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

Optimal. Leaf size=480 \[ -\frac{b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac{6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac{15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac{20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac{15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac{6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac{b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac{6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac{15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac{20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}+\frac{b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}-\frac{6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6} \]

[Out]

(15*b^2*d^4*n^2*(d + e*x^(1/3))^2)/(4*e^6) - (20*b^2*d^3*n^2*(d + e*x^(1/3))^3)/(9*e^6) + (15*b^2*d^2*n^2*(d +
e*x^(1/3))^4)/(16*e^6) - (6*b^2*d*n^2*(d + e*x^(1/3))^5)/(25*e^6) + (b^2*n^2*(d + e*x^(1/3))^6)/(36*e^6) - (6
*b^2*d^5*n^2*x^(1/3))/e^5 + (b^2*d^6*n^2*Log[d + e*x^(1/3)]^2)/(2*e^6) + (6*b*d^5*n*(d + e*x^(1/3))*(a + b*Log
[c*(d + e*x^(1/3))^n]))/e^6 - (15*b*d^4*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*e^6) + (20*b*
d^3*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^6) - (15*b*d^2*n*(d + e*x^(1/3))^4*(a + b*Log[c
*(d + e*x^(1/3))^n]))/(4*e^6) + (6*b*d*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(5*e^6) - (b*n*(d
+ e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(6*e^6) - (b*d^6*n*Log[d + e*x^(1/3)]*(a + b*Log[c*(d + e*x^
(1/3))^n]))/e^6 + (x^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/2

________________________________________________________________________________________

Rubi [A]  time = 0.461951, antiderivative size = 355, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ \frac{1}{60} b n \left (\frac{360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac{450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac{400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac{225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}-\frac{60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}+\frac{72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac{10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac{6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac{15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac{20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}+\frac{b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}-\frac{6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*b^2*d^4*n^2*(d + e*x^(1/3))^2)/(4*e^6) - (20*b^2*d^3*n^2*(d + e*x^(1/3))^3)/(9*e^6) + (15*b^2*d^2*n^2*(d +
e*x^(1/3))^4)/(16*e^6) - (6*b^2*d*n^2*(d + e*x^(1/3))^5)/(25*e^6) + (b^2*n^2*(d + e*x^(1/3))^6)/(36*e^6) - (6
*b^2*d^5*n^2*x^(1/3))/e^5 + (b^2*d^6*n^2*Log[d + e*x^(1/3)]^2)/(2*e^6) + (b*n*((360*d^5*(d + e*x^(1/3)))/e^6 -
(450*d^4*(d + e*x^(1/3))^2)/e^6 + (400*d^3*(d + e*x^(1/3))^3)/e^6 - (225*d^2*(d + e*x^(1/3))^4)/e^6 + (72*d*(
d + e*x^(1/3))^5)/e^6 - (10*(d + e*x^(1/3))^6)/e^6 - (60*d^6*Log[d + e*x^(1/3)])/e^6)*(a + b*Log[c*(d + e*x^(1
/3))^n]))/60 + (x^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/2

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 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx &=3 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(b e n) \operatorname{Subst}\left (\int \frac{x^6 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(b n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^6 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=\frac{1}{60} b n \left (\frac{360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac{450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac{400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac{225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac{72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac{10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{60 e^6 x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=\frac{1}{60} b n \left (\frac{360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac{450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac{400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac{225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac{72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac{10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac{1}{2} x^2 \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{x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{60 e^6}\\ &=\frac{1}{60} b n \left (\frac{360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac{450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac{400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac{225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac{72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac{10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac{1}{2} x^2 \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 (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5+\frac{60 d^6 \log (x)}{x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{60 e^6}\\ &=\frac{15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac{20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac{6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6}-\frac{6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac{1}{60} b n \left (\frac{360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac{450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac{400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac{225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac{72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac{10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{\left (b^2 d^6 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}\\ &=\frac{15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac{20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac{6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6}-\frac{6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac{b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac{1}{60} b n \left (\frac{360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac{450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac{400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac{225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac{72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac{10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\\ \end{align*}

Mathematica [A]  time = 0.308326, size = 301, normalized size = 0.63 \[ \frac{e \sqrt [3]{x} \left (1800 a^2 e^5 x^{5/3}+60 a b n \left (20 d^3 e^2 x^{2/3}-15 d^2 e^3 x-30 d^4 e \sqrt [3]{x}+60 d^5+12 d e^4 x^{4/3}-10 e^5 x^{5/3}\right )+b^2 n^2 \left (-1140 d^3 e^2 x^{2/3}+555 d^2 e^3 x+2610 d^4 e \sqrt [3]{x}-8820 d^5-264 d e^4 x^{4/3}+100 e^5 x^{5/3}\right )\right )-60 b \left (60 a \left (d^6-e^6 x^2\right )+b n \left (30 d^4 e^2 x^{2/3}+15 d^2 e^4 x^{4/3}-20 d^3 e^3 x-60 d^5 e \sqrt [3]{x}-147 d^6-12 d e^5 x^{5/3}+10 e^6 x^2\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-1800 b^2 \left (d^6-e^6 x^2\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3600 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x^(1/3)*(1800*a^2*e^5*x^(5/3) + 60*a*b*n*(60*d^5 - 30*d^4*e*x^(1/3) + 20*d^3*e^2*x^(2/3) - 15*d^2*e^3*x + 1
2*d*e^4*x^(4/3) - 10*e^5*x^(5/3)) + b^2*n^2*(-8820*d^5 + 2610*d^4*e*x^(1/3) - 1140*d^3*e^2*x^(2/3) + 555*d^2*e
^3*x - 264*d*e^4*x^(4/3) + 100*e^5*x^(5/3))) - 60*b*(60*a*(d^6 - e^6*x^2) + b*n*(-147*d^6 - 60*d^5*e*x^(1/3) +
30*d^4*e^2*x^(2/3) - 20*d^3*e^3*x + 15*d^2*e^4*x^(4/3) - 12*d*e^5*x^(5/3) + 10*e^6*x^2))*Log[c*(d + e*x^(1/3)
)^n] - 1800*b^2*(d^6 - e^6*x^2)*Log[c*(d + e*x^(1/3))^n]^2)/(3600*e^6)

________________________________________________________________________________________

Maple [F]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*ln(c*(d+e*x^(1/3))^n))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05463, size = 436, normalized size = 0.91 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{2} - \frac{1}{60} \, a b e n{\left (\frac{60 \, d^{6} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac{5}{3}} + 15 \, d^{2} e^{3} x^{\frac{4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac{2}{3}} - 60 \, d^{5} x^{\frac{1}{3}}}{e^{6}}\right )} + a b x^{2} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + \frac{1}{2} \, a^{2} x^{2} - \frac{1}{3600} \,{\left (60 \, e n{\left (\frac{60 \, d^{6} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac{5}{3}} + 15 \, d^{2} e^{3} x^{\frac{4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac{2}{3}} - 60 \, d^{5} x^{\frac{1}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) - \frac{{\left (100 \, e^{6} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 264 \, d e^{5} x^{\frac{5}{3}} + 555 \, d^{2} e^{4} x^{\frac{4}{3}} - 1140 \, d^{3} e^{3} x + 8820 \, d^{6} \log \left (e x^{\frac{1}{3}} + d\right ) + 2610 \, d^{4} e^{2} x^{\frac{2}{3}} - 8820 \, d^{5} e x^{\frac{1}{3}}\right )} n^{2}}{e^{6}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2*log((e*x^(1/3) + d)^n*c)^2 - 1/60*a*b*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*
x^(5/3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60*d^5*x^(1/3))/e^6) + a*b*x^2*log((e*x^(1/3)
+ d)^n*c) + 1/2*a^2*x^2 - 1/3600*(60*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*x^(5/3) + 15
*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60*d^5*x^(1/3))/e^6)*log((e*x^(1/3) + d)^n*c) - (100*e^6*
x^2 + 1800*d^6*log(e*x^(1/3) + d)^2 - 264*d*e^5*x^(5/3) + 555*d^2*e^4*x^(4/3) - 1140*d^3*e^3*x + 8820*d^6*log(
e*x^(1/3) + d) + 2610*d^4*e^2*x^(2/3) - 8820*d^5*e*x^(1/3))*n^2/e^6)*b^2

________________________________________________________________________________________

Fricas [A]  time = 2.18252, size = 1088, normalized size = 2.27 \begin{align*} \frac{1800 \, b^{2} e^{6} x^{2} \log \left (c\right )^{2} + 100 \,{\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n + 18 \, a^{2} e^{6}\right )} x^{2} + 1800 \,{\left (b^{2} e^{6} n^{2} x^{2} - b^{2} d^{6} n^{2}\right )} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 60 \,{\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x + 60 \,{\left (20 \, b^{2} d^{3} e^{3} n^{2} x + 147 \, b^{2} d^{6} n^{2} - 60 \, a b d^{6} n - 10 \,{\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n\right )} x^{2} + 60 \,{\left (b^{2} e^{6} n x^{2} - b^{2} d^{6} n\right )} \log \left (c\right ) + 6 \,{\left (2 \, b^{2} d e^{5} n^{2} x - 5 \, b^{2} d^{4} e^{2} n^{2}\right )} x^{\frac{2}{3}} - 15 \,{\left (b^{2} d^{2} e^{4} n^{2} x - 4 \, b^{2} d^{5} e n^{2}\right )} x^{\frac{1}{3}}\right )} \log \left (e x^{\frac{1}{3}} + d\right ) + 600 \,{\left (2 \, b^{2} d^{3} e^{3} n x -{\left (b^{2} e^{6} n - 6 \, a b e^{6}\right )} x^{2}\right )} \log \left (c\right ) + 6 \,{\left (435 \, b^{2} d^{4} e^{2} n^{2} - 300 \, a b d^{4} e^{2} n - 4 \,{\left (11 \, b^{2} d e^{5} n^{2} - 30 \, a b d e^{5} n\right )} x + 60 \,{\left (2 \, b^{2} d e^{5} n x - 5 \, b^{2} d^{4} e^{2} n\right )} \log \left (c\right )\right )} x^{\frac{2}{3}} - 15 \,{\left (588 \, b^{2} d^{5} e n^{2} - 240 \, a b d^{5} e n -{\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x + 60 \,{\left (b^{2} d^{2} e^{4} n x - 4 \, b^{2} d^{5} e n\right )} \log \left (c\right )\right )} x^{\frac{1}{3}}}{3600 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3600*(1800*b^2*e^6*x^2*log(c)^2 + 100*(b^2*e^6*n^2 - 6*a*b*e^6*n + 18*a^2*e^6)*x^2 + 1800*(b^2*e^6*n^2*x^2 -
b^2*d^6*n^2)*log(e*x^(1/3) + d)^2 - 60*(19*b^2*d^3*e^3*n^2 - 20*a*b*d^3*e^3*n)*x + 60*(20*b^2*d^3*e^3*n^2*x +
147*b^2*d^6*n^2 - 60*a*b*d^6*n - 10*(b^2*e^6*n^2 - 6*a*b*e^6*n)*x^2 + 60*(b^2*e^6*n*x^2 - b^2*d^6*n)*log(c) +
6*(2*b^2*d*e^5*n^2*x - 5*b^2*d^4*e^2*n^2)*x^(2/3) - 15*(b^2*d^2*e^4*n^2*x - 4*b^2*d^5*e*n^2)*x^(1/3))*log(e*x
^(1/3) + d) + 600*(2*b^2*d^3*e^3*n*x - (b^2*e^6*n - 6*a*b*e^6)*x^2)*log(c) + 6*(435*b^2*d^4*e^2*n^2 - 300*a*b*
d^4*e^2*n - 4*(11*b^2*d*e^5*n^2 - 30*a*b*d*e^5*n)*x + 60*(2*b^2*d*e^5*n*x - 5*b^2*d^4*e^2*n)*log(c))*x^(2/3) -
15*(588*b^2*d^5*e*n^2 - 240*a*b*d^5*e*n - (37*b^2*d^2*e^4*n^2 - 60*a*b*d^2*e^4*n)*x + 60*(b^2*d^2*e^4*n*x - 4
*b^2*d^5*e*n)*log(c))*x^(1/3))/e^6

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.33662, size = 1291, normalized size = 2.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3600*(1800*b^2*x^2*e*log(c)^2 + 3600*a*b*x^2*e*log(c) + (1800*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d)^2
- 10800*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d)^2 + 27000*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d
)^2 - 36000*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d)^2 + 27000*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)
*e + d)^2 - 10800*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d)^2 - 600*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e
+ d) + 4320*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d) - 13500*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e
+ d) + 24000*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d) - 27000*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*
e + d) + 21600*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d) + 100*(x^(1/3)*e + d)^6*e^(-5) - 864*(x^(1/3)*e +
d)^5*d*e^(-5) + 3375*(x^(1/3)*e + d)^4*d^2*e^(-5) - 8000*(x^(1/3)*e + d)^3*d^3*e^(-5) + 13500*(x^(1/3)*e + d)
^2*d^4*e^(-5) - 21600*(x^(1/3)*e + d)*d^5*e^(-5))*b^2*n^2 + 1800*a^2*x^2*e + 60*(60*(x^(1/3)*e + d)^6*e^(-5)*l
og(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x
^(1/3)*e + d) - 1200*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^
(1/3)*e + d) - 360*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d) - 10*(x^(1/3)*e + d)^6*e^(-5) + 72*(x^(1/3)*e
+ d)^5*d*e^(-5) - 225*(x^(1/3)*e + d)^4*d^2*e^(-5) + 400*(x^(1/3)*e + d)^3*d^3*e^(-5) - 450*(x^(1/3)*e + d)^2
*d^4*e^(-5) + 360*(x^(1/3)*e + d)*d^5*e^(-5))*b^2*n*log(c) + 60*(60*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d
) - 360*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d) -
1200*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d) - 3
60*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d) - 10*(x^(1/3)*e + d)^6*e^(-5) + 72*(x^(1/3)*e + d)^5*d*e^(-5)
- 225*(x^(1/3)*e + d)^4*d^2*e^(-5) + 400*(x^(1/3)*e + d)^3*d^3*e^(-5) - 450*(x^(1/3)*e + d)^2*d^4*e^(-5) + 36
0*(x^(1/3)*e + d)*d^5*e^(-5))*a*b*n)*e^(-1)