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

Optimal. Leaf size=231 \[ \frac {2 b e^3 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}+\frac {2 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d x^{2/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x}-\frac {2 b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}+\frac {b^2 e^3 n^2 \log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}-\frac {b^2 e^2 n^2}{d^2 \sqrt [3]{x}} \]

[Out]

-b^2*e^2*n^2/d^2/x^(1/3)+b^2*e^3*n^2*ln(d+e*x^(1/3))/d^3-b*e*n*(a+b*ln(c*(d+e*x^(1/3))^n))/d/x^(2/3)+2*b*e^2*n
*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/d^3/x^(1/3)+2*b*e^3*n*ln(1-d/(d+e*x^(1/3)))*(a+b*ln(c*(d+e*x^(1/3))
^n))/d^3-(a+b*ln(c*(d+e*x^(1/3))^n))^2/x-b^2*e^3*n^2*ln(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 = 0.50, antiderivative size = 253, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 31

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

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])

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 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 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 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 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 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 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 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 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])

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*}

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Mathematica [A]  time = 0.23, size = 274, normalized size = 1.19 \[ 3 \left (\frac {2}{3} b e n \left (-\frac {e^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 b d^3 n}+\frac {e^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}+\frac {e \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^2 \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 d x^{2/3}}+\frac {b e^2 n \text {Li}_2\left (\frac {d+e \sqrt [3]{x}}{d}\right )}{d^3}-\frac {b e^2 n \left (\frac {\log (x)}{3 d}-\frac {\log \left (d+e \sqrt [3]{x}\right )}{d}\right )}{d^2}-\frac {b e n \left (-\frac {e \log \left (d+e \sqrt [3]{x}\right )}{d^2}+\frac {e \log (x)}{3 d^2}+\frac {1}{d \sqrt [3]{x}}\right )}{2 d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*(-1/3*(a + b*Log[c*(d + e*x^(1/3))^n])^2/x + (2*b*e*n*(-1/2*(a + b*Log[c*(d + e*x^(1/3))^n])/(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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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a^{2}}{x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{2}}\,{d x} \]

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)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{\frac {1}{3}}+d \right )^{n}\right )+a \right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (\log \left (\frac {e x^{\frac {1}{3}}}{d} + 1\right ) \log \left (x^{\frac {1}{3}}\right ) + {\rm Li}_2\left (-\frac {e x^{\frac {1}{3}}}{d}\right )\right )} b^{2} e^{3} n^{2}}{d^{3}} - \frac {{\left (2 \, a b e^{3} n - {\left (3 \, e^{3} n^{2} - 2 \, e^{3} n \log \relax (c)\right )} b^{2}\right )} \log \left (e x^{\frac {1}{3}} + d\right )}{d^{3}} + \frac {2 \, {\left (b^{2} e^{3} n \log \relax (c) + a b e^{3} n\right )} \log \left (x^{\frac {1}{3}}\right )}{d^{3}} + \frac {b^{2} e^{6} n^{2} x - b^{2} d^{3} e^{3} n^{2} \log \relax (x)}{d^{6}} - \frac {12 \, b^{2} e^{8} n^{2} x^{\frac {5}{3}} - 15 \, b^{2} d e^{7} n^{2} x^{\frac {4}{3}} + 20 \, b^{2} d^{2} e^{6} n^{2} x - 40 \, b^{2} d^{3} e^{5} n^{2} x^{\frac {2}{3}} + 100 \, b^{2} d^{4} e^{4} n^{2} x^{\frac {1}{3}} + 20 \, {\left (b^{2} d^{3} e^{5} n^{2} x^{\frac {2}{3}} - 2 \, b^{2} d^{4} e^{4} n^{2} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3}}\right )}{20 \, d^{8}} + \frac {60 \, b^{2} d^{5} e^{3} n^{2} x^{\frac {5}{3}} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 45 \, b^{2} d e^{7} n^{2} x^{3} - 40 \, b^{2} d^{4} e^{4} n^{2} x^{2} \log \relax (x) + 300 \, b^{2} d^{4} e^{4} n^{2} x^{2} - 60 \, b^{2} d^{8} x^{\frac {2}{3}} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n}\right )^{2} - 60 \, {\left (b^{2} d^{7} e n \log \relax (c) + a b d^{7} e n\right )} x - 20 \, {\left (6 \, b^{2} d^{5} e^{3} n x^{\frac {5}{3}} \log \left (e x^{\frac {1}{3}} + d\right ) - 6 \, b^{2} d^{6} e^{2} n x^{\frac {4}{3}} + 3 \, b^{2} d^{7} e n x - 2 \, {\left (b^{2} d^{5} e^{3} n x \log \relax (x) - 3 \, b^{2} d^{8} \log \relax (c) - 3 \, a b d^{8}\right )} x^{\frac {2}{3}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n}\right ) - 60 \, {\left (b^{2} d^{8} \log \relax (c)^{2} + 2 \, a b d^{8} \log \relax (c) + a^{2} d^{8}\right )} x^{\frac {2}{3}} + 4 \, {\left (9 \, b^{2} e^{8} n^{2} x^{3} + 5 \, b^{2} d^{3} e^{5} n^{2} x^{2} \log \relax (x) - 15 \, b^{2} d^{3} e^{5} n^{2} x^{2} + 30 \, {\left (b^{2} d^{6} e^{2} n \log \relax (c) + a b d^{6} e^{2} n\right )} x\right )} x^{\frac {1}{3}} - \frac {60 \, {\left (b^{2} d^{3} e^{5} n^{2} x^{3} + b^{2} d^{6} e^{2} n^{2} x^{2}\right )}}{x^{\frac {2}{3}}}}{60 \, d^{8} x^{\frac {5}{3}}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^2}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}}{x^{2}}\, dx \]

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]

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

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