3.5.51 \(\int x (a+b \log (c (d+e \sqrt [3]{x})^n))^2 \, dx\) [451]

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

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 480, normalized size of antiderivative = 1.00, 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 = {2504, 2445, 2458, 45, 2372, 12, 14, 2338} \begin {gather*} -\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 {b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}-\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 {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} \end {gather*}

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) + (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

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

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Mathematica [A]
time = 0.26, size = 347, normalized size = 0.72 \begin {gather*} \frac {1800 b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )+180 b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (-20 a+49 b n-20 b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+e \sqrt [3]{x} \left (1800 a^2 e^5 x^{5/3}+60 a b n \left (60 d^5-30 d^4 e \sqrt [3]{x}+20 d^3 e^2 x^{2/3}-15 d^2 e^3 x+12 d e^4 x^{4/3}-10 e^5 x^{5/3}\right )+b^2 n^2 \left (-8820 d^5+2610 d^4 e \sqrt [3]{x}-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}\right )+60 b \left (60 a e^5 x^{5/3}+b n \left (60 d^5-30 d^4 e \sqrt [3]{x}+20 d^3 e^2 x^{2/3}-15 d^2 e^3 x+12 d e^4 x^{4/3}-10 e^5 x^{5/3}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+1800 b^2 e^5 x^{5/3} \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3600 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1800*b^2*d^6*n^2*Log[d + e*x^(1/3)]^2 + 180*b*d^6*n*Log[d + e*x^(1/3)]*(-20*a + 49*b*n - 20*b*Log[c*(d + e*x^
(1/3))^n]) + 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 + 12*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*e^5*x^(5/3) + 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 + 12*d*e^4*x^(4/3) - 10*e^5*x^(5/3)))*Log[c*(d + e*x^(1/3))^n] + 1800*
b^2*e^5*x^(5/3)*Log[c*(d + e*x^(1/3))^n]^2))/(3600*e^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(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)

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Maxima [A]
time = 0.31, size = 316, normalized size = 0.66 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )^{2} - \frac {1}{60} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (30 \, d^{4} x^{\frac {2}{3}} e - 60 \, d^{5} x^{\frac {1}{3}} - 20 \, d^{3} x e^{2} + 15 \, d^{2} x^{\frac {4}{3}} e^{3} - 12 \, d x^{\frac {5}{3}} e^{4} + 10 \, x^{2} e^{5}\right )} e^{\left (-6\right )}\right )} a b n e + a b x^{2} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{3600} \, {\left ({\left (1800 \, d^{6} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 8820 \, d^{6} \log \left (x^{\frac {1}{3}} e + d\right ) - 8820 \, d^{5} x^{\frac {1}{3}} e + 2610 \, d^{4} x^{\frac {2}{3}} e^{2} - 1140 \, d^{3} x e^{3} + 555 \, d^{2} x^{\frac {4}{3}} e^{4} - 264 \, d x^{\frac {5}{3}} e^{5} + 100 \, x^{2} e^{6}\right )} n^{2} e^{\left (-6\right )} - 60 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (30 \, d^{4} x^{\frac {2}{3}} e - 60 \, d^{5} x^{\frac {1}{3}} - 20 \, d^{3} x e^{2} + 15 \, d^{2} x^{\frac {4}{3}} e^{3} - 12 \, d x^{\frac {5}{3}} e^{4} + 10 \, x^{2} e^{5}\right )} e^{\left (-6\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*(a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.40, size = 452, normalized size = 0.94 \begin {gather*} \frac {1}{3600} \, {\left (1800 \, b^{2} x^{2} e^{6} \log \left (c\right )^{2} + 100 \, {\left (b^{2} n^{2} - 6 \, a b n + 18 \, a^{2}\right )} x^{2} e^{6} - 60 \, {\left (19 \, b^{2} d^{3} n^{2} - 20 \, a b d^{3} n\right )} x e^{3} - 1800 \, {\left (b^{2} d^{6} n^{2} - b^{2} n^{2} x^{2} e^{6}\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 60 \, {\left (147 \, b^{2} d^{6} n^{2} - 60 \, a b d^{6} n + 20 \, b^{2} d^{3} n^{2} x e^{3} - 10 \, {\left (b^{2} n^{2} - 6 \, a b n\right )} x^{2} e^{6} - 60 \, {\left (b^{2} d^{6} n - b^{2} n x^{2} e^{6}\right )} \log \left (c\right ) - 6 \, {\left (5 \, b^{2} d^{4} n^{2} e^{2} - 2 \, b^{2} d n^{2} x e^{5}\right )} x^{\frac {2}{3}} + 15 \, {\left (4 \, b^{2} d^{5} n^{2} e - b^{2} d^{2} n^{2} x e^{4}\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 600 \, {\left (2 \, b^{2} d^{3} n x e^{3} - {\left (b^{2} n - 6 \, a b\right )} x^{2} e^{6}\right )} \log \left (c\right ) - 6 \, {\left (4 \, {\left (11 \, b^{2} d n^{2} - 30 \, a b d n\right )} x e^{5} - 15 \, {\left (29 \, b^{2} d^{4} n^{2} - 20 \, a b d^{4} n\right )} e^{2} + 60 \, {\left (5 \, b^{2} d^{4} n e^{2} - 2 \, b^{2} d n x e^{5}\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} + 15 \, {\left ({\left (37 \, b^{2} d^{2} n^{2} - 60 \, a b d^{2} n\right )} x 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 e^{4}\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*(a+b*log(c*(d+e*x^(1/3))^n))^2,x, algorithm="fricas")

[Out]

1/3600*(1800*b^2*x^2*e^6*log(c)^2 + 100*(b^2*n^2 - 6*a*b*n + 18*a^2)*x^2*e^6 - 60*(19*b^2*d^3*n^2 - 20*a*b*d^3
*n)*x*e^3 - 1800*(b^2*d^6*n^2 - b^2*n^2*x^2*e^6)*log(x^(1/3)*e + d)^2 + 60*(147*b^2*d^6*n^2 - 60*a*b*d^6*n + 2
0*b^2*d^3*n^2*x*e^3 - 10*(b^2*n^2 - 6*a*b*n)*x^2*e^6 - 60*(b^2*d^6*n - b^2*n*x^2*e^6)*log(c) - 6*(5*b^2*d^4*n^
2*e^2 - 2*b^2*d*n^2*x*e^5)*x^(2/3) + 15*(4*b^2*d^5*n^2*e - b^2*d^2*n^2*x*e^4)*x^(1/3))*log(x^(1/3)*e + d) + 60
0*(2*b^2*d^3*n*x*e^3 - (b^2*n - 6*a*b)*x^2*e^6)*log(c) - 6*(4*(11*b^2*d*n^2 - 30*a*b*d*n)*x*e^5 - 15*(29*b^2*d
^4*n^2 - 20*a*b*d^4*n)*e^2 + 60*(5*b^2*d^4*n*e^2 - 2*b^2*d*n*x*e^5)*log(c))*x^(2/3) + 15*((37*b^2*d^2*n^2 - 60
*a*b*d^2*n)*x*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*e^4)*log(c))*x^(1/3
))*e^(-6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \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*(a+b*ln(c*(d+e*x**(1/3))**n))**2,x)

[Out]

Integral(x*(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. 956 vs. \(2 (419) = 838\).
time = 3.29, size = 956, normalized size = 1.99 \begin {gather*} \frac {1}{3600} \, {\left (1800 \, b^{2} x^{2} e \log \left (c\right )^{2} + 3600 \, a b x^{2} e \log \left (c\right ) + {\left (1800 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 10800 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 27000 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 36000 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 27000 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 10800 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 600 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 4320 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 13500 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 24000 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 27000 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 21600 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 100 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} - 864 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} + 3375 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} - 8000 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} + 13500 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} - 21600 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )}\right )} b^{2} n^{2} + 1800 \, a^{2} x^{2} e + 60 \, {\left (60 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 360 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 1200 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 360 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 10 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} + 72 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} - 225 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} + 400 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} - 450 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} + 360 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )}\right )} b^{2} n \log \left (c\right ) + 60 \, {\left (60 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 360 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 1200 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 360 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 10 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} e^{\left (-5\right )} + 72 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} - 225 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} + 400 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} - 450 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} + 360 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{5} e^{\left (-5\right )}\right )} a b n\right )} e^{\left (-1\right )} \end {gather*}

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)

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Mupad [B]
time = 1.72, size = 431, normalized size = 0.90 \begin {gather*} \frac {a^2\,x^2}{2}+\frac {b^2\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2}+\frac {b^2\,n^2\,x^2}{36}+a\,b\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )-\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^6}-\frac {a\,b\,n\,x^2}{6}-\frac {b^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{6}+\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+e\,x^{1/3}\right )}{20\,e^6}+\frac {37\,b^2\,d^2\,n^2\,x^{4/3}}{240\,e^2}+\frac {29\,b^2\,d^4\,n^2\,x^{2/3}}{40\,e^4}-\frac {49\,b^2\,d^5\,n^2\,x^{1/3}}{20\,e^5}-\frac {19\,b^2\,d^3\,n^2\,x}{60\,e^3}-\frac {11\,b^2\,d\,n^2\,x^{5/3}}{150\,e}-\frac {b^2\,d^2\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{4\,e^2}-\frac {b^2\,d^4\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2\,e^4}+\frac {b^2\,d^5\,n\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^5}+\frac {a\,b\,d^3\,n\,x}{3\,e^3}+\frac {a\,b\,d\,n\,x^{5/3}}{5\,e}-\frac {a\,b\,d^6\,n\,\ln \left (d+e\,x^{1/3}\right )}{e^6}+\frac {b^2\,d^3\,n\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^3}+\frac {b^2\,d\,n\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{5\,e}-\frac {a\,b\,d^2\,n\,x^{4/3}}{4\,e^2}-\frac {a\,b\,d^4\,n\,x^{2/3}}{2\,e^4}+\frac {a\,b\,d^5\,n\,x^{1/3}}{e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*x^2)/2 + (b^2*x^2*log(c*(d + e*x^(1/3))^n)^2)/2 + (b^2*n^2*x^2)/36 + a*b*x^2*log(c*(d + e*x^(1/3))^n) - (
b^2*d^6*log(c*(d + e*x^(1/3))^n)^2)/(2*e^6) - (a*b*n*x^2)/6 - (b^2*n*x^2*log(c*(d + e*x^(1/3))^n))/6 + (49*b^2
*d^6*n^2*log(d + e*x^(1/3)))/(20*e^6) + (37*b^2*d^2*n^2*x^(4/3))/(240*e^2) + (29*b^2*d^4*n^2*x^(2/3))/(40*e^4)
- (49*b^2*d^5*n^2*x^(1/3))/(20*e^5) - (19*b^2*d^3*n^2*x)/(60*e^3) - (11*b^2*d*n^2*x^(5/3))/(150*e) - (b^2*d^2
*n*x^(4/3)*log(c*(d + e*x^(1/3))^n))/(4*e^2) - (b^2*d^4*n*x^(2/3)*log(c*(d + e*x^(1/3))^n))/(2*e^4) + (b^2*d^5
*n*x^(1/3)*log(c*(d + e*x^(1/3))^n))/e^5 + (a*b*d^3*n*x)/(3*e^3) + (a*b*d*n*x^(5/3))/(5*e) - (a*b*d^6*n*log(d
+ e*x^(1/3)))/e^6 + (b^2*d^3*n*x*log(c*(d + e*x^(1/3))^n))/(3*e^3) + (b^2*d*n*x^(5/3)*log(c*(d + e*x^(1/3))^n)
)/(5*e) - (a*b*d^2*n*x^(4/3))/(4*e^2) - (a*b*d^4*n*x^(2/3))/(2*e^4) + (a*b*d^5*n*x^(1/3))/e^5

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