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

Optimal. Leaf size=238 \[ -\frac{b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{d}{d+e x^{2/3}}\right )}{d^3}+\frac{b e^3 n \log \left (1-\frac{d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}+\frac{b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}-\frac{b^2 e^2 n^2}{2 d^2 x^{2/3}}+\frac{b^2 e^3 n^2 \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.500007, antiderivative size = 261, 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} \[ \frac{b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )}{d^3}-\frac{e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}+\frac{b e^3 n \log \left (-\frac{e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}+\frac{b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3 x^{2/3}}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d x^{4/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 x^2}-\frac{b^2 e^2 n^2}{2 d^2 x^{2/3}}+\frac{b^2 e^3 n^2 \log \left (d+e x^{2/3}\right )}{2 d^3}-\frac{b^2 e^3 n^2 \log (x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.303298, size = 264, normalized size = 1.11 \[ -\frac{\frac{e x^{2/3} \left (-6 b e^2 n x^{4/3} \left (b n \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )+\log \left (-\frac{e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )\right )+3 b d^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+3 e^2 x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-6 b d e n x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-2 b^2 e^2 n^2 x^{4/3} \left (3 \log \left (d+e x^{2/3}\right )-2 \log (x)\right )+b^2 e n^2 x^{2/3} \left (-3 e x^{2/3} \log \left (d+e x^{2/3}\right )+3 d+2 e x^{2/3} \log (x)\right )\right )}{d^3}+3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*(a + b*Log[c*(d + e*x^(2/3))^n])^2 + (e*x^(2/3)*(3*b*d^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]) - 6*b*d*e*n*x^
(2/3)*(a + b*Log[c*(d + e*x^(2/3))^n]) + 3*e^2*x^(4/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2 - 2*b^2*e^2*n^2*x^(4
/3)*(3*Log[d + e*x^(2/3)] - 2*Log[x]) + b^2*e*n^2*x^(2/3)*(3*d - 3*e*x^(2/3)*Log[d + e*x^(2/3)] + 2*e*x^(2/3)*
Log[x]) - 6*b*e^2*n*x^(4/3)*((a + b*Log[c*(d + e*x^(2/3))^n])*Log[-((e*x^(2/3))/d)] + b*n*PolyLog[2, 1 + (e*x^
(2/3))/d])))/d^3)/(6*x^2)

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Maple [F]  time = 0.338, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n}\right )^{2}}{2 \, x^{2}} + \int \frac{3 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x + 2 \,{\left (b^{2} e n x + 3 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x + 3 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{\frac{1}{3}}\right )} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n}\right ) + 3 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x^{\frac{1}{3}}}{3 \,{\left (e x^{4} + d x^{\frac{10}{3}}\right )}}\,{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^3,x, algorithm="maxima")

[Out]

-1/2*b^2*log((e*x^(2/3) + d)^n)^2/x^2 + integrate(1/3*(3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x + 2*(b^2*
e*n*x + 3*(b^2*e*log(c) + a*b*e)*x + 3*(b^2*d*log(c) + a*b*d)*x^(1/3))*log((e*x^(2/3) + d)^n) + 3*(b^2*d*log(c
)^2 + 2*a*b*d*log(c) + a^2*d)*x^(1/3))/(e*x^4 + d*x^(10/3)), x)

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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^{3}}, 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^3,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^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**(2/3))**n))**2/x**3,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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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