\(\int \frac {(a+b \log (c (d+\frac {e}{x^{2/3}})^n))^2}{x^3} \, dx\) [519]

Optimal result

Integrand size = 24, antiderivative size = 276 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2} \]

[Out]

3/4*b^2*d*n^2*(d+e/x^(2/3))^2/e^3-1/9*b^2*n^2*(d+e/x^(2/3))^3/e^3-3*b^2*d^2*n^2/e^2/x^(2/3)+1/2*b^2*d^3*n^2*ln
(d+e/x^(2/3))^2/e^3+3*b*d^2*n*(d+e/x^(2/3))*(a+b*ln(c*(d+e/x^(2/3))^n))/e^3-3/2*b*d*n*(d+e/x^(2/3))^2*(a+b*ln(
c*(d+e/x^(2/3))^n))/e^3+1/3*b*n*(d+e/x^(2/3))^3*(a+b*ln(c*(d+e/x^(2/3))^n))/e^3-b*d^3*n*ln(d+e/x^(2/3))*(a+b*l
n(c*(d+e/x^(2/3))^n))/e^3-1/2*(a+b*ln(c*(d+e/x^(2/3))^n))^2/x^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2445, 2458, 45, 2372, 12, 14, 2338} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3} \]

[In]

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

[Out]

(3*b^2*d*n^2*(d + e/x^(2/3))^2)/(4*e^3) - (b^2*n^2*(d + e/x^(2/3))^3)/(9*e^3) - (3*b^2*d^2*n^2)/(e^2*x^(2/3))
+ (b^2*d^3*n^2*Log[d + e/x^(2/3)]^2)/(2*e^3) + (3*b*d^2*n*(d + e/x^(2/3))*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^
3 - (3*b*d*n*(d + e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*e^3) + (b*n*(d + e/x^(2/3))^3*(a + b*Log[c
*(d + e/x^(2/3))^n]))/(3*e^3) - (b*d^3*n*Log[d + e/x^(2/3)]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^3 - (a + b*Log
[c*(d + e/x^(2/3))^n])^2/(2*x^2)

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 45

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 2338

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 2372

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]}, Dist[a + b*Log[c*x^n], u, 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 2445

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 2458

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 2504

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*} \text {integral}& = -\left (\frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac {1}{x^{2/3}}\right )\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+(b e n) \text {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}{x^{2/3}}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+(b n) \text {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}{x^{2/3}}\right ) \\ & = \frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\left (b^2 n^2\right ) \text {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}{x^{2/3}}\right ) \\ & = \frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 n^2\right ) \text {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}{x^{2/3}}\right )}{6 e^3} \\ & = \frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 n^2\right ) \text {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}{x^{2/3}}\right )}{6 e^3} \\ & = \frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2}+\frac {\left (b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{e^3} \\ & = \frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{4 e^3}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^3}-\frac {3 b^2 d^2 n^2}{e^2 x^{2/3}}+\frac {b^2 d^3 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{2 e^3}+\frac {3 b d^2 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^3}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^3}-\frac {b d^3 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^3}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 x^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.38 (sec) , antiderivative size = 691, normalized size of antiderivative = 2.50 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {-18 e^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+b n \left (9 b d n x^{2/3} \left (e \left (e-2 d x^{2/3}\right )+2 d^2 x^{4/3} \log \left (d+\frac {e}{x^{2/3}}\right )\right )-2 b n \left (e \left (2 e^2-3 d e x^{2/3}+6 d^2 x^{4/3}\right )-6 d^3 x^2 \log \left (d+\frac {e}{x^{2/3}}\right )\right )+12 e^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-18 d e^2 x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+36 d^2 x^{4/3} \left (e (a-b n)+b \left (e+d x^{2/3}\right ) \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )-36 d^3 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )-36 d^3 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )-36 d^3 x^2 \left (\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )+b n \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right )\right )+18 b d^3 n x^2 \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )\right )+18 b d^3 n x^2 \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )-4 \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )\right )}{36 e^3 x^2} \]

[In]

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

[Out]

(-18*e^3*(a + b*Log[c*(d + e/x^(2/3))^n])^2 + b*n*(9*b*d*n*x^(2/3)*(e*(e - 2*d*x^(2/3)) + 2*d^2*x^(4/3)*Log[d
+ e/x^(2/3)]) - 2*b*n*(e*(2*e^2 - 3*d*e*x^(2/3) + 6*d^2*x^(4/3)) - 6*d^3*x^2*Log[d + e/x^(2/3)]) + 12*e^3*(a +
b*Log[c*(d + e/x^(2/3))^n]) - 18*d*e^2*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n]) + 36*d^2*x^(4/3)*(e*(a - b*n)
+ b*(e + d*x^(2/3))*Log[c*(d + e/x^(2/3))^n]) - 36*d^3*x^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqr
t[-d]*x^(1/3)] - 36*d^3*x^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] - 36*d^3*x^2*((a
+ b*Log[c*(d + e/x^(2/3))^n])*Log[-(e/(d*x^(2/3)))] + b*n*PolyLog[2, 1 + e/(d*x^(2/3))]) + 18*b*d^3*n*x^2*(Log
[Sqrt[e] - Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 2*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*
Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]]) - 4*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] + 2*PolyLog[2, 1/2 - (Sqrt[-d]
*x^(1/3))/(2*Sqrt[e])]) + 18*b*d^3*n*x^2*(Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 2
*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] - 4*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])]) + 2*PolyLog[2, (1 + (Sqrt[-
d]*x^(1/3))/Sqrt[e])/2] - 4*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]])))/(36*e^3*x^2)

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{2}}{x^{3}}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {4 \, b^{2} e^{3} n^{2} + 18 \, b^{2} e^{3} \log \left (c\right )^{2} - 12 \, a b e^{3} n + 18 \, a^{2} e^{3} + 18 \, {\left (b^{2} d^{3} n^{2} x^{2} + b^{2} e^{3} n^{2}\right )} \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{2} - 12 \, {\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} \log \left (c\right ) - 6 \, {\left (6 \, b^{2} d^{2} e n^{2} x^{\frac {4}{3}} - 3 \, b^{2} d e^{2} n^{2} x^{\frac {2}{3}} + 2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + {\left (11 \, b^{2} d^{3} n^{2} - 6 \, a b d^{3} n\right )} x^{2} - 6 \, {\left (b^{2} d^{3} n x^{2} + b^{2} e^{3} n\right )} \log \left (c\right )\right )} \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) - 3 \, {\left (5 \, b^{2} d e^{2} n^{2} - 6 \, b^{2} d e^{2} n \log \left (c\right ) - 6 \, a b d e^{2} n\right )} x^{\frac {2}{3}} - 6 \, {\left (6 \, b^{2} d^{2} e n x \log \left (c\right ) - {\left (11 \, b^{2} d^{2} e n^{2} - 6 \, a b d^{2} e n\right )} x\right )} x^{\frac {1}{3}}}{36 \, e^{3} x^{2}} \]

[In]

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

[Out]

-1/36*(4*b^2*e^3*n^2 + 18*b^2*e^3*log(c)^2 - 12*a*b*e^3*n + 18*a^2*e^3 + 18*(b^2*d^3*n^2*x^2 + b^2*e^3*n^2)*lo
g((d*x + e*x^(1/3))/x)^2 - 12*(b^2*e^3*n - 3*a*b*e^3)*log(c) - 6*(6*b^2*d^2*e*n^2*x^(4/3) - 3*b^2*d*e^2*n^2*x^
(2/3) + 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^2 - 6*(b^2*d^3*n*x^2 + b^2*e^3*n)*log(c
))*log((d*x + e*x^(1/3))/x) - 3*(5*b^2*d*e^2*n^2 - 6*b^2*d*e^2*n*log(c) - 6*a*b*d*e^2*n)*x^(2/3) - 6*(6*b^2*d^
2*e*n*x*log(c) - (11*b^2*d^2*e*n^2 - 6*a*b*d^2*e*n)*x)*x^(1/3))/(e^3*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {1}{6} \, a b e n {\left (\frac {6 \, d^{3} \log \left (d x^{\frac {2}{3}} + e\right )}{e^{4}} - \frac {6 \, d^{3} \log \left (x^{\frac {2}{3}}\right )}{e^{4}} - \frac {6 \, d^{2} x^{\frac {4}{3}} - 3 \, d e x^{\frac {2}{3}} + 2 \, e^{2}}{e^{3} x^{2}}\right )} - \frac {1}{36} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (d x^{\frac {2}{3}} + e\right )}{e^{4}} - \frac {6 \, d^{3} \log \left (x^{\frac {2}{3}}\right )}{e^{4}} - \frac {6 \, d^{2} x^{\frac {4}{3}} - 3 \, d e x^{\frac {2}{3}} + 2 \, e^{2}}{e^{3} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) - \frac {{\left (18 \, d^{3} x^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 8 \, d^{3} x^{2} \log \left (x\right )^{2} - 44 \, d^{3} x^{2} \log \left (x\right ) - 66 \, d^{2} e x^{\frac {4}{3}} + 15 \, d e^{2} x^{\frac {2}{3}} - 4 \, e^{3} - 6 \, {\left (4 \, d^{3} x^{2} \log \left (x\right ) - 11 \, d^{3} x^{2}\right )} \log \left (d x^{\frac {2}{3}} + e\right )\right )} n^{2}}{e^{3} x^{2}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]

[In]

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

[Out]

-1/6*a*b*e*n*(6*d^3*log(d*x^(2/3) + e)/e^4 - 6*d^3*log(x^(2/3))/e^4 - (6*d^2*x^(4/3) - 3*d*e*x^(2/3) + 2*e^2)/
(e^3*x^2)) - 1/36*(6*e*n*(6*d^3*log(d*x^(2/3) + e)/e^4 - 6*d^3*log(x^(2/3))/e^4 - (6*d^2*x^(4/3) - 3*d*e*x^(2/
3) + 2*e^2)/(e^3*x^2))*log(c*(d + e/x^(2/3))^n) - (18*d^3*x^2*log(d*x^(2/3) + e)^2 + 8*d^3*x^2*log(x)^2 - 44*d
^3*x^2*log(x) - 66*d^2*e*x^(4/3) + 15*d*e^2*x^(2/3) - 4*e^3 - 6*(4*d^3*x^2*log(x) - 11*d^3*x^2)*log(d*x^(2/3)
+ e))*n^2/(e^3*x^2))*b^2 - 1/2*b^2*log(c*(d + e/x^(2/3))^n)^2/x^2 - a*b*log(c*(d + e/x^(2/3))^n)/x^2 - 1/2*a^2
/x^2

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.65 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{4\,e}}{x^{4/3}}-{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^2\,\left (\frac {b^2}{2\,x^2}+\frac {b^2\,d^3}{2\,e^3}\right )-\frac {\frac {a^2}{2}-\frac {a\,b\,n}{3}+\frac {b^2\,n^2}{9}}{x^2}-\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\,\left (\frac {b\,\left (3\,a-b\,n\right )}{3\,x^2}-\frac {\frac {b\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {3\,a\,b\,d}{2\,e}}{x^{4/3}}+\frac {d\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}\right )}{e\,x^{2/3}}\right )-\frac {\frac {d\,\left (\frac {d\,\left (\frac {3\,a^2}{2}-a\,b\,n+\frac {b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{e^2}}{x^{2/3}}+\frac {\ln \left (d+\frac {e}{x^{2/3}}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{6\,e^3} \]

[In]

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

[Out]

((d*((3*a^2)/2 + (b^2*n^2)/3 - a*b*n))/(2*e) - (d*(3*a^2 - b^2*n^2))/(4*e))/x^(4/3) - log(c*(d + e/x^(2/3))^n)
^2*(b^2/(2*x^2) + (b^2*d^3)/(2*e^3)) - (a^2/2 + (b^2*n^2)/9 - (a*b*n)/3)/x^2 - log(c*(d + e/x^(2/3))^n)*((b*(3
*a - b*n))/(3*x^2) - ((b*d*(3*a - b*n))/(2*e) - (3*a*b*d)/(2*e))/x^(4/3) + (d*((b*d*(3*a - b*n))/e - (3*a*b*d)
/e))/(e*x^(2/3))) - ((d*((d*((3*a^2)/2 + (b^2*n^2)/3 - a*b*n))/e - (d*(3*a^2 - b^2*n^2))/(2*e)))/e + (b^2*d^2*
n^2)/e^2)/x^(2/3) + (log(d + e/x^(2/3))*(11*b^2*d^3*n^2 - 6*a*b*d^3*n))/(6*e^3)