3.471 \(\int x^3 (a+b \log (c (d+e x^{2/3})^n))^2 \, dx\)

Optimal. Leaf size=482 \[ -\frac{b d^6 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^6}+\frac{3 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^6}-\frac{15 b d^4 n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^6}+\frac{10 b d^3 n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^6}-\frac{15 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 e^6}+\frac{3 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 e^6}-\frac{b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{12 e^6}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{3 b^2 d^5 n^2 x^{2/3}}{e^5}+\frac{15 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2}{8 e^6}-\frac{10 b^2 d^3 n^2 \left (d+e x^{2/3}\right )^3}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4}{32 e^6}+\frac{b^2 d^6 n^2 \log ^2\left (d+e x^{2/3}\right )}{4 e^6}-\frac{3 b^2 d n^2 \left (d+e x^{2/3}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e x^{2/3}\right )^6}{72 e^6} \]

[Out]

(15*b^2*d^4*n^2*(d + e*x^(2/3))^2)/(8*e^6) - (10*b^2*d^3*n^2*(d + e*x^(2/3))^3)/(9*e^6) + (15*b^2*d^2*n^2*(d +
e*x^(2/3))^4)/(32*e^6) - (3*b^2*d*n^2*(d + e*x^(2/3))^5)/(25*e^6) + (b^2*n^2*(d + e*x^(2/3))^6)/(72*e^6) - (3
*b^2*d^5*n^2*x^(2/3))/e^5 + (b^2*d^6*n^2*Log[d + e*x^(2/3)]^2)/(4*e^6) + (3*b*d^5*n*(d + e*x^(2/3))*(a + b*Log
[c*(d + e*x^(2/3))^n]))/e^6 - (15*b*d^4*n*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(4*e^6) + (10*b*
d^3*n*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^6) - (15*b*d^2*n*(d + e*x^(2/3))^4*(a + b*Log[c
*(d + e*x^(2/3))^n]))/(8*e^6) + (3*b*d*n*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*e^6) - (b*n*(d
+ e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n]))/(12*e^6) - (b*d^6*n*Log[d + e*x^(2/3)]*(a + b*Log[c*(d + e*x
^(2/3))^n]))/(2*e^6) + (x^4*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/4

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Rubi [A]  time = 0.488, antiderivative size = 355, normalized size of antiderivative = 0.74, 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} \[ \frac{1}{120} b n \left (\frac{360 d^5 \left (d+e x^{2/3}\right )}{e^6}-\frac{450 d^4 \left (d+e x^{2/3}\right )^2}{e^6}+\frac{400 d^3 \left (d+e x^{2/3}\right )^3}{e^6}-\frac{225 d^2 \left (d+e x^{2/3}\right )^4}{e^6}-\frac{60 d^6 \log \left (d+e x^{2/3}\right )}{e^6}+\frac{72 d \left (d+e x^{2/3}\right )^5}{e^6}-\frac{10 \left (d+e x^{2/3}\right )^6}{e^6}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{3 b^2 d^5 n^2 x^{2/3}}{e^5}+\frac{15 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2}{8 e^6}-\frac{10 b^2 d^3 n^2 \left (d+e x^{2/3}\right )^3}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4}{32 e^6}+\frac{b^2 d^6 n^2 \log ^2\left (d+e x^{2/3}\right )}{4 e^6}-\frac{3 b^2 d n^2 \left (d+e x^{2/3}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e x^{2/3}\right )^6}{72 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*b^2*d^4*n^2*(d + e*x^(2/3))^2)/(8*e^6) - (10*b^2*d^3*n^2*(d + e*x^(2/3))^3)/(9*e^6) + (15*b^2*d^2*n^2*(d +
e*x^(2/3))^4)/(32*e^6) - (3*b^2*d*n^2*(d + e*x^(2/3))^5)/(25*e^6) + (b^2*n^2*(d + e*x^(2/3))^6)/(72*e^6) - (3
*b^2*d^5*n^2*x^(2/3))/e^5 + (b^2*d^6*n^2*Log[d + e*x^(2/3)]^2)/(4*e^6) + (b*n*((360*d^5*(d + e*x^(2/3)))/e^6 -
(450*d^4*(d + e*x^(2/3))^2)/e^6 + (400*d^3*(d + e*x^(2/3))^3)/e^6 - (225*d^2*(d + e*x^(2/3))^4)/e^6 + (72*d*(
d + e*x^(2/3))^5)/e^6 - (10*(d + e*x^(2/3))^6)/e^6 - (60*d^6*Log[d + e*x^(2/3)])/e^6)*(a + b*Log[c*(d + e*x^(2
/3))^n]))/120 + (x^4*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/4

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^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx &=\frac{3}{2} \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,x^{2/3}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{1}{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,x^{2/3}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{1}{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 x^{2/3}\right )\\ &=\frac{1}{120} b n \left (\frac{360 d^5 \left (d+e x^{2/3}\right )}{e^6}-\frac{450 d^4 \left (d+e x^{2/3}\right )^2}{e^6}+\frac{400 d^3 \left (d+e x^{2/3}\right )^3}{e^6}-\frac{225 d^2 \left (d+e x^{2/3}\right )^4}{e^6}+\frac{72 d \left (d+e x^{2/3}\right )^5}{e^6}-\frac{10 \left (d+e x^{2/3}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e x^{2/3}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{1}{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 x^{2/3}\right )\\ &=\frac{1}{120} b n \left (\frac{360 d^5 \left (d+e x^{2/3}\right )}{e^6}-\frac{450 d^4 \left (d+e x^{2/3}\right )^2}{e^6}+\frac{400 d^3 \left (d+e x^{2/3}\right )^3}{e^6}-\frac{225 d^2 \left (d+e x^{2/3}\right )^4}{e^6}+\frac{72 d \left (d+e x^{2/3}\right )^5}{e^6}-\frac{10 \left (d+e x^{2/3}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e x^{2/3}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\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 x^{2/3}\right )}{120 e^6}\\ &=\frac{1}{120} b n \left (\frac{360 d^5 \left (d+e x^{2/3}\right )}{e^6}-\frac{450 d^4 \left (d+e x^{2/3}\right )^2}{e^6}+\frac{400 d^3 \left (d+e x^{2/3}\right )^3}{e^6}-\frac{225 d^2 \left (d+e x^{2/3}\right )^4}{e^6}+\frac{72 d \left (d+e x^{2/3}\right )^5}{e^6}-\frac{10 \left (d+e x^{2/3}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e x^{2/3}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\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 x^{2/3}\right )}{120 e^6}\\ &=\frac{15 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2}{8 e^6}-\frac{10 b^2 d^3 n^2 \left (d+e x^{2/3}\right )^3}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4}{32 e^6}-\frac{3 b^2 d n^2 \left (d+e x^{2/3}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e x^{2/3}\right )^6}{72 e^6}-\frac{3 b^2 d^5 n^2 x^{2/3}}{e^5}+\frac{1}{120} b n \left (\frac{360 d^5 \left (d+e x^{2/3}\right )}{e^6}-\frac{450 d^4 \left (d+e x^{2/3}\right )^2}{e^6}+\frac{400 d^3 \left (d+e x^{2/3}\right )^3}{e^6}-\frac{225 d^2 \left (d+e x^{2/3}\right )^4}{e^6}+\frac{72 d \left (d+e x^{2/3}\right )^5}{e^6}-\frac{10 \left (d+e x^{2/3}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e x^{2/3}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\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 x^{2/3}\right )}{2 e^6}\\ &=\frac{15 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2}{8 e^6}-\frac{10 b^2 d^3 n^2 \left (d+e x^{2/3}\right )^3}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4}{32 e^6}-\frac{3 b^2 d n^2 \left (d+e x^{2/3}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e x^{2/3}\right )^6}{72 e^6}-\frac{3 b^2 d^5 n^2 x^{2/3}}{e^5}+\frac{b^2 d^6 n^2 \log ^2\left (d+e x^{2/3}\right )}{4 e^6}+\frac{1}{120} b n \left (\frac{360 d^5 \left (d+e x^{2/3}\right )}{e^6}-\frac{450 d^4 \left (d+e x^{2/3}\right )^2}{e^6}+\frac{400 d^3 \left (d+e x^{2/3}\right )^3}{e^6}-\frac{225 d^2 \left (d+e x^{2/3}\right )^4}{e^6}+\frac{72 d \left (d+e x^{2/3}\right )^5}{e^6}-\frac{10 \left (d+e x^{2/3}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+e x^{2/3}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2\\ \end{align*}

Mathematica [A]  time = 0.368755, size = 328, normalized size = 0.68 \[ \frac{e x^{2/3} \left (1800 a^2 e^5 x^{10/3}+60 a b n \left (20 d^3 e^2 x^{4/3}-15 d^2 e^3 x^2-30 d^4 e x^{2/3}+60 d^5+12 d e^4 x^{8/3}-10 e^5 x^{10/3}\right )+b^2 n^2 \left (-1140 d^3 e^2 x^{4/3}+555 d^2 e^3 x^2+2610 d^4 e x^{2/3}-8820 d^5-264 d e^4 x^{8/3}+100 e^5 x^{10/3}\right )\right )+60 b \left (b n \left (-30 d^4 e^2 x^{4/3}+20 d^3 e^3 x^2-15 d^2 e^4 x^{8/3}+60 d^5 e x^{2/3}+60 d^6+12 d e^5 x^{10/3}-10 e^6 x^4\right )-60 a \left (d^6-e^6 x^4\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )-1800 b^2 \left (d^6-e^6 x^4\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+5220 b^2 d^6 n^2 \log \left (d+e x^{2/3}\right )}{7200 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x^(2/3)*(1800*a^2*e^5*x^(10/3) + 60*a*b*n*(60*d^5 - 30*d^4*e*x^(2/3) + 20*d^3*e^2*x^(4/3) - 15*d^2*e^3*x^2
+ 12*d*e^4*x^(8/3) - 10*e^5*x^(10/3)) + b^2*n^2*(-8820*d^5 + 2610*d^4*e*x^(2/3) - 1140*d^3*e^2*x^(4/3) + 555*d
^2*e^3*x^2 - 264*d*e^4*x^(8/3) + 100*e^5*x^(10/3))) + 5220*b^2*d^6*n^2*Log[d + e*x^(2/3)] + 60*b*(b*n*(60*d^6
+ 60*d^5*e*x^(2/3) - 30*d^4*e^2*x^(4/3) + 20*d^3*e^3*x^2 - 15*d^2*e^4*x^(8/3) + 12*d*e^5*x^(10/3) - 10*e^6*x^4
) - 60*a*(d^6 - e^6*x^4))*Log[c*(d + e*x^(2/3))^n] - 1800*b^2*(d^6 - e^6*x^4)*Log[c*(d + e*x^(2/3))^n]^2)/(720
0*e^6)

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Maple [F]  time = 0.336, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05741, size = 446, normalized size = 0.93 \begin{align*} \frac{1}{4} \, b^{2} x^{4} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + \frac{1}{2} \, a b x^{4} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + \frac{1}{4} \, a^{2} x^{4} - \frac{1}{120} \, a b e n{\left (\frac{60 \, d^{6} \log \left (e x^{\frac{2}{3}} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac{10}{3}} + 15 \, d^{2} e^{3} x^{\frac{8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac{4}{3}} - 60 \, d^{5} x^{\frac{2}{3}}}{e^{6}}\right )} - \frac{1}{7200} \,{\left (60 \, e n{\left (\frac{60 \, d^{6} \log \left (e x^{\frac{2}{3}} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac{10}{3}} + 15 \, d^{2} e^{3} x^{\frac{8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac{4}{3}} - 60 \, d^{5} x^{\frac{2}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) - \frac{{\left (100 \, e^{6} x^{4} - 264 \, d e^{5} x^{\frac{10}{3}} + 555 \, d^{2} e^{4} x^{\frac{8}{3}} - 1140 \, d^{3} e^{3} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac{2}{3}} + d\right )^{2} + 2610 \, d^{4} e^{2} x^{\frac{4}{3}} + 8820 \, d^{6} \log \left (e x^{\frac{2}{3}} + d\right ) - 8820 \, d^{5} e x^{\frac{2}{3}}\right )} n^{2}}{e^{6}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4*log((e*x^(2/3) + d)^n*c)^2 + 1/2*a*b*x^4*log((e*x^(2/3) + d)^n*c) + 1/4*a^2*x^4 - 1/120*a*b*e*n*(6
0*d^6*log(e*x^(2/3) + d)/e^7 + (10*e^5*x^4 - 12*d*e^4*x^(10/3) + 15*d^2*e^3*x^(8/3) - 20*d^3*e^2*x^2 + 30*d^4*
e*x^(4/3) - 60*d^5*x^(2/3))/e^6) - 1/7200*(60*e*n*(60*d^6*log(e*x^(2/3) + d)/e^7 + (10*e^5*x^4 - 12*d*e^4*x^(1
0/3) + 15*d^2*e^3*x^(8/3) - 20*d^3*e^2*x^2 + 30*d^4*e*x^(4/3) - 60*d^5*x^(2/3))/e^6)*log((e*x^(2/3) + d)^n*c)
- (100*e^6*x^4 - 264*d*e^5*x^(10/3) + 555*d^2*e^4*x^(8/3) - 1140*d^3*e^3*x^2 + 1800*d^6*log(e*x^(2/3) + d)^2 +
2610*d^4*e^2*x^(4/3) + 8820*d^6*log(e*x^(2/3) + d) - 8820*d^5*e*x^(2/3))*n^2/e^6)*b^2

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Fricas [A]  time = 2.41827, size = 1125, normalized size = 2.33 \begin{align*} \frac{1800 \, b^{2} e^{6} x^{4} \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^{4} - 60 \,{\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x^{2} + 1800 \,{\left (b^{2} e^{6} n^{2} x^{4} - b^{2} d^{6} n^{2}\right )} \log \left (e x^{\frac{2}{3}} + d\right )^{2} + 60 \,{\left (20 \, b^{2} d^{3} e^{3} n^{2} x^{2} + 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^{4} + 60 \,{\left (b^{2} e^{6} n x^{4} - b^{2} d^{6} n\right )} \log \left (c\right ) - 15 \,{\left (b^{2} d^{2} e^{4} n^{2} x^{2} - 4 \, b^{2} d^{5} e n^{2}\right )} x^{\frac{2}{3}} + 6 \,{\left (2 \, b^{2} d e^{5} n^{2} x^{3} - 5 \, b^{2} d^{4} e^{2} n^{2} x\right )} x^{\frac{1}{3}}\right )} \log \left (e x^{\frac{2}{3}} + d\right ) + 600 \,{\left (2 \, b^{2} d^{3} e^{3} n x^{2} -{\left (b^{2} e^{6} n - 6 \, a b e^{6}\right )} x^{4}\right )} \log \left (c\right ) - 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^{2} + 60 \,{\left (b^{2} d^{2} e^{4} n x^{2} - 4 \, b^{2} d^{5} e n\right )} \log \left (c\right )\right )} x^{\frac{2}{3}} - 6 \,{\left (4 \,{\left (11 \, b^{2} d e^{5} n^{2} - 30 \, a b d e^{5} n\right )} x^{3} - 15 \,{\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x - 60 \,{\left (2 \, b^{2} d e^{5} n x^{3} - 5 \, b^{2} d^{4} e^{2} n x\right )} \log \left (c\right )\right )} x^{\frac{1}{3}}}{7200 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7200*(1800*b^2*e^6*x^4*log(c)^2 + 100*(b^2*e^6*n^2 - 6*a*b*e^6*n + 18*a^2*e^6)*x^4 - 60*(19*b^2*d^3*e^3*n^2
- 20*a*b*d^3*e^3*n)*x^2 + 1800*(b^2*e^6*n^2*x^4 - b^2*d^6*n^2)*log(e*x^(2/3) + d)^2 + 60*(20*b^2*d^3*e^3*n^2*x
^2 + 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^4 + 60*(b^2*e^6*n*x^4 - b^2*d^6*n)*log(
c) - 15*(b^2*d^2*e^4*n^2*x^2 - 4*b^2*d^5*e*n^2)*x^(2/3) + 6*(2*b^2*d*e^5*n^2*x^3 - 5*b^2*d^4*e^2*n^2*x)*x^(1/3
))*log(e*x^(2/3) + d) + 600*(2*b^2*d^3*e^3*n*x^2 - (b^2*e^6*n - 6*a*b*e^6)*x^4)*log(c) - 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^2 + 60*(b^2*d^2*e^4*n*x^2 - 4*b^2*d^5*e*n)*log(
c))*x^(2/3) - 6*(4*(11*b^2*d*e^5*n^2 - 30*a*b*d*e^5*n)*x^3 - 15*(29*b^2*d^4*e^2*n^2 - 20*a*b*d^4*e^2*n)*x - 60
*(2*b^2*d*e^5*n*x^3 - 5*b^2*d^4*e^2*n*x)*log(c))*x^(1/3))/e^6

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

[Out]

Timed out

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Giac [B]  time = 1.51409, size = 1287, normalized size = 2.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4*log(c)^2 + 1/2*a*b*x^4*log(c) + 1/4*a^2*x^4 + 1/7200*(1800*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e
+ d)^2 - 10800*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3
)*e + d)^2 - 36000*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d)^2 + 27000*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(
x^(2/3)*e + d)^2 - 10800*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d)^2 - 600*(x^(2/3)*e + d)^6*e^(-5)*log(x^
(2/3)*e + d) + 4320*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d) - 13500*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(
2/3)*e + d) + 24000*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d) - 27000*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x
^(2/3)*e + d) + 21600*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d) + 100*(x^(2/3)*e + d)^6*e^(-5) - 864*(x^(2
/3)*e + d)^5*d*e^(-5) + 3375*(x^(2/3)*e + d)^4*d^2*e^(-5) - 8000*(x^(2/3)*e + d)^3*d^3*e^(-5) + 13500*(x^(2/3)
*e + d)^2*d^4*e^(-5) - 21600*(x^(2/3)*e + d)*d^5*e^(-5))*b^2*n^2*e^(-1) + 1/120*(60*(x^(2/3)*e + d)^6*e^(-5)*l
og(x^(2/3)*e + d) - 360*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x
^(2/3)*e + d) - 1200*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^
(2/3)*e + d) - 360*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-5) + 72*(x^(2/3)*e
+ d)^5*d*e^(-5) - 225*(x^(2/3)*e + d)^4*d^2*e^(-5) + 400*(x^(2/3)*e + d)^3*d^3*e^(-5) - 450*(x^(2/3)*e + d)^2
*d^4*e^(-5) + 360*(x^(2/3)*e + d)*d^5*e^(-5))*b^2*n*e^(-1)*log(c) + 1/120*(60*(x^(2/3)*e + d)^6*e^(-5)*log(x^(
2/3)*e + d) - 360*(x^(2/3)*e + d)^5*d*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)
*e + d) - 1200*(x^(2/3)*e + d)^3*d^3*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*
e + d) - 360*(x^(2/3)*e + d)*d^5*e^(-5)*log(x^(2/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-5) + 72*(x^(2/3)*e + d)^
5*d*e^(-5) - 225*(x^(2/3)*e + d)^4*d^2*e^(-5) + 400*(x^(2/3)*e + d)^3*d^3*e^(-5) - 450*(x^(2/3)*e + d)^2*d^4*e
^(-5) + 360*(x^(2/3)*e + d)*d^5*e^(-5))*a*b*n*e^(-1)