3.5.71 \(\int x^3 (a+b \log (c (d+e x^{2/3})^n))^2 \, dx\) [471]

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

[Out]

15/8*b^2*d^4*n^2*(d+e*x^(2/3))^2/e^6-10/9*b^2*d^3*n^2*(d+e*x^(2/3))^3/e^6+15/32*b^2*d^2*n^2*(d+e*x^(2/3))^4/e^
6-3/25*b^2*d*n^2*(d+e*x^(2/3))^5/e^6+1/72*b^2*n^2*(d+e*x^(2/3))^6/e^6-3*b^2*d^5*n^2*x^(2/3)/e^5+1/4*b^2*d^6*n^
2*ln(d+e*x^(2/3))^2/e^6+3*b*d^5*n*(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6-15/4*b*d^4*n*(d+e*x^(2/3))^2*(
a+b*ln(c*(d+e*x^(2/3))^n))/e^6+10/3*b*d^3*n*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6-15/8*b*d^2*n*(d+e*
x^(2/3))^4*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6+3/5*b*d*n*(d+e*x^(2/3))^5*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6-1/12*b*n*
(d+e*x^(2/3))^6*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6-1/2*b*d^6*n*ln(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6+1/
4*x^4*(a+b*ln(c*(d+e*x^(2/3))^n))^2

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Rubi [A]
time = 0.33, antiderivative size = 482, 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 {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 {b^2 d^6 n^2 \log ^2\left (d+e x^{2/3}\right )}{4 e^6}-\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 {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} \end {gather*}

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) + (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

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

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Mathematica [A]
time = 0.26, size = 328, normalized size = 0.68 \begin {gather*} \frac {e x^{2/3} \left (1800 a^2 e^5 x^{10/3}+60 a b n \left (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}\right )+b^2 n^2 \left (-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}\right )\right )+5220 b^2 d^6 n^2 \log \left (d+e x^{2/3}\right )+60 b \left (b n \left (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\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 )}{7200 e^6} \end {gather*}

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

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 = 0.32, size = 323, normalized size = 0.67 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )^{2} + \frac {1}{2} \, a b x^{4} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right ) + \frac {1}{4} \, a^{2} x^{4} - \frac {1}{120} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (30 \, d^{4} x^{\frac {4}{3}} e - 20 \, d^{3} x^{2} e^{2} - 60 \, d^{5} x^{\frac {2}{3}} + 15 \, d^{2} x^{\frac {8}{3}} e^{3} - 12 \, d x^{\frac {10}{3}} e^{4} + 10 \, x^{4} e^{5}\right )} e^{\left (-6\right )}\right )} a b n e + \frac {1}{7200} \, {\left ({\left (1800 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 8820 \, d^{6} \log \left (x^{\frac {2}{3}} e + d\right ) - 8820 \, d^{5} x^{\frac {2}{3}} e + 2610 \, d^{4} x^{\frac {4}{3}} e^{2} - 1140 \, d^{3} x^{2} e^{3} + 555 \, d^{2} x^{\frac {8}{3}} e^{4} - 264 \, d x^{\frac {10}{3}} e^{5} + 100 \, x^{4} e^{6}\right )} n^{2} e^{\left (-6\right )} - 60 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (30 \, d^{4} x^{\frac {4}{3}} e - 20 \, d^{3} x^{2} e^{2} - 60 \, d^{5} x^{\frac {2}{3}} + 15 \, d^{2} x^{\frac {8}{3}} e^{3} - 12 \, d x^{\frac {10}{3}} e^{4} + 10 \, x^{4} e^{5}\right )} e^{\left (-6\right )}\right )} n e \log \left ({\left (x^{\frac {2}{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^3*(a+b*log(c*(d+e*x^(2/3))^n))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.44, size = 473, normalized size = 0.98 \begin {gather*} \frac {1}{7200} \, {\left (1800 \, b^{2} x^{4} e^{6} \log \left (c\right )^{2} + 100 \, {\left (b^{2} n^{2} - 6 \, a b n + 18 \, a^{2}\right )} x^{4} e^{6} - 60 \, {\left (19 \, b^{2} d^{3} n^{2} - 20 \, a b d^{3} n\right )} x^{2} e^{3} - 1800 \, {\left (b^{2} d^{6} n^{2} - b^{2} n^{2} x^{4} e^{6}\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 60 \, {\left (147 \, b^{2} d^{6} n^{2} + 20 \, b^{2} d^{3} n^{2} x^{2} e^{3} - 60 \, a b d^{6} n - 10 \, {\left (b^{2} n^{2} - 6 \, a b n\right )} x^{4} e^{6} - 60 \, {\left (b^{2} d^{6} n - b^{2} n x^{4} e^{6}\right )} \log \left (c\right ) + 15 \, {\left (4 \, b^{2} d^{5} n^{2} e - b^{2} d^{2} n^{2} x^{2} e^{4}\right )} x^{\frac {2}{3}} - 6 \, {\left (5 \, b^{2} d^{4} n^{2} x e^{2} - 2 \, b^{2} d n^{2} x^{3} e^{5}\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 600 \, {\left (2 \, b^{2} d^{3} n x^{2} e^{3} - {\left (b^{2} n - 6 \, a b\right )} x^{4} e^{6}\right )} \log \left (c\right ) + 15 \, {\left ({\left (37 \, b^{2} d^{2} n^{2} - 60 \, a b d^{2} n\right )} x^{2} e^{4} - 12 \, {\left (49 \, b^{2} d^{5} n^{2} - 20 \, a b d^{5} n\right )} e + 60 \, {\left (4 \, b^{2} d^{5} n e - b^{2} d^{2} n x^{2} e^{4}\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} - 6 \, {\left (4 \, {\left (11 \, b^{2} d n^{2} - 30 \, a b d n\right )} x^{3} e^{5} - 15 \, {\left (29 \, b^{2} d^{4} n^{2} - 20 \, a b d^{4} n\right )} x e^{2} + 60 \, {\left (5 \, b^{2} d^{4} n x e^{2} - 2 \, b^{2} d n x^{3} e^{5}\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}\right )} e^{\left (-6\right )} \end {gather*}

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*x^4*e^6*log(c)^2 + 100*(b^2*n^2 - 6*a*b*n + 18*a^2)*x^4*e^6 - 60*(19*b^2*d^3*n^2 - 20*a*b*d^3
*n)*x^2*e^3 - 1800*(b^2*d^6*n^2 - b^2*n^2*x^4*e^6)*log(x^(2/3)*e + d)^2 + 60*(147*b^2*d^6*n^2 + 20*b^2*d^3*n^2
*x^2*e^3 - 60*a*b*d^6*n - 10*(b^2*n^2 - 6*a*b*n)*x^4*e^6 - 60*(b^2*d^6*n - b^2*n*x^4*e^6)*log(c) + 15*(4*b^2*d
^5*n^2*e - b^2*d^2*n^2*x^2*e^4)*x^(2/3) - 6*(5*b^2*d^4*n^2*x*e^2 - 2*b^2*d*n^2*x^3*e^5)*x^(1/3))*log(x^(2/3)*e
+ d) + 600*(2*b^2*d^3*n*x^2*e^3 - (b^2*n - 6*a*b)*x^4*e^6)*log(c) + 15*((37*b^2*d^2*n^2 - 60*a*b*d^2*n)*x^2*e
^4 - 12*(49*b^2*d^5*n^2 - 20*a*b*d^5*n)*e + 60*(4*b^2*d^5*n*e - b^2*d^2*n*x^2*e^4)*log(c))*x^(2/3) - 6*(4*(11*
b^2*d*n^2 - 30*a*b*d*n)*x^3*e^5 - 15*(29*b^2*d^4*n^2 - 20*a*b*d^4*n)*x*e^2 + 60*(5*b^2*d^4*n*x*e^2 - 2*b^2*d*n
*x^3*e^5)*log(c))*x^(1/3))*e^(-6)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (419) = 838\).
time = 4.44, size = 933, normalized size = 1.94 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} \log \left (c\right )^{2} + \frac {1}{2} \, a b x^{4} \log \left (c\right ) + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{7200} \, {\left (1800 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 10800 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 27000 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 36000 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 27000 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 600 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 4320 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 13500 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 24000 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 27000 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 100 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-6\right )} - 864 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-6\right )} + 3375 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-6\right )} - 8000 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-6\right )} + 13500 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-6\right )} - 10800 \, {\left ({\left (x^{\frac {2}{3}} e + d\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 2 \, {\left (x^{\frac {2}{3}} e + d\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 2 \, x^{\frac {2}{3}} e + 2 \, d\right )} d^{5} e^{\left (-6\right )}\right )} b^{2} n^{2} + \frac {1}{120} \, {\left (60 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 360 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 900 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 1200 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 900 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 10 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-6\right )} + 72 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-6\right )} - 225 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-6\right )} + 400 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-6\right )} - 450 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-6\right )} - 360 \, {\left ({\left (x^{\frac {2}{3}} e + d\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - x^{\frac {2}{3}} e - d\right )} d^{5} e^{\left (-6\right )}\right )} b^{2} n \log \left (c\right ) + \frac {1}{120} \, {\left (60 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 360 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 900 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 1200 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 900 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-6\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 10 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-6\right )} + 72 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-6\right )} - 225 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-6\right )} + 400 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-6\right )} - 450 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-6\right )} - 360 \, {\left ({\left (x^{\frac {2}{3}} e + d\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - x^{\frac {2}{3}} e - d\right )} d^{5} e^{\left (-6\right )}\right )} a b n \end {gather*}

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

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Mupad [B]
time = 1.75, size = 440, normalized size = 0.91 \begin {gather*} \frac {a^2\,x^4}{4}+\frac {b^2\,x^4\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{4}+\frac {b^2\,n^2\,x^4}{72}+\frac {a\,b\,x^4\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{2}-\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2}{4\,e^6}-\frac {a\,b\,n\,x^4}{12}-\frac {b^2\,n\,x^4\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{12}+\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+e\,x^{2/3}\right )}{40\,e^6}-\frac {19\,b^2\,d^3\,n^2\,x^2}{120\,e^3}+\frac {37\,b^2\,d^2\,n^2\,x^{8/3}}{480\,e^2}+\frac {29\,b^2\,d^4\,n^2\,x^{4/3}}{80\,e^4}-\frac {49\,b^2\,d^5\,n^2\,x^{2/3}}{40\,e^5}-\frac {11\,b^2\,d\,n^2\,x^{10/3}}{300\,e}+\frac {b^2\,d^3\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{6\,e^3}-\frac {b^2\,d^2\,n\,x^{8/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{8\,e^2}-\frac {b^2\,d^4\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{4\,e^4}+\frac {b^2\,d^5\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{2\,e^5}+\frac {a\,b\,d\,n\,x^{10/3}}{10\,e}-\frac {a\,b\,d^6\,n\,\ln \left (d+e\,x^{2/3}\right )}{2\,e^6}+\frac {b^2\,d\,n\,x^{10/3}\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{10\,e}+\frac {a\,b\,d^3\,n\,x^2}{6\,e^3}-\frac {a\,b\,d^2\,n\,x^{8/3}}{8\,e^2}-\frac {a\,b\,d^4\,n\,x^{4/3}}{4\,e^4}+\frac {a\,b\,d^5\,n\,x^{2/3}}{2\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*x^4)/4 + (b^2*x^4*log(c*(d + e*x^(2/3))^n)^2)/4 + (b^2*n^2*x^4)/72 + (a*b*x^4*log(c*(d + e*x^(2/3))^n))/2
- (b^2*d^6*log(c*(d + e*x^(2/3))^n)^2)/(4*e^6) - (a*b*n*x^4)/12 - (b^2*n*x^4*log(c*(d + e*x^(2/3))^n))/12 + (
49*b^2*d^6*n^2*log(d + e*x^(2/3)))/(40*e^6) - (19*b^2*d^3*n^2*x^2)/(120*e^3) + (37*b^2*d^2*n^2*x^(8/3))/(480*e
^2) + (29*b^2*d^4*n^2*x^(4/3))/(80*e^4) - (49*b^2*d^5*n^2*x^(2/3))/(40*e^5) - (11*b^2*d*n^2*x^(10/3))/(300*e)
+ (b^2*d^3*n*x^2*log(c*(d + e*x^(2/3))^n))/(6*e^3) - (b^2*d^2*n*x^(8/3)*log(c*(d + e*x^(2/3))^n))/(8*e^2) - (b
^2*d^4*n*x^(4/3)*log(c*(d + e*x^(2/3))^n))/(4*e^4) + (b^2*d^5*n*x^(2/3)*log(c*(d + e*x^(2/3))^n))/(2*e^5) + (a
*b*d*n*x^(10/3))/(10*e) - (a*b*d^6*n*log(d + e*x^(2/3)))/(2*e^6) + (b^2*d*n*x^(10/3)*log(c*(d + e*x^(2/3))^n))
/(10*e) + (a*b*d^3*n*x^2)/(6*e^3) - (a*b*d^2*n*x^(8/3))/(8*e^2) - (a*b*d^4*n*x^(4/3))/(4*e^4) + (a*b*d^5*n*x^(
2/3))/(2*e^5)

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