - 840*b^2*d^6*e^3*n^2*x - 7129*b^2
*d^9*n^2 + 2520*a*b*d^9*n - 280*(b^2*e^9*n^2 - 9*a*b*e^9*n)*x^3 + 2520*(b^2*e^9*n*x^3 + b^2*d^9*n)*log(c) + 63
*(5*b^2*d*e^8*n^2*x^2 - 8*b^2*d^4*e^5*n^2*x + 20*b^2*d^7*e^2*n^2)*x^(2/3) - 90*(4*b^2*d^2*e^7*n^2*x^2 - 7*b^2*
d^5*e^4*n^2*x + 28*b^2*d^8*e*n^2)*x^(1/3))*log(e*x^(1/3) + d) + 352800*(3*b^2*d^3*e^6*n*x^2 - 6*b^2*d^6*e^3*n*
x - 2*(b^2*e^9*n - 9*a*b*e^9)*x^3)*log(c) - 63*(92180*b^2*d^7*e^2*n^2 - 50400*a*b*d^7*e^2*n + 175*(17*b^2*d*e^
8*n^2 - 72*a*b*d*e^8*n)*x^2 - 8*(1879*b^2*d^4*e^5*n^2 - 2520*a*b*d^4*e^5*n)*x - 2520*(5*b^2*d*e^8*n*x^2 - 8*b^
2*d^4*e^5*n*x + 20*b^2*d^7*e^2*n)*log(c))*x^(2/3) + 90*(199612*b^2*d^8*e*n^2 - 70560*a*b*d^8*e*n + 20*(191*b^2
*d^2*e^7*n^2 - 504*a*b*d^2*e^7*n)*x^2 - 7*(2509*b^2*d^5*e^4*n^2 - 2520*a*b*d^5*e^4*n)*x - 2520*(4*b^2*d^2*e^7*
n*x^2 - 7*b^2*d^5*e^4*n*x + 28*b^2*d^8*e*n)*log(c))*x^(1/3))/e^9

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.34386, size = 1926, normalized size = 2.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="giac")

[Out]

1/9525600*(3175200*b^2*x^3*e*log(c)^2 + 6350400*a*b*x^3*e*log(c) + 3175200*a^2*x^3*e + (3175200*(x^(1/3)*e + d
)^9*e^(-8)*log(x^(1/3)*e + d)^2 - 28576800*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d)^2 + 114307200*(x^(1/3
)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^2 - 266716800*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d)^2 + 400
075200*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d)^2 - 400075200*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*
e + d)^2 + 266716800*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d)^2 - 114307200*(x^(1/3)*e + d)^2*d^7*e^(-8
)*log(x^(1/3)*e + d)^2 + 28576800*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d)^2 - 705600*(x^(1/3)*e + d)^9*e
^(-8)*log(x^(1/3)*e + d) + 7144200*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) - 32659200*(x^(1/3)*e + d)^7*
d^2*e^(-8)*log(x^(1/3)*e + d) + 88905600*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) - 160030080*(x^(1/3)*
e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) + 200037600*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d) - 177811200
*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) + 114307200*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) -
57153600*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) + 78400*(x^(1/3)*e + d)^9*e^(-8) - 893025*(x^(1/3)*e +
d)^8*d*e^(-8) + 4665600*(x^(1/3)*e + d)^7*d^2*e^(-8) - 14817600*(x^(1/3)*e + d)^6*d^3*e^(-8) + 32006016*(x^(1
/3)*e + d)^5*d^4*e^(-8) - 50009400*(x^(1/3)*e + d)^4*d^5*e^(-8) + 59270400*(x^(1/3)*e + d)^3*d^6*e^(-8) - 5715
3600*(x^(1/3)*e + d)^2*d^7*e^(-8) + 57153600*(x^(1/3)*e + d)*d^8*e^(-8))*b^2*n^2 + 2520*(2520*(x^(1/3)*e + d)^
9*e^(-8)*log(x^(1/3)*e + d) - 22680*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) + 90720*(x^(1/3)*e + d)^7*d^
2*e^(-8)*log(x^(1/3)*e + d) - 211680*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) + 317520*(x^(1/3)*e + d)^
5*d^4*e^(-8)*log(x^(1/3)*e + d) - 317520*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d) + 211680*(x^(1/3)*e +
d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) - 90720*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) + 22680*(x^(1/3)*e
+ d)*d^8*e^(-8)*log(x^(1/3)*e + d) - 280*(x^(1/3)*e + d)^9*e^(-8) + 2835*(x^(1/3)*e + d)^8*d*e^(-8) - 12960*(
x^(1/3)*e + d)^7*d^2*e^(-8) + 35280*(x^(1/3)*e + d)^6*d^3*e^(-8) - 63504*(x^(1/3)*e + d)^5*d^4*e^(-8) + 79380*
(x^(1/3)*e + d)^4*d^5*e^(-8) - 70560*(x^(1/3)*e + d)^3*d^6*e^(-8) + 45360*(x^(1/3)*e + d)^2*d^7*e^(-8) - 22680
*(x^(1/3)*e + d)*d^8*e^(-8))*b^2*n*log(c) + 2520*(2520*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) - 22680*(x^
(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) + 90720*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) - 211680*(x
^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) + 317520*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) - 31752
0*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d) + 211680*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) - 9
0720*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) + 22680*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) - 2
80*(x^(1/3)*e + d)^9*e^(-8) + 2835*(x^(1/3)*e + d)^8*d*e^(-8) - 12960*(x^(1/3)*e + d)^7*d^2*e^(-8) + 35280*(x^
(1/3)*e + d)^6*d^3*e^(-8) - 63504*(x^(1/3)*e + d)^5*d^4*e^(-8) + 79380*(x^(1/3)*e + d)^4*d^5*e^(-8) - 70560*(x
^(1/3)*e + d)^3*d^6*e^(-8) + 45360*(x^(1/3)*e + d)^2*d^7*e^(-8) - 22680*(x^(1/3)*e + d)*d^8*e^(-8))*a*b*n)*e^(
-1)