3.5.77 \(\int (a+b \log (c (d+e x^{2/3})^n))^2 \, dx\) [477]

Optimal. Leaf size=364 \[ \frac {4 a b d n \sqrt [3]{x}}{e}-\frac {32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac {8}{9} b^2 n^2 x+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}-\frac {8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac {4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac {4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {4 i b^2 d^{3/2} n^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}} \]

[Out]

4*a*b*d*n*x^(1/3)/e-32/3*b^2*d*n^2*x^(1/3)/e+8/9*b^2*n^2*x+32/3*b^2*d^(3/2)*n^2*arctan(x^(1/3)*e^(1/2)/d^(1/2)
)/e^(3/2)-4*I*b^2*d^(3/2)*n^2*arctan(x^(1/3)*e^(1/2)/d^(1/2))^2/e^(3/2)+4*b^2*d*n*x^(1/3)*ln(c*(d+e*x^(2/3))^n
)/e-4/3*b*n*x*(a+b*ln(c*(d+e*x^(2/3))^n))-4*b*d^(3/2)*n*arctan(x^(1/3)*e^(1/2)/d^(1/2))*(a+b*ln(c*(d+e*x^(2/3)
)^n))/e^(3/2)+x*(a+b*ln(c*(d+e*x^(2/3))^n))^2-8*b^2*d^(3/2)*n^2*arctan(x^(1/3)*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(
d^(1/2)+I*x^(1/3)*e^(1/2)))/e^(3/2)-4*I*b^2*d^(3/2)*n^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x^(1/3)*e^(1/2)))/e^(
3/2)

________________________________________________________________________________________

Rubi [A]
time = 0.30, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2501, 2507, 2526, 2498, 327, 211, 2505, 308, 2520, 12, 5040, 4964, 2449, 2352} \begin {gather*} -\frac {4 i b^2 d^{3/2} n^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {4 b d^{3/2} n \text {ArcTan}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}-\frac {4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {4 a b d n \sqrt [3]{x}}{e}-\frac {4 i b^2 d^{3/2} n^2 \text {ArcTan}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}+\frac {32 b^2 d^{3/2} n^2 \text {ArcTan}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {8 b^2 d^{3/2} n^2 \text {ArcTan}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac {4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac {32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac {8}{9} b^2 n^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*b*d*n*x^(1/3))/e - (32*b^2*d*n^2*x^(1/3))/(3*e) + (8*b^2*n^2*x)/9 + (32*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[e]*x
^(1/3))/Sqrt[d]])/(3*e^(3/2)) - ((4*I)*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(3/2) - (8*b^2*d
^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(3/2) + (4*b^2*
d*n*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/e - (4*b*n*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/3 - (4*b*d^(3/2)*n*ArcTan
[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^(3/2) + x*(a + b*Log[c*(d + e*x^(2/3))^n])^2 -
((4*I)*b^2*d^(3/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(3/2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2501

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p,
q}, x] && FractionQ[n]

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2507

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)
^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])^q/(f*(m + 1))), x] - Dist[b*e*n*p*(q/(f^n*(m + 1))), Int[(f*x)^(m + n)*
((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1]
&& IntegerQ[n] && NeQ[m, -1]

Rule 2520

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[u*(x^(n - 1)/(d + e*x^n)
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x
^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx &=3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(4 b e n) \text {Subst}\left (\int \frac {x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(4 b e n) \text {Subst}\left (\int \left (-\frac {d \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac {d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-(4 b n) \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )+\frac {(4 b d n) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{e}-\frac {\left (4 b d^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac {4 a b d n \sqrt [3]{x}}{e}-\frac {4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {\left (4 b^2 d n\right ) \text {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{e}+\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac {1}{3} \left (8 b^2 e n^2\right ) \text {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {4 a b d n \sqrt [3]{x}}{e}+\frac {4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac {4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\left (8 b^2 d n^2\right ) \text {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^{3/2} n^2\right ) \text {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {e}}+\frac {1}{3} \left (8 b^2 e n^2\right ) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {4 a b d n \sqrt [3]{x}}{e}-\frac {32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac {8}{9} b^2 n^2 x-\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}+\frac {4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac {4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {\left (8 b^2 d n^2\right ) \text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac {\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e}+\frac {\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac {4 a b d n \sqrt [3]{x}}{e}-\frac {32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac {8}{9} b^2 n^2 x+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}-\frac {8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac {4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac {4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {\left (8 b^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac {4 a b d n \sqrt [3]{x}}{e}-\frac {32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac {8}{9} b^2 n^2 x+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}-\frac {8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac {4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac {4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {\left (8 i b^2 d^{3/2} n^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{e^{3/2}}\\ &=\frac {4 a b d n \sqrt [3]{x}}{e}-\frac {32 b^2 d n^2 \sqrt [3]{x}}{3 e}+\frac {8}{9} b^2 n^2 x+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}-\frac {8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac {4 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}-\frac {4}{3} b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {4 i b^2 d^{3/2} n^2 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{e^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 319, normalized size = 0.88 \begin {gather*} \frac {-36 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2-12 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (3 a-8 b n+6 b n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )+3 b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\sqrt {e} \sqrt [3]{x} \left (12 a b n \left (3 d-e x^{2/3}\right )+8 b^2 n^2 \left (-12 d+e x^{2/3}\right )+9 a^2 e x^{2/3}+6 b \left (6 b d n+3 a e x^{2/3}-2 b e n x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+9 b^2 e x^{2/3} \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )\right )-36 i b^2 d^{3/2} n^2 \text {Li}_2\left (\frac {i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}{-i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}\right )}{9 e^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-36*I)*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2 - 12*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]
]*(3*a - 8*b*n + 6*b*n*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))] + 3*b*Log[c*(d + e*x^(2/3))^n]) + Sqrt[e
]*x^(1/3)*(12*a*b*n*(3*d - e*x^(2/3)) + 8*b^2*n^2*(-12*d + e*x^(2/3)) + 9*a^2*e*x^(2/3) + 6*b*(6*b*d*n + 3*a*e
*x^(2/3) - 2*b*e*n*x^(2/3))*Log[c*(d + e*x^(2/3))^n] + 9*b^2*e*x^(2/3)*Log[c*(d + e*x^(2/3))^n]^2) - (36*I)*b^
2*d^(3/2)*n^2*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x^(1/3))/((-I)*Sqrt[d] + Sqrt[e]*x^(1/3))])/(9*e^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(2*(3*d^(3/2)*arctan(x^(1/3)*e^(1/2)/sqrt(d))*e^(-5/2) + (x*e - 3*d*x^(1/3))*e^(-2))*n*e - 3*x*log((x^(2/
3)*e + d)^n*c))*a*b + (n^2*x*log(x^(2/3)*e + d)^2 + integrate(1/3*(3*x*e*log(c)^2 + 3*d*x^(1/3)*log(c)^2 - 2*(
2*n*x*e - 3*x*e*log(c) - 3*d*x^(1/3)*log(c))*n*log(x^(2/3)*e + d))/(x*e + d*x^(1/3)), x))*b^2 + a^2*x

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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