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

Optimal. Leaf size=269 \[ -\frac{2 b d^3 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac{6 b d^2 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}-\frac{3 b d n \left (d+\frac{e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^3}+\frac{2 b n \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}-\frac{6 b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac{b^2 d^3 n^2 \log ^2\left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}+\frac{3 b^2 d n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}{2 e^3}-\frac{2 b^2 n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^3}{9 e^3} \]

[Out]

(3*b^2*d*n^2*(d + e/x^(1/3))^2)/(2*e^3) - (2*b^2*n^2*(d + e/x^(1/3))^3)/(9*e^3) - (6*b^2*d^2*n^2)/(e^2*x^(1/3)
) + (b^2*d^3*n^2*Log[d + e/x^(1/3)]^2)/e^3 + (6*b*d^2*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n]))/e^3
- (3*b*d*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/e^3 + (2*b*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d
+ e/x^(1/3))^n]))/(3*e^3) - (2*b*d^3*n*Log[d + e/x^(1/3)]*(a + b*Log[c*(d + e/x^(1/3))^n]))/e^3 - (a + b*Log[c
*(d + e/x^(1/3))^n])^2/x

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Rubi [A]  time = 0.311076, antiderivative size = 212, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ \frac{1}{3} b n \left (\frac{18 d^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}-\frac{6 d^3 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}-\frac{9 d \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}{e^3}+\frac{2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^3}{e^3}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x}-\frac{6 b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac{b^2 d^3 n^2 \log ^2\left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}+\frac{3 b^2 d n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}{2 e^3}-\frac{2 b^2 n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^3}{9 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^2*d*n^2*(d + e/x^(1/3))^2)/(2*e^3) - (2*b^2*n^2*(d + e/x^(1/3))^3)/(9*e^3) - (6*b^2*d^2*n^2)/(e^2*x^(1/3)
) + (b^2*d^3*n^2*Log[d + e/x^(1/3)]^2)/e^3 + (b*n*((18*d^2*(d + e/x^(1/3)))/e^3 - (9*d*(d + e/x^(1/3))^2)/e^3
+ (2*(d + e/x^(1/3))^3)/e^3 - (6*d^3*Log[d + e/x^(1/3)])/e^3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/3 - (a + b*Log
[c*(d + e/x^(1/3))^n])^2/x

Rule 2454

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])

Rule 2398

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 2411

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 43

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 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 2301

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]

Rubi steps

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

Mathematica [C]  time = 0.360091, size = 374, normalized size = 1.39 \[ \frac{\frac{b n \left (-36 b d^3 n x \text{PolyLog}\left (2,\frac{e}{d \sqrt [3]{x}}+1\right )-36 b d^3 n x \text{PolyLog}\left (2,\frac{d \sqrt [3]{x}}{e}+1\right )-36 d^3 x \log \left (d \sqrt [3]{x}+e\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )-36 d^3 x \log \left (-\frac{e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )-18 d e^2 \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )+12 e^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )+36 a d^2 e x^{2/3}+36 b d^2 x^{2/3} \left (d \sqrt [3]{x}+e\right ) \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-2 b e n \left (6 d^2 x^{2/3}-3 d e \sqrt [3]{x}+2 e^2\right )-36 b d^2 e n x^{2/3}+30 b d^3 n x \log \left (d+\frac{e}{\sqrt [3]{x}}\right )+18 b d^3 n x \log \left (d \sqrt [3]{x}+e\right ) \left (\log \left (d \sqrt [3]{x}+e\right )-2 \log \left (-\frac{d \sqrt [3]{x}}{e}\right )\right )+9 b d e n \sqrt [3]{x} \left (e-2 d \sqrt [3]{x}\right )\right )}{e^3}-18 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{18 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*(a + b*Log[c*(d + e/x^(1/3))^n])^2 + (b*n*(-2*b*e*n*(2*e^2 - 3*d*e*x^(1/3) + 6*d^2*x^(2/3)) + 9*b*d*e*n*(
e - 2*d*x^(1/3))*x^(1/3) + 36*a*d^2*e*x^(2/3) - 36*b*d^2*e*n*x^(2/3) + 30*b*d^3*n*x*Log[d + e/x^(1/3)] + 36*b*
d^2*(e + d*x^(1/3))*x^(2/3)*Log[c*(d + e/x^(1/3))^n] + 12*e^3*(a + b*Log[c*(d + e/x^(1/3))^n]) - 18*d*e^2*x^(1
/3)*(a + b*Log[c*(d + e/x^(1/3))^n]) - 36*d^3*x*(a + b*Log[c*(d + e/x^(1/3))^n])*Log[e + d*x^(1/3)] - 36*d^3*x
*(a + b*Log[c*(d + e/x^(1/3))^n])*Log[-(e/(d*x^(1/3)))] + 18*b*d^3*n*x*Log[e + d*x^(1/3)]*(Log[e + d*x^(1/3)]
- 2*Log[-((d*x^(1/3))/e)]) - 36*b*d^3*n*x*PolyLog[2, 1 + e/(d*x^(1/3))] - 36*b*d^3*n*x*PolyLog[2, 1 + (d*x^(1/
3))/e]))/e^3)/(18*x)

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Maple [F]  time = 0.576, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt [3]{x}}}} \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^(1/3))^n))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.06979, size = 383, normalized size = 1.42 \begin{align*} -\frac{1}{3} \, a b e n{\left (\frac{6 \, d^{3} \log \left (d x^{\frac{1}{3}} + e\right )}{e^{4}} - \frac{2 \, d^{3} \log \left (x\right )}{e^{4}} - \frac{6 \, d^{2} x^{\frac{2}{3}} - 3 \, d e x^{\frac{1}{3}} + 2 \, e^{2}}{e^{3} x}\right )} - \frac{1}{18} \,{\left (6 \, e n{\left (\frac{6 \, d^{3} \log \left (d x^{\frac{1}{3}} + e\right )}{e^{4}} - \frac{2 \, d^{3} \log \left (x\right )}{e^{4}} - \frac{6 \, d^{2} x^{\frac{2}{3}} - 3 \, d e x^{\frac{1}{3}} + 2 \, e^{2}}{e^{3} x}\right )} \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right ) - \frac{{\left (18 \, d^{3} x \log \left (d x^{\frac{1}{3}} + e\right )^{2} + 2 \, d^{3} x \log \left (x\right )^{2} - 22 \, d^{3} x \log \left (x\right ) - 66 \, d^{2} e x^{\frac{2}{3}} + 15 \, d e^{2} x^{\frac{1}{3}} - 4 \, e^{3} - 6 \,{\left (2 \, d^{3} x \log \left (x\right ) - 11 \, d^{3} x\right )} \log \left (d x^{\frac{1}{3}} + e\right )\right )} n^{2}}{e^{3} x}\right )} b^{2} - \frac{b^{2} \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right )^{2}}{x} - \frac{2 \, a b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right )}{x} - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^n))^2/x^2,x, algorithm="maxima")

[Out]

-1/3*a*b*e*n*(6*d^3*log(d*x^(1/3) + e)/e^4 - 2*d^3*log(x)/e^4 - (6*d^2*x^(2/3) - 3*d*e*x^(1/3) + 2*e^2)/(e^3*x
)) - 1/18*(6*e*n*(6*d^3*log(d*x^(1/3) + e)/e^4 - 2*d^3*log(x)/e^4 - (6*d^2*x^(2/3) - 3*d*e*x^(1/3) + 2*e^2)/(e
^3*x))*log(c*(d + e/x^(1/3))^n) - (18*d^3*x*log(d*x^(1/3) + e)^2 + 2*d^3*x*log(x)^2 - 22*d^3*x*log(x) - 66*d^2
*e*x^(2/3) + 15*d*e^2*x^(1/3) - 4*e^3 - 6*(2*d^3*x*log(x) - 11*d^3*x)*log(d*x^(1/3) + e))*n^2/(e^3*x))*b^2 - b
^2*log(c*(d + e/x^(1/3))^n)^2/x - 2*a*b*log(c*(d + e/x^(1/3))^n)/x - a^2/x

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Fricas [A]  time = 1.90434, size = 798, normalized size = 2.97 \begin{align*} -\frac{4 \, b^{2} e^{3} n^{2} - 12 \, a b e^{3} n + 18 \, a^{2} e^{3} - 18 \,{\left (b^{2} e^{3} x - b^{2} e^{3}\right )} \log \left (c\right )^{2} + 18 \,{\left (b^{2} d^{3} n^{2} x + b^{2} e^{3} n^{2}\right )} \log \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right )^{2} - 2 \,{\left (2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + 9 \, a^{2} e^{3}\right )} x - 12 \,{\left (b^{2} e^{3} n - 3 \, a b e^{3} -{\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} x\right )} \log \left (c\right ) - 6 \,{\left (6 \, b^{2} d^{2} e n^{2} x^{\frac{2}{3}} - 3 \, b^{2} d e^{2} n^{2} x^{\frac{1}{3}} + 2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n +{\left (11 \, b^{2} d^{3} n^{2} - 6 \, a b d^{3} n\right )} x - 6 \,{\left (b^{2} d^{3} n x + b^{2} e^{3} n\right )} \log \left (c\right )\right )} \log \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right ) + 6 \,{\left (11 \, b^{2} d^{2} e n^{2} - 6 \, b^{2} d^{2} e n \log \left (c\right ) - 6 \, a b d^{2} e n\right )} x^{\frac{2}{3}} - 3 \,{\left (5 \, b^{2} d e^{2} n^{2} - 6 \, b^{2} d e^{2} n \log \left (c\right ) - 6 \, a b d e^{2} n\right )} x^{\frac{1}{3}}}{18 \, e^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^n))^2/x^2,x, algorithm="fricas")

[Out]

-1/18*(4*b^2*e^3*n^2 - 12*a*b*e^3*n + 18*a^2*e^3 - 18*(b^2*e^3*x - b^2*e^3)*log(c)^2 + 18*(b^2*d^3*n^2*x + b^2
*e^3*n^2)*log((d*x + e*x^(2/3))/x)^2 - 2*(2*b^2*e^3*n^2 - 6*a*b*e^3*n + 9*a^2*e^3)*x - 12*(b^2*e^3*n - 3*a*b*e
^3 - (b^2*e^3*n - 3*a*b*e^3)*x)*log(c) - 6*(6*b^2*d^2*e*n^2*x^(2/3) - 3*b^2*d*e^2*n^2*x^(1/3) + 2*b^2*e^3*n^2
- 6*a*b*e^3*n + (11*b^2*d^3*n^2 - 6*a*b*d^3*n)*x - 6*(b^2*d^3*n*x + b^2*e^3*n)*log(c))*log((d*x + e*x^(2/3))/x
) + 6*(11*b^2*d^2*e*n^2 - 6*b^2*d^2*e*n*log(c) - 6*a*b*d^2*e*n)*x^(2/3) - 3*(5*b^2*d*e^2*n^2 - 6*b^2*d*e^2*n*l
og(c) - 6*a*b*d*e^2*n)*x^(1/3))/(e^3*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**(1/3))**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 (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\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^(1/3))^n))^2/x^2,x, algorithm="giac")

[Out]

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