\(\int (a+b \log (c (d+\frac {e}{x^{2/3}})^n))^2 \, dx\) [522]

Optimal result

Integrand size = 20, antiderivative size = 309 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\frac {4 a b e n \sqrt [3]{x}}{d}+\frac {8 b^2 e^{3/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{d^{3/2}}+\frac {4 i b^2 e^{3/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{d^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {4 b^2 e n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}-\frac {4 b e^{3/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{d^{3/2}}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {4 i b^2 e^{3/2} n^2 \operatorname {PolyLog}\left (2,-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}} \] Output:

4*a*b*e*n*x^(1/3)/d+8*b^2*e^(3/2)*n^2*arctan(d^(1/2)*x^(1/3)/e^(1/2))/d^(3 
/2)+4*I*b^2*e^(3/2)*n^2*arctan(d^(1/2)*x^(1/3)/e^(1/2))^2/d^(3/2)-8*b^2*e^ 
(3/2)*n^2*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)+4*b^2*e*n*x^(1/3)*ln(c*(d+e/x^(2/3))^n)/d-4*b*e^(3/2)* 
n*arctan(d^(1/2)*x^(1/3)/e^(1/2))*(a+b*ln(c*(d+e/x^(2/3))^n))/d^(3/2)+x*(a 
+b*ln(c*(d+e/x^(2/3))^n))^2+4*I*b^2*e^(3/2)*n^2*polylog(2,-1+2*e^(1/2)/(e^ 
(1/2)-I*d^(1/2)*x^(1/3)))/d^(3/2)
 

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.69 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+b e n \left (\frac {4 a \sqrt [3]{x}}{d}-\frac {8 b \sqrt {e} n \arctan \left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {4 b \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{d}-\frac {2 \sqrt {e} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {2 \sqrt {e} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {b \sqrt {e} n \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )\right )}{(-d)^{3/2}}+\frac {b d \sqrt {e} n \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )-4 \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-4 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{(-d)^{5/2}}\right ) \] Input:

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

Output:

x*(a + b*Log[c*(d + e/x^(2/3))^n])^2 + b*e*n*((4*a*x^(1/3))/d - (8*b*Sqrt[ 
e]*n*ArcTan[Sqrt[e]/(Sqrt[d]*x^(1/3))])/d^(3/2) + (4*b*x^(1/3)*Log[c*(d + 
e/x^(2/3))^n])/d - (2*Sqrt[e]*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] 
 - Sqrt[-d]*x^(1/3)])/(-d)^(3/2) + (2*Sqrt[e]*(a + b*Log[c*(d + e/x^(2/3)) 
^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)])/(-d)^(3/2) + (b*Sqrt[e]*n*(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])]))/(-d)^(3/2) + (b*d*Sqrt[e]*n*(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*S 
qrt[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]])) 
/(-d)^(5/2))
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.

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

\(\Big \downarrow \) 2901

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

\(\Big \downarrow \) 2907

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

\(\Big \downarrow \) 2921

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

3*((x*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/3 + (4*b*e*n*((a*x^(1/3))/d + (2 
*b*Sqrt[e]*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/d^(3/2) + (I*b*Sqrt[e]*n*A 
rcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/d^(3/2) - (2*b*Sqrt[e]*n*ArcTan[(Sqrt[ 
d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d 
^(3/2) + (b*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/d - (Sqrt[e]*ArcTan[(Sqrt[d] 
*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/d^(3/2) + (I*b*Sqrt[e 
]*n*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(3/2)))/ 
3)
 

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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \, dx=\frac {-12 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) a b e n +24 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) b^{2} e \,n^{2}+3 x^{\frac {1}{3}} {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{2} d e +12 x^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d e n +12 x^{\frac {1}{3}} a b d e n -\left (\int \frac {{\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2}}{x^{\frac {2}{3}}}d x \right ) b^{2} d e +3 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{2} d^{2} x +6 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a b \,d^{2} x +3 a^{2} d^{2} x}{3 d^{2}} \] Input:

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

Output:

( - 12*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*a*b*e*n + 24*s 
qrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*b**2*e*n**2 + 3*x**(1/ 
3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**2*d*e + 12*x**(1/3)*log 
(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d*e*n + 12*x**(1/3)*a*b*d*e*n 
- int(log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2/x**(2/3),x)*b**2*d*e + 
3*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**2*d**2*x + 6*log(((x**(2 
/3)*d + e)**n*c)/x**((2*n)/3))*a*b*d**2*x + 3*a**2*d**2*x)/(3*d**2)