\(\int (a+b \log (c (d+e x^{2/3})^n))^3 \, dx\) [486]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=-\frac {32 a b^2 d n^2 \sqrt [3]{x}}{e}+\frac {208 b^3 d n^3 \sqrt [3]{x}}{3 e}-\frac {16}{9} b^3 n^3 x-\frac {208 b^3 d^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {32 i b^3 d^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}+\frac {64 b^3 d^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {32 b^3 d n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}+\frac {8}{3} b^2 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {32 b^2 d^{3/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+\frac {6 b d n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e}-2 b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {32 i b^3 d^{3/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {2 b d^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^{2/3}},x\right )}{e} \] Output:

-32*a*b^2*d*n^2*x^(1/3)/e+208/3*b^3*d*n^3*x^(1/3)/e-16/9*b^3*n^3*x-208/3*b 
^3*d^(3/2)*n^3*arctan(e^(1/2)*x^(1/3)/d^(1/2))/e^(3/2)+32*I*b^3*d^(3/2)*n^ 
3*arctan(e^(1/2)*x^(1/3)/d^(1/2))^2/e^(3/2)+64*b^3*d^(3/2)*n^3*arctan(e^(1 
/2)*x^(1/3)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*e^(1/2)*x^(1/3)))/e^(3/2)-32* 
b^3*d*n^2*x^(1/3)*ln(c*(d+e*x^(2/3))^n)/e+8/3*b^2*n^2*x*(a+b*ln(c*(d+e*x^( 
2/3))^n))+32*b^2*d^(3/2)*n^2*arctan(e^(1/2)*x^(1/3)/d^(1/2))*(a+b*ln(c*(d+ 
e*x^(2/3))^n))/e^(3/2)+6*b*d*n*x^(1/3)*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e-2*b 
*n*x*(a+b*ln(c*(d+e*x^(2/3))^n))^2+x*(a+b*ln(c*(d+e*x^(2/3))^n))^3+32*I*b^ 
3*d^(3/2)*n^3*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*e^(1/2)*x^(1/3)))/e^(3/2)-2 
*b*d^2*n*Defer(Int)((a+b*ln(c*(d+e*x^(2/3))^n))^2/(d+e*x^(2/3))/x^(2/3),x) 
/e
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1299\) vs. \(2(486)=972\).

Time = 9.40 (sec) , antiderivative size = 1299, normalized size of antiderivative = 64.95 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(6*b*d*n*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n]) 
^2)/e - (6*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a - b*n*Log[d + 
e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e^(3/2) + 3*b*n*x*Log[d + e*x^ 
(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 + x*(a 
- b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*(a - 2*b*n - b*n* 
Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n]) + (b^2*n^2*x^(1/3)*(a - b 
*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])*((-96*d^(3/2)*ArcSin[S 
qrt[d]/Sqrt[d + e*x^(2/3)]])/(Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e* 
x^(2/3))]) - d*(104 - 48*Log[d + e*x^(2/3)] + 9*Log[d + e*x^(2/3)]^2) + (d 
 + e*x^(2/3))*(8 - 12*Log[d + e*x^(2/3)] + 9*Log[d + e*x^(2/3)]^2) + (36*( 
-d)^(3/2)*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + 
(e*x^(2/3))/d]))/Sqrt[e*x^(2/3)] + (9*d*(2*Log[(1 + Sqrt[-((e*x^(2/3))/d)] 
)/2]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*Log[1 + (e*x^(2/3))/d] + Lo 
g[1 + (e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/Sq 
rt[-((e*x^(2/3))/d)]))/(3*e) + (b^3*n^3*(624*d*e*x^(2/3) - 16*e^2*x^(4/3) 
+ 624*d^(3/2)*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*ArcSin 
[Sqrt[d]/Sqrt[d + e*x^(2/3)]] + 432*d^2*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]* 
HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3)) 
] + 144*d^2*Sqrt[-((e*x^(2/3))/d)]*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 
 288*d*e*x^(2/3)*Log[d + e*x^(2/3)] + 24*e^2*x^(4/3)*Log[d + e*x^(2/3)]...
 

Rubi [N/A]

Not integrable

Time = 1.90 (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, 2926, 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.

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

\(\Big \downarrow \) 2901

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

\(\Big \downarrow \) 2907

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {1}{3} x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-2 b e n \left (\frac {d^2 \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d+e x^{2/3}}d\sqrt [3]{x}}{e^2}-\frac {16 b d^{3/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^{5/2}}-\frac {d \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^2}-\frac {4 b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e}+\frac {x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e}+\frac {16 a b d n \sqrt [3]{x}}{3 e^2}-\frac {16 i b^2 d^{3/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{5/2}}+\frac {104 b^2 d^{3/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{9 e^{5/2}}-\frac {32 b^2 d^{3/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{5/2}}+\frac {16 b^2 d n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^2}-\frac {16 i b^2 d^{3/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{5/2}}-\frac {104 b^2 d n^2 \sqrt [3]{x}}{9 e^2}+\frac {8 b^2 n^2 x}{27 e}\right )\right )\)

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 2926

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b 
*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e 
, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] & 
& IntegerQ[s]
 

Maple [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

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 = 62, normalized size of antiderivative = 3.10 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3} \,d x } \] Input:

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

Output:

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

Sympy [N/A]

Not integrable

Time = 61.68 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int \left (a + b \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}\right )^{3}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Exception raised: ValueError} \] 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 = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 25.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3 \,d x \] 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 = 0.19 (sec) , antiderivative size = 504, normalized size of antiderivative = 25.20 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\frac {-54 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) a^{2} b d n +288 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) a \,b^{2} d \,n^{2}-624 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) b^{3} d \,n^{3}+54 x^{\frac {1}{3}} {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2} b^{3} d e n +108 x^{\frac {1}{3}} \mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a \,b^{2} d e n -288 x^{\frac {1}{3}} \mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{3} d e \,n^{2}+54 x^{\frac {1}{3}} a^{2} b d e n -288 x^{\frac {1}{3}} a \,b^{2} d e \,n^{2}+624 x^{\frac {1}{3}} b^{3} d e \,n^{3}-18 \left (\int \frac {{\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2}}{x^{\frac {2}{3}} d +x^{\frac {4}{3}} e}d x \right ) b^{3} d^{2} e n -36 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}{x^{\frac {2}{3}} d +x^{\frac {4}{3}} e}d x \right ) a \,b^{2} d^{2} e n +96 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}{x^{\frac {2}{3}} d +x^{\frac {4}{3}} e}d x \right ) b^{3} d^{2} e \,n^{2}+9 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{3} b^{3} e^{2} x +27 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2} a \,b^{2} e^{2} x -18 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2} b^{3} e^{2} n x +27 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a^{2} b \,e^{2} x -36 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a \,b^{2} e^{2} n x +24 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{3} e^{2} n^{2} x +9 a^{3} e^{2} x -18 a^{2} b \,e^{2} n x +24 a \,b^{2} e^{2} n^{2} x -16 b^{3} e^{2} n^{3} x}{9 e^{2}} \] Input:

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

Output:

( - 54*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*a**2*b*d*n + 2 
88*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*a*b**2*d*n**2 - 62 
4*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*b**3*d*n**3 + 54*x* 
*(1/3)*log((x**(2/3)*e + d)**n*c)**2*b**3*d*e*n + 108*x**(1/3)*log((x**(2/ 
3)*e + d)**n*c)*a*b**2*d*e*n - 288*x**(1/3)*log((x**(2/3)*e + d)**n*c)*b** 
3*d*e*n**2 + 54*x**(1/3)*a**2*b*d*e*n - 288*x**(1/3)*a*b**2*d*e*n**2 + 624 
*x**(1/3)*b**3*d*e*n**3 - 18*int(log((x**(2/3)*e + d)**n*c)**2/(x**(2/3)*d 
 + x**(1/3)*e*x),x)*b**3*d**2*e*n - 36*int(log((x**(2/3)*e + d)**n*c)/(x** 
(2/3)*d + x**(1/3)*e*x),x)*a*b**2*d**2*e*n + 96*int(log((x**(2/3)*e + d)** 
n*c)/(x**(2/3)*d + x**(1/3)*e*x),x)*b**3*d**2*e*n**2 + 9*log((x**(2/3)*e + 
 d)**n*c)**3*b**3*e**2*x + 27*log((x**(2/3)*e + d)**n*c)**2*a*b**2*e**2*x 
- 18*log((x**(2/3)*e + d)**n*c)**2*b**3*e**2*n*x + 27*log((x**(2/3)*e + d) 
**n*c)*a**2*b*e**2*x - 36*log((x**(2/3)*e + d)**n*c)*a*b**2*e**2*n*x + 24* 
log((x**(2/3)*e + d)**n*c)*b**3*e**2*n**2*x + 9*a**3*e**2*x - 18*a**2*b*e* 
*2*n*x + 24*a*b**2*e**2*n**2*x - 16*b**3*e**2*n**3*x)/(9*e**2)