\(\int x^2 (a+b \log (c (d+e x^{2/3})^n))^2 \, dx\) [476]

Optimal result

Integrand size = 24, antiderivative size = 547 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3-\frac {4504 b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}+\frac {8 b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {4 i b^2 d^{9/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{9/2}} \]

[Out]

-4/3*a*b*d^4*n*x^(1/3)/e^4+4504/945*b^2*d^4*n^2*x^(1/3)/e^4-1984/2835*b^2*d^3*n^2*x/e^3+1144/4725*b^2*d^2*n^2*
x^(5/3)/e^2-128/1323*b^2*d*n^2*x^(7/3)/e+8/243*b^2*n^2*x^3-4504/945*b^2*d^(9/2)*n^2*arctan(x^(1/3)*e^(1/2)/d^(
1/2))/e^(9/2)+4/3*I*b^2*d^(9/2)*n^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x^(1/3)*e^(1/2)))/e^(9/2)-4/3*b^2*d^4*n*x
^(1/3)*ln(c*(d+e*x^(2/3))^n)/e^4+4/9*b*d^3*n*x*(a+b*ln(c*(d+e*x^(2/3))^n))/e^3-4/15*b*d^2*n*x^(5/3)*(a+b*ln(c*
(d+e*x^(2/3))^n))/e^2+4/21*b*d*n*x^(7/3)*(a+b*ln(c*(d+e*x^(2/3))^n))/e-4/27*b*n*x^3*(a+b*ln(c*(d+e*x^(2/3))^n)
)+4/3*b*d^(9/2)*n*arctan(x^(1/3)*e^(1/2)/d^(1/2))*(a+b*ln(c*(d+e*x^(2/3))^n))/e^(9/2)+1/3*x^3*(a+b*ln(c*(d+e*x
^(2/3))^n))^2+8/3*b^2*d^(9/2)*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^
(9/2)+4/3*I*b^2*d^(9/2)*n^2*arctan(x^(1/3)*e^(1/2)/d^(1/2))^2/e^(9/2)

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2508, 2507, 2526, 2498, 327, 211, 2505, 308, 2520, 12, 5040, 4964, 2449, 2352} \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {4 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^{9/2}}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4 i b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}-\frac {4504 b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {8 b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 i b^2 d^{9/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{9/2}}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3 \]

[In]

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

[Out]

(-4*a*b*d^4*n*x^(1/3))/(3*e^4) + (4504*b^2*d^4*n^2*x^(1/3))/(945*e^4) - (1984*b^2*d^3*n^2*x)/(2835*e^3) + (114
4*b^2*d^2*n^2*x^(5/3))/(4725*e^2) - (128*b^2*d*n^2*x^(7/3))/(1323*e) + (8*b^2*n^2*x^3)/243 - (4504*b^2*d^(9/2)
*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(945*e^(9/2)) + (((4*I)/3)*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sq
rt[d]]^2)/e^(9/2) + (8*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*
x^(1/3))])/(3*e^(9/2)) - (4*b^2*d^4*n*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/(3*e^4) + (4*b*d^3*n*x*(a + b*Log[c*(d
+ e*x^(2/3))^n]))/(9*e^3) - (4*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(15*e^2) + (4*b*d*n*x^(7/3)*
(a + b*Log[c*(d + e*x^(2/3))^n]))/(21*e) - (4*b*n*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/27 + (4*b*d^(9/2)*n*Ar
cTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^(9/2)) + (x^3*(a + b*Log[c*(d + e*x^(2/
3))^n])^2)/3 + (((4*I)/3)*b^2*d^(9/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(9/2)

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 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 2508

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

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*} \text {integral}& = 3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {1}{3} (4 b e n) \text {Subst}\left (\int \frac {x^{10} \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 ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {1}{3} (4 b e n) \text {Subst}\left (\int \left (\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5}-\frac {d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac {d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {1}{3} (4 b n) \text {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )-\frac {\left (4 b d^4 n\right ) \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 )}{3 e^4}+\frac {\left (4 b d^5 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 )}{3 e^4}+\frac {\left (4 b d^3 n\right ) \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 )}{3 e^3}-\frac {\left (4 b d^2 n\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac {(4 b d n) \text {Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e} \\ & = -\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {\left (4 b^2 d^4 n\right ) \text {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}-\frac {1}{21} \left (8 b^2 d n^2\right ) \text {Subst}\left (\int \frac {x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )-\frac {\left (8 b^2 d^5 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 )}{3 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e}+\frac {1}{27} \left (8 b^2 e n^2\right ) \text {Subst}\left (\int \frac {x^{10}}{d+e x^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {1}{21} \left (8 b^2 d n^2\right ) \text {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^4}{e^2}+\frac {x^6}{e}+\frac {d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )-\frac {\left (8 b^2 d^{9/2} 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 )}{3 e^{7/2}}+\frac {\left (8 b^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac {\left (8 b^2 d^3 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 )}{9 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x^2}{e^2}+\frac {x^4}{e}-\frac {d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{15 e}+\frac {1}{27} \left (8 b^2 e n^2\right ) \text {Subst}\left (\int \left (\frac {d^4}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^4}{e^3}-\frac {d x^6}{e^2}+\frac {x^8}{e}-\frac {d^5}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {\left (8 b^2 d^4 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 )}{3 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4} \\ & = -\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}+\frac {8 b^2 d^{9/2} 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 )}{3 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {\left (8 b^2 d^4 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 )}{3 e^4} \\ & = -\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}+\frac {8 b^2 d^{9/2} 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 )}{3 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {\left (8 i b^2 d^{9/2} 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 )}{3 e^{9/2}} \\ & = -\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}+\frac {8 b^2 d^{9/2} 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 )}{3 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {4 i b^2 d^{9/2} n^2 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{3 e^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.80 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\frac {396900 i b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2+1260 b d^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (315 a-1126 b n+630 b n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )+315 b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\sqrt {e} \sqrt [3]{x} \left (99225 a^2 e^4 x^{8/3}-1260 a b n \left (315 d^4-105 d^3 e x^{2/3}+63 d^2 e^2 x^{4/3}-45 d e^3 x^2+35 e^4 x^{8/3}\right )+8 b^2 n^2 \left (177345 d^4-26040 d^3 e x^{2/3}+9009 d^2 e^2 x^{4/3}-3600 d e^3 x^2+1225 e^4 x^{8/3}\right )-630 b \left (-315 a e^4 x^{8/3}+2 b n \left (315 d^4-105 d^3 e x^{2/3}+63 d^2 e^2 x^{4/3}-45 d e^3 x^2+35 e^4 x^{8/3}\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+99225 b^2 e^4 x^{8/3} \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )\right )+396900 i b^2 d^{9/2} n^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}{-i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}\right )}{297675 e^{9/2}} \]

[In]

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

[Out]

((396900*I)*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2 + 1260*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sq
rt[d]]*(315*a - 1126*b*n + 630*b*n*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))] + 315*b*Log[c*(d + e*x^(2/3)
)^n]) + Sqrt[e]*x^(1/3)*(99225*a^2*e^4*x^(8/3) - 1260*a*b*n*(315*d^4 - 105*d^3*e*x^(2/3) + 63*d^2*e^2*x^(4/3)
- 45*d*e^3*x^2 + 35*e^4*x^(8/3)) + 8*b^2*n^2*(177345*d^4 - 26040*d^3*e*x^(2/3) + 9009*d^2*e^2*x^(4/3) - 3600*d
*e^3*x^2 + 1225*e^4*x^(8/3)) - 630*b*(-315*a*e^4*x^(8/3) + 2*b*n*(315*d^4 - 105*d^3*e*x^(2/3) + 63*d^2*e^2*x^(
4/3) - 45*d*e^3*x^2 + 35*e^4*x^(8/3)))*Log[c*(d + e*x^(2/3))^n] + 99225*b^2*e^4*x^(8/3)*Log[c*(d + e*x^(2/3))^
n]^2) + (396900*I)*b^2*d^(9/2)*n^2*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x^(1/3))/((-I)*Sqrt[d] + Sqrt[e]*x^(1/3))])
/(297675*e^(9/2))

Maple [F]

\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2 \,d x \]

[In]

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

[Out]

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