\(\int \frac {(a+b \log (c (d+e \sqrt [3]{x})^n))^2}{x} \, dx\) [453]

Optimal result

Integrand size = 24, antiderivative size = 93 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x} \, dx=3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )-6 b^2 n^2 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt [3]{x}}{d}\right ) \] Output:

3*(a+b*ln(c*(d+e*x^(1/3))^n))^2*ln(-e*x^(1/3)/d)+6*b*n*(a+b*ln(c*(d+e*x^(1 
/3))^n))*polylog(2,1+e*x^(1/3)/d)-6*b^2*n^2*polylog(3,1+e*x^(1/3)/d)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(195\) vs. \(2(93)=186\).

Time = 0.13 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.10 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x} \, dx=\left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log (x)+2 b n \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (\left (\log \left (d+e \sqrt [3]{x}\right )-\log \left (1+\frac {e \sqrt [3]{x}}{d}\right )\right ) \log (x)-3 \operatorname {PolyLog}\left (2,-\frac {e \sqrt [3]{x}}{d}\right )\right )+3 b^2 n^2 \left (\log ^2\left (d+e \sqrt [3]{x}\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+2 \log \left (d+e \sqrt [3]{x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt [3]{x}}{d}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.77 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2904, 2843, 2881, 2821, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2904

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

\(\displaystyle 3 \left (\log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d+e \sqrt [3]{x}}{d}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )-\operatorname {PolyLog}\left (2,\frac {d+e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )\right )\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle 3 \left (\log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-2 b n \left (b n \operatorname {PolyLog}\left (3,\frac {d+e \sqrt [3]{x}}{d}\right )-\operatorname {PolyLog}\left (2,\frac {d+e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )\right )\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2843

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

rule 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* 
((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, 
 f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
 

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{2}}{x}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \] Input:

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

Output:

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, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \] Input:

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

Output:

b^2*log((e*x^(1/3) + d)^n)^2*log(x) + 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*log(x) - 3*(b^2*e*log(c) + a*b*e)* 
x - 3*(b^2*d*log(c) + a*b*d)*x^(2/3))*log((e*x^(1/3) + d)^n) + 3*(b^2*d*lo 
g(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^(2/3))/(e*x^2 + d*x^(5/3)), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^2}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x} \, dx=\frac {\left (\int \frac {{\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right )}^{2}}{x^{\frac {4}{3}} e +d x}d x \right ) b^{2} d n +2 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right )}{x^{\frac {4}{3}} e +d x}d x \right ) a b d n +{\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right )}^{3} b^{2}+3 {\mathrm {log}\left (\left (x^{\frac {1}{3}} e +d \right )^{n} c \right )}^{2} a b +\mathrm {log}\left (x \right ) a^{2} n}{n} \] Input:

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

Output:

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