3.5.72 \(\int x (a+b \log (c (d+e x^{2/3})^n))^2 \, dx\) [472]

Optimal. Leaf size=275 \[ -\frac {3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e x^{2/3}\right )}{2 e^3}-\frac {3 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}+\frac {b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \]

[Out]

-3/4*b^2*d*n^2*(d+e*x^(2/3))^2/e^3+1/9*b^2*n^2*(d+e*x^(2/3))^3/e^3+3*b^2*d^2*n^2*x^(2/3)/e^2-1/2*b^2*d^3*n^2*l
n(d+e*x^(2/3))^2/e^3-3*b*d^2*n*(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))/e^3+3/2*b*d*n*(d+e*x^(2/3))^2*(a+b*ln
(c*(d+e*x^(2/3))^n))/e^3-1/3*b*n*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e*x^(2/3))^n))/e^3+b*d^3*n*ln(d+e*x^(2/3))*(a+b*
ln(c*(d+e*x^(2/3))^n))/e^3+1/2*x^2*(a+b*ln(c*(d+e*x^(2/3))^n))^2

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 275, 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^3 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 )}{e^3}-\frac {3 b d^2 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^3}+\frac {3 b d 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 )}{2 e^3}-\frac {b 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^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {b^2 d^3 n^2 \log ^2\left (d+e x^{2/3}\right )}{2 e^3}+\frac {3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^2*d*n^2*(d + e*x^(2/3))^2)/(4*e^3) + (b^2*n^2*(d + e*x^(2/3))^3)/(9*e^3) + (3*b^2*d^2*n^2*x^(2/3))/e^2 -
(b^2*d^3*n^2*Log[d + e*x^(2/3)]^2)/(2*e^3) - (3*b*d^2*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^3
+ (3*b*d*n*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*e^3) - (b*n*(d + e*x^(2/3))^3*(a + b*Log[c*
(d + e*x^(2/3))^n]))/(3*e^3) + (b*d^3*n*Log[d + e*x^(2/3)]*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^3 + (x^2*(a + b
*Log[c*(d + e*x^(2/3))^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 x^{2/3}\right )^n\right )\right )^2 \, dx &=\frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,x^{2/3}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(b e n) \text {Subst}\left (\int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,x^{2/3}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x^{2/3}\right )\\ &=-\frac {1}{6} b n \left (\frac {18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac {9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac {2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e x^{2/3}\right )\\ &=-\frac {1}{6} b n \left (\frac {18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac {9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac {2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {1}{2} x^2 \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 {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e x^{2/3}\right )}{6 e^3}\\ &=-\frac {1}{6} b n \left (\frac {18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac {9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac {2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {1}{2} x^2 \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 (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+e x^{2/3}\right )}{6 e^3}\\ &=-\frac {3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {1}{6} b n \left (\frac {18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac {9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac {2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {\left (b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x^{2/3}\right )}{e^3}\\ &=-\frac {3 b^2 d n^2 \left (d+e x^{2/3}\right )^2}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {3 b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e x^{2/3}\right )}{2 e^3}-\frac {1}{6} b n \left (\frac {18 d^2 \left (d+e x^{2/3}\right )}{e^3}-\frac {9 d \left (d+e x^{2/3}\right )^2}{e^3}+\frac {2 \left (d+e x^{2/3}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e x^{2/3}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.11, size = 239, normalized size = 0.87 \begin {gather*} \frac {18 a^2 d^3-36 a b d^2 e n x^{2/3}+66 b^2 d^2 e n^2 x^{2/3}+18 a b d e^2 n x^{4/3}-15 b^2 d e^2 n^2 x^{4/3}+18 a^2 e^3 x^2-12 a b e^3 n x^2+4 b^2 e^3 n^2 x^2-30 b^2 d^3 n^2 \log \left (d+e x^{2/3}\right )+6 b \left (6 a \left (d^3+e^3 x^2\right )-b n \left (6 d^3+6 d^2 e x^{2/3}-3 d e^2 x^{4/3}+2 e^3 x^2\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+18 b^2 \left (d^3+e^3 x^2\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )}{36 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(18*a^2*d^3 - 36*a*b*d^2*e*n*x^(2/3) + 66*b^2*d^2*e*n^2*x^(2/3) + 18*a*b*d*e^2*n*x^(4/3) - 15*b^2*d*e^2*n^2*x^
(4/3) + 18*a^2*e^3*x^2 - 12*a*b*e^3*n*x^2 + 4*b^2*e^3*n^2*x^2 - 30*b^2*d^3*n^2*Log[d + e*x^(2/3)] + 6*b*(6*a*(
d^3 + e^3*x^2) - b*n*(6*d^3 + 6*d^2*e*x^(2/3) - 3*d*e^2*x^(4/3) + 2*e^3*x^2))*Log[c*(d + e*x^(2/3))^n] + 18*b^
2*(d^3 + e^3*x^2)*Log[c*(d + e*x^(2/3))^n]^2)/(36*e^3)

________________________________________________________________________________________

Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 232, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )^{2} + \frac {1}{6} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} a b n e + a b x^{2} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{2} x^{2} - \frac {1}{36} \, {\left ({\left (18 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 66 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right ) - 66 \, d^{2} x^{\frac {2}{3}} e + 15 \, d x^{\frac {4}{3}} e^{2} - 4 \, x^{2} e^{3}\right )} n^{2} e^{\left (-3\right )} - 6 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\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*(a+b*log(c*(d+e*x^(2/3))^n))^2,x, algorithm="maxima")

[Out]

1/2*b^2*x^2*log((x^(2/3)*e + d)^n*c)^2 + 1/6*(6*d^3*e^(-4)*log(x^(2/3)*e + d) + (3*d*x^(4/3)*e - 2*x^2*e^2 - 6
*d^2*x^(2/3))*e^(-3))*a*b*n*e + a*b*x^2*log((x^(2/3)*e + d)^n*c) + 1/2*a^2*x^2 - 1/36*((18*d^3*log(x^(2/3)*e +
d)^2 + 66*d^3*log(x^(2/3)*e + d) - 66*d^2*x^(2/3)*e + 15*d*x^(4/3)*e^2 - 4*x^2*e^3)*n^2*e^(-3) - 6*(6*d^3*e^(
-4)*log(x^(2/3)*e + d) + (3*d*x^(4/3)*e - 2*x^2*e^2 - 6*d^2*x^(2/3))*e^(-3))*n*e*log((x^(2/3)*e + d)^n*c))*b^2

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 286, normalized size = 1.04 \begin {gather*} \frac {1}{36} \, {\left (18 \, b^{2} x^{2} e^{3} \log \left (c\right )^{2} - 12 \, {\left (b^{2} n - 3 \, a b\right )} x^{2} e^{3} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} n^{2} - 6 \, a b n + 9 \, a^{2}\right )} x^{2} e^{3} + 18 \, {\left (b^{2} d^{3} n^{2} + b^{2} n^{2} x^{2} e^{3}\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 6 \, {\left (6 \, b^{2} d^{2} n^{2} x^{\frac {2}{3}} e - 3 \, b^{2} d n^{2} x^{\frac {4}{3}} e^{2} + 11 \, b^{2} d^{3} n^{2} - 6 \, a b d^{3} n + 2 \, {\left (b^{2} n^{2} - 3 \, a b n\right )} x^{2} e^{3} - 6 \, {\left (b^{2} d^{3} n + b^{2} n x^{2} e^{3}\right )} \log \left (c\right )\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 6 \, {\left (6 \, b^{2} d^{2} n e \log \left (c\right ) - {\left (11 \, b^{2} d^{2} n^{2} - 6 \, a b d^{2} n\right )} e\right )} x^{\frac {2}{3}} + 3 \, {\left (6 \, b^{2} d n x e^{2} \log \left (c\right ) - {\left (5 \, b^{2} d n^{2} - 6 \, a b d n\right )} x e^{2}\right )} x^{\frac {1}{3}}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(18*b^2*x^2*e^3*log(c)^2 - 12*(b^2*n - 3*a*b)*x^2*e^3*log(c) + 2*(2*b^2*n^2 - 6*a*b*n + 9*a^2)*x^2*e^3 +
18*(b^2*d^3*n^2 + b^2*n^2*x^2*e^3)*log(x^(2/3)*e + d)^2 - 6*(6*b^2*d^2*n^2*x^(2/3)*e - 3*b^2*d*n^2*x^(4/3)*e^2
+ 11*b^2*d^3*n^2 - 6*a*b*d^3*n + 2*(b^2*n^2 - 3*a*b*n)*x^2*e^3 - 6*(b^2*d^3*n + b^2*n*x^2*e^3)*log(c))*log(x^
(2/3)*e + d) - 6*(6*b^2*d^2*n*e*log(c) - (11*b^2*d^2*n^2 - 6*a*b*d^2*n)*e)*x^(2/3) + 3*(6*b^2*d*n*x*e^2*log(c)
- (5*b^2*d*n^2 - 6*a*b*d*n)*x*e^2)*x^(1/3))*e^(-3)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 4.88, size = 316, normalized size = 1.15 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} \log \left (c\right )^{2} + \frac {1}{36} \, {\left (18 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + {\left (18 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 12 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} \log \left (x^{\frac {2}{3}} e + d\right ) + 54 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d \log \left (x^{\frac {2}{3}} e + d\right ) - 108 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + 4 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} - 27 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d + 108 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2}\right )} e^{\left (-3\right )}\right )} b^{2} n^{2} + \frac {1}{6} \, {\left (6 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x^{\frac {2}{3}} e + d \right |}\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} e\right )} b^{2} n \log \left (c\right ) + a b x^{2} \log \left (c\right ) + \frac {1}{6} \, {\left (6 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x^{\frac {2}{3}} e + d \right |}\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} e\right )} a b n + \frac {1}{2} \, a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2*log(c)^2 + 1/36*(18*x^2*log(x^(2/3)*e + d)^2 + (18*d^3*log(x^(2/3)*e + d)^2 - 12*(x^(2/3)*e + d)^3
*log(x^(2/3)*e + d) + 54*(x^(2/3)*e + d)^2*d*log(x^(2/3)*e + d) - 108*(x^(2/3)*e + d)*d^2*log(x^(2/3)*e + d) +
4*(x^(2/3)*e + d)^3 - 27*(x^(2/3)*e + d)^2*d + 108*(x^(2/3)*e + d)*d^2)*e^(-3))*b^2*n^2 + 1/6*(6*x^2*log(x^(2
/3)*e + d) + (6*d^3*e^(-4)*log(abs(x^(2/3)*e + d)) + (3*d*x^(4/3)*e - 2*x^2*e^2 - 6*d^2*x^(2/3))*e^(-3))*e)*b^
2*n*log(c) + a*b*x^2*log(c) + 1/6*(6*x^2*log(x^(2/3)*e + d) + (6*d^3*e^(-4)*log(abs(x^(2/3)*e + d)) + (3*d*x^(
4/3)*e - 2*x^2*e^2 - 6*d^2*x^(2/3))*e^(-3))*e)*a*b*n + 1/2*a^2*x^2

________________________________________________________________________________________

Mupad [B]
time = 0.53, size = 299, normalized size = 1.09 \begin {gather*} {\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2\,\left (\frac {b^2\,x^2}{2}+\frac {b^2\,d^3}{2\,e^3}\right )-x^{4/3}\,\left (\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{4\,e}\right )+x^2\,\left (\frac {a^2}{2}-\frac {a\,b\,n}{3}+\frac {b^2\,n^2}{9}\right )+\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\,\left (\frac {b\,x^2\,\left (3\,a-b\,n\right )}{3}-x^{4/3}\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {3\,a\,b\,d}{2\,e}\right )+\frac {d\,x^{2/3}\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}\right )}{e}\right )+x^{2/3}\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{e^2}\right )-\frac {\ln \left (d+e\,x^{2/3}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{6\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*(d + e*x^(2/3))^n)^2*((b^2*x^2)/2 + (b^2*d^3)/(2*e^3)) - x^(4/3)*((d*((3*a^2)/2 + (b^2*n^2)/3 - a*b*n))/
(2*e) - (d*(3*a^2 - b^2*n^2))/(4*e)) + x^2*(a^2/2 + (b^2*n^2)/9 - (a*b*n)/3) + log(c*(d + e*x^(2/3))^n)*((b*x^
2*(3*a - b*n))/3 - x^(4/3)*((b*d*(3*a - b*n))/(2*e) - (3*a*b*d)/(2*e)) + (d*x^(2/3)*((b*d*(3*a - b*n))/e - (3*
a*b*d)/e))/e) + x^(2/3)*((d*((d*((3*a^2)/2 + (b^2*n^2)/3 - a*b*n))/e - (d*(3*a^2 - b^2*n^2))/(2*e)))/e + (b^2*
d^2*n^2)/e^2) - (log(d + e*x^(2/3))*(11*b^2*d^3*n^2 - 6*a*b*d^3*n))/(6*e^3)

________________________________________________________________________________________