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

Optimal. Leaf size=231 \[ -\frac{2 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt [3]{x}}\right )}{d^3}+\frac{2 b e^3 n \log \left (1-\frac{d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}+\frac{2 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x}-\frac{b^2 e^2 n^2}{d^2 \sqrt [3]{x}}+\frac{b^2 e^3 n^2 \log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.501207, antiderivative size = 253, normalized size of antiderivative = 1.1, number of steps used = 14, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac{2 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right )}{d^3}-\frac{e^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3}+\frac{2 b e^3 n \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}+\frac{2 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x}-\frac{b^2 e^2 n^2}{d^2 \sqrt [3]{x}}+\frac{b^2 e^3 n^2 \log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

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]

Rule 2317

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

Rule 2391

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

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.222512, size = 274, normalized size = 1.19 \[ 3 \left (\frac{2}{3} b e n \left (\frac{b e^2 n \text{PolyLog}\left (2,\frac{d+e \sqrt [3]{x}}{d}\right )}{d^3}-\frac{e^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 b d^3 n}+\frac{e^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}+\frac{e \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^2 \sqrt [3]{x}}-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 d x^{2/3}}-\frac{b e^2 n \left (\frac{\log (x)}{3 d}-\frac{\log \left (d+e \sqrt [3]{x}\right )}{d}\right )}{d^2}-\frac{b e n \left (-\frac{e \log \left (d+e \sqrt [3]{x}\right )}{d^2}+\frac{e \log (x)}{3 d^2}+\frac{1}{d \sqrt [3]{x}}\right )}{2 d}\right )-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+e\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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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]

-2*(log(e*x^(1/3)/d + 1)*log(x^(1/3)) + dilog(-e*x^(1/3)/d))*b^2*e^3*n^2/d^3 - (2*a*b*e^3*n - (3*e^3*n^2 - 2*e
^3*n*log(c))*b^2)*log(e*x^(1/3) + d)/d^3 + 2*(b^2*e^3*n*log(c) + a*b*e^3*n)*log(x^(1/3))/d^3 + integrate((b^2*
e^6*n^2*x - b^2*d^3*e^3*n^2)/x, x)/d^6 - 1/20*(12*b^2*e^8*n^2*x^(5/3) - 15*b^2*d*e^7*n^2*x^(4/3) + 20*b^2*d^2*
e^6*n^2*x - 40*b^2*d^3*e^5*n^2*x^(2/3) + 100*b^2*d^4*e^4*n^2*x^(1/3) + 20*(b^2*d^3*e^5*n^2*x^(2/3) - 2*b^2*d^4
*e^4*n^2*x^(1/3))*log(x^(1/3)))/d^8 + 1/60*(60*b^2*d^5*e^3*n^2*x^(5/3)*log(e*x^(1/3) + d)^2 - 45*b^2*d*e^7*n^2
*x^3 - 40*b^2*d^4*e^4*n^2*x^2*log(x) + 300*b^2*d^4*e^4*n^2*x^2 - 60*b^2*d^8*x^(2/3)*log((e*x^(1/3) + d)^n)^2 -
60*(b^2*d^7*e*n*log(c) + a*b*d^7*e*n)*x - 20*(6*b^2*d^5*e^3*n*x^(5/3)*log(e*x^(1/3) + d) - 6*b^2*d^6*e^2*n*x^
(4/3) + 3*b^2*d^7*e*n*x - 2*(b^2*d^5*e^3*n*x*log(x) - 3*b^2*d^8*log(c) - 3*a*b*d^8)*x^(2/3))*log((e*x^(1/3) +
d)^n) - 60*(b^2*d^8*log(c)^2 + 2*a*b*d^8*log(c) + a^2*d^8)*x^(2/3) + 4*(9*b^2*e^8*n^2*x^3 + 5*b^2*d^3*e^5*n^2*
x^2*log(x) - 15*b^2*d^3*e^5*n^2*x^2 + 30*(b^2*d^6*e^2*n*log(c) + a*b*d^6*e^2*n)*x)*x^(1/3) - 60*(b^2*d^3*e^5*n
^2*x^3 + b^2*d^6*e^2*n^2*x^2)/x^(2/3))/(d^8*x^(5/3))

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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{1}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x^{\frac{1}{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^(1/3))^n))^2/x^2,x, algorithm="fricas")

[Out]

integral((b^2*log((e*x^(1/3) + d)^n*c)^2 + 2*a*b*log((e*x^(1/3) + 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**(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 ({\left (e x^{\frac{1}{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^(1/3))^n))^2/x^2,x, algorithm="giac")

[Out]

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