\(\int (a+b \log (c (d+\frac {e}{x^{2/3}})^n))^3 \, dx\) [529]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=-\frac {24 i b^3 e^{3/2} n^3 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{3/2}}+\frac {48 b^3 e^{3/2} n^3 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {24 b^2 e^{3/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {24 i b^3 e^{3/2} n^3 \operatorname {PolyLog}\left (2,-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac {2 b e^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{\left (e+d x^{2/3}\right ) x^{2/3}},x\right )}{d} \] Output:

-24*I*b^3*e^(3/2)*n^3*arctan(d^(1/2)*x^(1/3)/e^(1/2))^2/d^(3/2)+48*b^3*e^( 
3/2)*n^3*arctan(d^(1/2)*x^(1/3)/e^(1/2))*ln(2-2*e^(1/2)/(e^(1/2)-I*d^(1/2) 
*x^(1/3)))/d^(3/2)+24*b^2*e^(3/2)*n^2*arctan(d^(1/2)*x^(1/3)/e^(1/2))*(a+b 
*ln(c*(d+e/x^(2/3))^n))/d^(3/2)+6*b*e*n*x^(1/3)*(a+b*ln(c*(d+e/x^(2/3))^n) 
)^2/d+x*(a+b*ln(c*(d+e/x^(2/3))^n))^3-24*I*b^3*e^(3/2)*n^3*polylog(2,-1+2* 
e^(1/2)/(e^(1/2)-I*d^(1/2)*x^(1/3)))/d^(3/2)-2*b*e^2*n*Defer(Int)((a+b*ln( 
c*(d+e/x^(2/3))^n))^2/(e+d*x^(2/3))/x^(2/3),x)/d
 

Mathematica [N/A]

Not integrable

Time = 6.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {2901, 2907, 2921, 2009}

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.

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

\(\Big \downarrow \) 2901

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

\(\Big \downarrow \) 2907

\(\displaystyle 3 \left (2 b e n \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d+\frac {e}{x^{2/3}}}d\sqrt [3]{x}+\frac {1}{3} x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2921

\(\displaystyle 3 \left (2 b e n \int \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}-\frac {e \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d \left (x^{2/3} d+e\right )}\right )d\sqrt [3]{x}+\frac {1}{3} x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {1}{3} x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+2 b e n \left (-\frac {e \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^{2/3} d+e}d\sqrt [3]{x}}{d}+\frac {4 b \sqrt {e} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+\frac {\sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}-\frac {4 i b^2 \sqrt {e} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{3/2}}+\frac {8 b^2 \sqrt {e} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac {4 i b^2 \sqrt {e} n^2 \operatorname {PolyLog}\left (2,\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right )}{d^{3/2}}\right )\right )\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2901

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k   Subst[Int[x^(k - 1)*(a + b*Log[c* 
(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, 
 x] && FractionQ[n]
 

rule 2907

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

rule 2921

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + 
 (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{t = ExpandIntegrand[(a + b*Log[ 
c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, 
 b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && Integ 
erQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 
0] && LtQ[r, 0]))
 

Maple [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 25.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3 \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 451, normalized size of antiderivative = 22.55 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\frac {-18 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) a^{2} b e n +72 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) a \,b^{2} e \,n^{2}+3 x^{\frac {1}{3}} {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{3} b^{3} d e +18 x^{\frac {1}{3}} {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{3} d e n +36 x^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a \,b^{2} d e n +18 x^{\frac {1}{3}} a^{2} b d e n -\left (\int \frac {{\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{3}}{x^{\frac {2}{3}} e +x^{\frac {4}{3}} d}d x \right ) b^{3} d \,e^{2}-12 \left (\int \frac {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}{x^{\frac {2}{3}} e +x^{\frac {4}{3}} d}d x \right ) a \,b^{2} d \,e^{2} n +24 \left (\int \frac {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}{x^{\frac {2}{3}} e +x^{\frac {4}{3}} d}d x \right ) b^{3} d \,e^{2} n^{2}-\left (\int \frac {x^{\frac {1}{3}} {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{3}}{x^{\frac {1}{3}} e +d x}d x \right ) b^{3} d^{2} e +3 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{3} b^{3} d^{2} x +9 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} a \,b^{2} d^{2} x +9 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a^{2} b \,d^{2} x +3 a^{3} d^{2} x}{3 d^{2}} \] Input:

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

Output:

( - 18*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*a**2*b*e*n + 7 
2*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*a*b**2*e*n**2 + 3*x 
**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**3*b**3*d*e + 18*x**(1/3 
)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**3*d*e*n + 36*x**(1/3)*lo 
g(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a*b**2*d*e*n + 18*x**(1/3)*a**2*b* 
d*e*n - int(log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**3/(x**(2/3)*e + x** 
(1/3)*d*x),x)*b**3*d*e**2 - 12*int(log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3 
))/(x**(2/3)*e + x**(1/3)*d*x),x)*a*b**2*d*e**2*n + 24*int(log(((x**(2/3)* 
d + e)**n*c)/x**((2*n)/3))/(x**(2/3)*e + x**(1/3)*d*x),x)*b**3*d*e**2*n**2 
 - int((x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**3)/(x**(1/3)*e 
 + d*x),x)*b**3*d**2*e + 3*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**3*b* 
*3*d**2*x + 9*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*a*b**2*d**2*x + 
 9*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a**2*b*d**2*x + 3*a**3*d**2*x 
)/(3*d**2)