3.5.50 \(\int x^2 (a+b \log (c (d+e \sqrt [3]{x})^n))^2 \, dx\) [450]

Optimal. Leaf size=680 \[ -\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 {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 {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 {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.47, antiderivative size = 680, normalized size of antiderivative = 1.00, 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 = {2504, 2445, 2458, 45, 2372, 12, 14, 2338} \begin {gather*} \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 {b^2 d^9 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{3 e^9}+\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 {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} \end {gather*}

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) - (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

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 45

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 2338

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]

Rule 2372

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]}, Dist[a + b*Log[c*x^n], u, 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 2445

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 2458

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 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 x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx &=3 \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.38, size = 458, normalized size = 0.67 \begin {gather*} \frac {-3175200 b^2 d^9 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )+2520 b d^9 n \log \left (d+e \sqrt [3]{x}\right ) \left (2520 a-7129 b n+2520 b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+e \sqrt [3]{x} \left (3175200 a^2 e^8 x^{8/3}-2520 a b n \left (2520 d^8-1260 d^7 e \sqrt [3]{x}+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}\right )+b^2 n^2 \left (17965080 d^8-5807340 d^7 e \sqrt [3]{x}+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}\right )-2520 b \left (-2520 a e^8 x^{8/3}+b n \left (2520 d^8-1260 d^7 e \sqrt [3]{x}+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}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+3175200 b^2 e^8 x^{8/3} \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{9525600 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3175200*b^2*d^9*n^2*Log[d + e*x^(1/3)]^2 + 2520*b*d^9*n*Log[d + e*x^(1/3)]*(2520*a - 7129*b*n + 2520*b*Log[c
*(d + e*x^(1/3))^n]) + 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*(-2520*a*e^8*x^(8/3) + 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)))*Log
[c*(d + e*x^(1/3))^n] + 3175200*b^2*e^8*x^(8/3)*Log[c*(d + e*x^(1/3))^n]^2))/(9525600*e^9)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )^{2}\, dx\]

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 = 0.29, size = 407, normalized size = 0.60 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )^{2} + \frac {2}{3} \, a b x^{3} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{2} x^{3} + \frac {1}{3780} \, {\left (2520 \, d^{9} e^{\left (-10\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (1260 \, d^{7} x^{\frac {2}{3}} e - 2520 \, d^{8} x^{\frac {1}{3}} - 840 \, d^{6} x e^{2} + 630 \, d^{5} x^{\frac {4}{3}} e^{3} - 504 \, d^{4} x^{\frac {5}{3}} e^{4} + 420 \, d^{3} x^{2} e^{5} - 360 \, d^{2} x^{\frac {7}{3}} e^{6} + 315 \, d x^{\frac {8}{3}} e^{7} - 280 \, x^{3} e^{8}\right )} e^{\left (-9\right )}\right )} a b n e - \frac {1}{9525600} \, {\left ({\left (3175200 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 17965080 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right ) - 17965080 \, d^{8} x^{\frac {1}{3}} e + 5807340 \, d^{7} x^{\frac {2}{3}} e^{2} - 2813160 \, d^{6} x e^{3} + 1580670 \, d^{5} x^{\frac {4}{3}} e^{4} - 947016 \, d^{4} x^{\frac {5}{3}} e^{5} + 577500 \, d^{3} x^{2} e^{6} - 343800 \, d^{2} x^{\frac {7}{3}} e^{7} + 187425 \, d x^{\frac {8}{3}} e^{8} - 78400 \, x^{3} e^{9}\right )} n^{2} e^{\left (-9\right )} - 2520 \, {\left (2520 \, d^{9} e^{\left (-10\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (1260 \, d^{7} x^{\frac {2}{3}} e - 2520 \, d^{8} x^{\frac {1}{3}} - 840 \, d^{6} x e^{2} + 630 \, d^{5} x^{\frac {4}{3}} e^{3} - 504 \, d^{4} x^{\frac {5}{3}} e^{4} + 420 \, d^{3} x^{2} e^{5} - 360 \, d^{2} x^{\frac {7}{3}} e^{6} + 315 \, d x^{\frac {8}{3}} e^{7} - 280 \, x^{3} e^{8}\right )} e^{\left (-9\right )}\right )} n e \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )\right )} b^{2} \end {gather*}

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((x^(1/3)*e + d)^n*c)^2 + 2/3*a*b*x^3*log((x^(1/3)*e + d)^n*c) + 1/3*a^2*x^3 + 1/3780*(2520*d^9
*e^(-10)*log(x^(1/3)*e + d) + (1260*d^7*x^(2/3)*e - 2520*d^8*x^(1/3) - 840*d^6*x*e^2 + 630*d^5*x^(4/3)*e^3 - 5
04*d^4*x^(5/3)*e^4 + 420*d^3*x^2*e^5 - 360*d^2*x^(7/3)*e^6 + 315*d*x^(8/3)*e^7 - 280*x^3*e^8)*e^(-9))*a*b*n*e
- 1/9525600*((3175200*d^9*log(x^(1/3)*e + d)^2 + 17965080*d^9*log(x^(1/3)*e + d) - 17965080*d^8*x^(1/3)*e + 58
07340*d^7*x^(2/3)*e^2 - 2813160*d^6*x*e^3 + 1580670*d^5*x^(4/3)*e^4 - 947016*d^4*x^(5/3)*e^5 + 577500*d^3*x^2*
e^6 - 343800*d^2*x^(7/3)*e^7 + 187425*d*x^(8/3)*e^8 - 78400*x^3*e^9)*n^2*e^(-9) - 2520*(2520*d^9*e^(-10)*log(x
^(1/3)*e + d) + (1260*d^7*x^(2/3)*e - 2520*d^8*x^(1/3) - 840*d^6*x*e^2 + 630*d^5*x^(4/3)*e^3 - 504*d^4*x^(5/3)
*e^4 + 420*d^3*x^2*e^5 - 360*d^2*x^(7/3)*e^6 + 315*d*x^(8/3)*e^7 - 280*x^3*e^8)*e^(-9))*n*e*log((x^(1/3)*e + d
)^n*c))*b^2

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Fricas [A]
time = 0.43, size = 623, normalized size = 0.92 \begin {gather*} \frac {1}{9525600} \, {\left (3175200 \, b^{2} x^{3} e^{9} \log \left (c\right )^{2} + 39200 \, {\left (2 \, b^{2} n^{2} - 18 \, a b n + 81 \, a^{2}\right )} x^{3} e^{9} - 2100 \, {\left (275 \, b^{2} d^{3} n^{2} - 504 \, a b d^{3} n\right )} x^{2} e^{6} + 840 \, {\left (3349 \, b^{2} d^{6} n^{2} - 2520 \, a b d^{6} n\right )} x e^{3} + 3175200 \, {\left (b^{2} d^{9} n^{2} + b^{2} n^{2} x^{3} e^{9}\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 2520 \, {\left (7129 \, b^{2} d^{9} n^{2} - 2520 \, a b d^{9} n + 840 \, b^{2} d^{6} n^{2} x e^{3} - 420 \, b^{2} d^{3} n^{2} x^{2} e^{6} + 280 \, {\left (b^{2} n^{2} - 9 \, a b n\right )} x^{3} e^{9} - 2520 \, {\left (b^{2} d^{9} n + b^{2} n x^{3} e^{9}\right )} \log \left (c\right ) - 63 \, {\left (20 \, b^{2} d^{7} n^{2} e^{2} - 8 \, b^{2} d^{4} n^{2} x e^{5} + 5 \, b^{2} d n^{2} x^{2} e^{8}\right )} x^{\frac {2}{3}} + 90 \, {\left (28 \, b^{2} d^{8} n^{2} e - 7 \, b^{2} d^{5} n^{2} x e^{4} + 4 \, b^{2} d^{2} n^{2} x^{2} e^{7}\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 352800 \, {\left (6 \, b^{2} d^{6} n x e^{3} - 3 \, b^{2} d^{3} n x^{2} e^{6} + 2 \, {\left (b^{2} n - 9 \, a b\right )} x^{3} e^{9}\right )} \log \left (c\right ) - 63 \, {\left (175 \, {\left (17 \, b^{2} d n^{2} - 72 \, a b d n\right )} x^{2} e^{8} - 8 \, {\left (1879 \, b^{2} d^{4} n^{2} - 2520 \, a b d^{4} n\right )} x e^{5} + 20 \, {\left (4609 \, b^{2} d^{7} n^{2} - 2520 \, a b d^{7} n\right )} e^{2} - 2520 \, {\left (20 \, b^{2} d^{7} n e^{2} - 8 \, b^{2} d^{4} n x e^{5} + 5 \, b^{2} d n x^{2} e^{8}\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} + 90 \, {\left (20 \, {\left (191 \, b^{2} d^{2} n^{2} - 504 \, a b d^{2} n\right )} x^{2} e^{7} - 7 \, {\left (2509 \, b^{2} d^{5} n^{2} - 2520 \, a b d^{5} n\right )} x e^{4} + 28 \, {\left (7129 \, b^{2} d^{8} n^{2} - 2520 \, a b d^{8} n\right )} e - 2520 \, {\left (28 \, b^{2} d^{8} n e - 7 \, b^{2} d^{5} n x e^{4} + 4 \, b^{2} d^{2} n x^{2} e^{7}\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}\right )} e^{\left (-9\right )} \end {gather*}

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*x^3*e^9*log(c)^2 + 39200*(2*b^2*n^2 - 18*a*b*n + 81*a^2)*x^3*e^9 - 2100*(275*b^2*d^3*n^
2 - 504*a*b*d^3*n)*x^2*e^6 + 840*(3349*b^2*d^6*n^2 - 2520*a*b*d^6*n)*x*e^3 + 3175200*(b^2*d^9*n^2 + b^2*n^2*x^
3*e^9)*log(x^(1/3)*e + d)^2 - 2520*(7129*b^2*d^9*n^2 - 2520*a*b*d^9*n + 840*b^2*d^6*n^2*x*e^3 - 420*b^2*d^3*n^
2*x^2*e^6 + 280*(b^2*n^2 - 9*a*b*n)*x^3*e^9 - 2520*(b^2*d^9*n + b^2*n*x^3*e^9)*log(c) - 63*(20*b^2*d^7*n^2*e^2
- 8*b^2*d^4*n^2*x*e^5 + 5*b^2*d*n^2*x^2*e^8)*x^(2/3) + 90*(28*b^2*d^8*n^2*e - 7*b^2*d^5*n^2*x*e^4 + 4*b^2*d^2
*n^2*x^2*e^7)*x^(1/3))*log(x^(1/3)*e + d) - 352800*(6*b^2*d^6*n*x*e^3 - 3*b^2*d^3*n*x^2*e^6 + 2*(b^2*n - 9*a*b
)*x^3*e^9)*log(c) - 63*(175*(17*b^2*d*n^2 - 72*a*b*d*n)*x^2*e^8 - 8*(1879*b^2*d^4*n^2 - 2520*a*b*d^4*n)*x*e^5
+ 20*(4609*b^2*d^7*n^2 - 2520*a*b*d^7*n)*e^2 - 2520*(20*b^2*d^7*n*e^2 - 8*b^2*d^4*n*x*e^5 + 5*b^2*d*n*x^2*e^8)
*log(c))*x^(2/3) + 90*(20*(191*b^2*d^2*n^2 - 504*a*b*d^2*n)*x^2*e^7 - 7*(2509*b^2*d^5*n^2 - 2520*a*b*d^5*n)*x*
e^4 + 28*(7129*b^2*d^8*n^2 - 2520*a*b*d^8*n)*e - 2520*(28*b^2*d^8*n*e - 7*b^2*d^5*n*x*e^4 + 4*b^2*d^2*n*x^2*e^
7)*log(c))*x^(1/3))*e^(-9)

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

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]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (596) = 1192\).
time = 3.77, size = 1427, normalized size = 2.10 \begin {gather*} \text {Too large to display} \end {gather*}

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)

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Mupad [B]
time = 4.70, size = 608, normalized size = 0.89 \begin {gather*} \frac {a^2\,x^3}{3}+\frac {b^2\,x^3\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{3}+\frac {2\,b^2\,n^2\,x^3}{243}+\frac {2\,a\,b\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3}+\frac {b^2\,d^9\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{3\,e^9}-\frac {2\,a\,b\,n\,x^3}{27}-\frac {2\,b^2\,n\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{27}-\frac {7129\,b^2\,d^9\,n^2\,\ln \left (d+e\,x^{1/3}\right )}{3780\,e^9}-\frac {275\,b^2\,d^3\,n^2\,x^2}{4536\,e^3}+\frac {191\,b^2\,d^2\,n^2\,x^{7/3}}{5292\,e^2}+\frac {1879\,b^2\,d^4\,n^2\,x^{5/3}}{18900\,e^4}-\frac {2509\,b^2\,d^5\,n^2\,x^{4/3}}{15120\,e^5}-\frac {4609\,b^2\,d^7\,n^2\,x^{2/3}}{7560\,e^7}+\frac {7129\,b^2\,d^8\,n^2\,x^{1/3}}{3780\,e^8}-\frac {17\,b^2\,d\,n^2\,x^{8/3}}{864\,e}+\frac {3349\,b^2\,d^6\,n^2\,x}{11340\,e^6}+\frac {b^2\,d^3\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{9\,e^3}-\frac {2\,b^2\,d^2\,n\,x^{7/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{21\,e^2}-\frac {2\,b^2\,d^4\,n\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{15\,e^4}+\frac {b^2\,d^5\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{6\,e^5}+\frac {b^2\,d^7\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^7}-\frac {2\,b^2\,d^8\,n\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^8}+\frac {a\,b\,d\,n\,x^{8/3}}{12\,e}-\frac {2\,a\,b\,d^6\,n\,x}{9\,e^6}+\frac {2\,a\,b\,d^9\,n\,\ln \left (d+e\,x^{1/3}\right )}{3\,e^9}+\frac {b^2\,d\,n\,x^{8/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{12\,e}-\frac {2\,b^2\,d^6\,n\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{9\,e^6}+\frac {a\,b\,d^3\,n\,x^2}{9\,e^3}-\frac {2\,a\,b\,d^2\,n\,x^{7/3}}{21\,e^2}-\frac {2\,a\,b\,d^4\,n\,x^{5/3}}{15\,e^4}+\frac {a\,b\,d^5\,n\,x^{4/3}}{6\,e^5}+\frac {a\,b\,d^7\,n\,x^{2/3}}{3\,e^7}-\frac {2\,a\,b\,d^8\,n\,x^{1/3}}{3\,e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*x^3)/3 + (b^2*x^3*log(c*(d + e*x^(1/3))^n)^2)/3 + (2*b^2*n^2*x^3)/243 + (2*a*b*x^3*log(c*(d + e*x^(1/3))^
n))/3 + (b^2*d^9*log(c*(d + e*x^(1/3))^n)^2)/(3*e^9) - (2*a*b*n*x^3)/27 - (2*b^2*n*x^3*log(c*(d + e*x^(1/3))^n
))/27 - (7129*b^2*d^9*n^2*log(d + e*x^(1/3)))/(3780*e^9) - (275*b^2*d^3*n^2*x^2)/(4536*e^3) + (191*b^2*d^2*n^2
*x^(7/3))/(5292*e^2) + (1879*b^2*d^4*n^2*x^(5/3))/(18900*e^4) - (2509*b^2*d^5*n^2*x^(4/3))/(15120*e^5) - (4609
*b^2*d^7*n^2*x^(2/3))/(7560*e^7) + (7129*b^2*d^8*n^2*x^(1/3))/(3780*e^8) - (17*b^2*d*n^2*x^(8/3))/(864*e) + (3
349*b^2*d^6*n^2*x)/(11340*e^6) + (b^2*d^3*n*x^2*log(c*(d + e*x^(1/3))^n))/(9*e^3) - (2*b^2*d^2*n*x^(7/3)*log(c
*(d + e*x^(1/3))^n))/(21*e^2) - (2*b^2*d^4*n*x^(5/3)*log(c*(d + e*x^(1/3))^n))/(15*e^4) + (b^2*d^5*n*x^(4/3)*l
og(c*(d + e*x^(1/3))^n))/(6*e^5) + (b^2*d^7*n*x^(2/3)*log(c*(d + e*x^(1/3))^n))/(3*e^7) - (2*b^2*d^8*n*x^(1/3)
*log(c*(d + e*x^(1/3))^n))/(3*e^8) + (a*b*d*n*x^(8/3))/(12*e) - (2*a*b*d^6*n*x)/(9*e^6) + (2*a*b*d^9*n*log(d +
e*x^(1/3)))/(3*e^9) + (b^2*d*n*x^(8/3)*log(c*(d + e*x^(1/3))^n))/(12*e) - (2*b^2*d^6*n*x*log(c*(d + e*x^(1/3)
)^n))/(9*e^6) + (a*b*d^3*n*x^2)/(9*e^3) - (2*a*b*d^2*n*x^(7/3))/(21*e^2) - (2*a*b*d^4*n*x^(5/3))/(15*e^4) + (a
*b*d^5*n*x^(4/3))/(6*e^5) + (a*b*d^7*n*x^(2/3))/(3*e^7) - (2*a*b*d^8*n*x^(1/3))/(3*e^8)

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