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

Optimal. Leaf size=298 \[ -\frac{4 i b^2 e^{3/2} n^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{3/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}-\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{8 b^2 e^{3/2} n^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.405929, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {2458, 2457, 2476, 2455, 205, 2470, 12, 4920, 4854, 2402, 2315} \[ -\frac{4 i b^2 e^{3/2} n^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac{4 b e^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{3/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}-\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{3/2}}+\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{8 b^2 e^{3/2} n^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2458

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 2457

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 2476

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 2455

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 205

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

Rule 2470

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 12

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

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[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 2402

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 2315

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}+(4 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}+(4 b e n) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^2}-\frac{e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}+\frac{(4 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac{\left (4 b e^2 n\right ) \operatorname{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 )}{d}\\ &=-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d \sqrt [3]{x}}-\frac{4 b e^{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 )}{d^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{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 )}{d}\\ &=\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d \sqrt [3]{x}}-\frac{4 b e^{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 )}{d^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}+\frac{\left (8 b^2 e^{5/2} n^2\right ) \operatorname{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 )}{d^{3/2}}\\ &=\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d \sqrt [3]{x}}-\frac{4 b e^{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 )}{d^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}-\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{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 )}{d^2}\\ &=\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{8 b^2 e^{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 )}{d^{3/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d \sqrt [3]{x}}-\frac{4 b e^{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 )}{d^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{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 )}{d^2}\\ &=\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{8 b^2 e^{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 )}{d^{3/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d \sqrt [3]{x}}-\frac{4 b e^{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 )}{d^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}-\frac{\left (8 i b^2 e^{3/2} n^2\right ) \operatorname{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 )}{d^{3/2}}\\ &=\frac{8 b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{3/2}}-\frac{4 i b^2 e^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{3/2}}-\frac{8 b^2 e^{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 )}{d^{3/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d \sqrt [3]{x}}-\frac{4 b e^{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 )}{d^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x}-\frac{4 i b^2 e^{3/2} n^2 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.187435, size = 247, normalized size = 0.83 \[ \frac{-4 i b^2 e^{3/2} n^2 x \text{PolyLog}\left (2,\frac{\sqrt{e} \sqrt [3]{x}+i \sqrt{d}}{\sqrt{e} \sqrt [3]{x}-i \sqrt{d}}\right )-4 b e^{3/2} n x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )+2 b n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )-2 b n\right )-\sqrt{d} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (a d+b d \log \left (c \left (d+e x^{2/3}\right )^n\right )+4 b e n x^{2/3}\right )-4 i b^2 e^{3/2} n^2 x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{d^{3/2} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.347, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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