\(\int \frac {(a+b \log (c (d+e x^{2/3})^n))^2}{x^4} \, dx\) [479]

Optimal result

Integrand size = 24, antiderivative size = 476 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {8 b^2 e^2 n^2}{105 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2}{105 d^3 x}-\frac {568 b^2 e^4 n^2}{315 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{315 d^{9/2}}+\frac {4 i b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 d^{9/2}}+\frac {8 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}+\frac {4 b e^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {4 i b^2 e^{9/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}} \] Output:

-8/105*b^2*e^2*n^2/d^2/x^(5/3)+32/105*b^2*e^3*n^2/d^3/x-568/315*b^2*e^4*n^ 
2/d^4/x^(1/3)-1408/315*b^2*e^(9/2)*n^2*arctan(e^(1/2)*x^(1/3)/d^(1/2))/d^( 
9/2)+4/3*I*b^2*e^(9/2)*n^2*arctan(e^(1/2)*x^(1/3)/d^(1/2))^2/d^(9/2)+8/3*b 
^2*e^(9/2)*n^2*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)))/d^(9/2)-4/21*b*e*n*(a+b*ln(c*(d+e*x^(2/3))^n))/d/x^(7/3)+4/ 
15*b*e^2*n*(a+b*ln(c*(d+e*x^(2/3))^n))/d^2/x^(5/3)-4/9*b*e^3*n*(a+b*ln(c*( 
d+e*x^(2/3))^n))/d^3/x+4/3*b*e^4*n*(a+b*ln(c*(d+e*x^(2/3))^n))/d^4/x^(1/3) 
+4/3*b*e^(9/2)*n*arctan(e^(1/2)*x^(1/3)/d^(1/2))*(a+b*ln(c*(d+e*x^(2/3))^n 
))/d^(9/2)-1/3*(a+b*ln(c*(d+e*x^(2/3))^n))^2/x^3+4/3*I*b^2*e^(9/2)*n^2*pol 
ylog(2,1-2*d^(1/2)/(d^(1/2)+I*e^(1/2)*x^(1/3)))/d^(9/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.63 (sec) , antiderivative size = 473, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {4}{3} b e n \left (-\frac {2 b e^{7/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{d^{9/2}}-\frac {2 b e n \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\frac {e x^{2/3}}{d}\right )}{35 d^2 x^{5/3}}+\frac {2 b e^2 n \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {e x^{2/3}}{d}\right )}{15 d^3 x}-\frac {2 b e^3 n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^{2/3}}{d}\right )}{3 d^4 \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{7 d x^{7/3}}+\frac {e \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^2 x^{5/3}}-\frac {e^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^3 x}+\frac {e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^4 \sqrt [3]{x}}+\frac {e^{7/2} \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{9/2}}+\frac {i b e^{7/2} n \left (\arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (\arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )-2 i \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}{-i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}\right )\right )}{d^{9/2}}\right ) \] Input:

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

Output:

-1/3*(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^3 + (4*b*e*n*((-2*b*e^(7/2)*n*Ar 
cTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/d^(9/2) - (2*b*e*n*Hypergeometric2F1[-5/2 
, 1, -3/2, -((e*x^(2/3))/d)])/(35*d^2*x^(5/3)) + (2*b*e^2*n*Hypergeometric 
2F1[-3/2, 1, -1/2, -((e*x^(2/3))/d)])/(15*d^3*x) - (2*b*e^3*n*Hypergeometr 
ic2F1[-1/2, 1, 1/2, -((e*x^(2/3))/d)])/(3*d^4*x^(1/3)) - (a + b*Log[c*(d + 
 e*x^(2/3))^n])/(7*d*x^(7/3)) + (e*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*d^ 
2*x^(5/3)) - (e^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*d^3*x) + (e^3*(a + 
b*Log[c*(d + e*x^(2/3))^n]))/(d^4*x^(1/3)) + (e^(7/2)*ArcTan[(Sqrt[e]*x^(1 
/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/d^(9/2) + (I*b*e^(7/2)*n*( 
ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]] - (2* 
I)*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))]) + PolyLog[2, (I*Sqrt[d] 
 + Sqrt[e]*x^(1/3))/((-I)*Sqrt[d] + Sqrt[e]*x^(1/3))]))/d^(9/2)))/3
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2908, 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.

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

\(\Big \downarrow \) 2908

\(\displaystyle 3 \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{10/3}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 2907

\(\displaystyle 3 \left (\frac {4}{9} b e n \int \frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{\left (d+e x^{2/3}\right ) x^{8/3}}d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{9 x^3}\right )\)

\(\Big \downarrow \) 2926

\(\displaystyle 3 \left (\frac {4}{9} b e n \int \left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) e^4}{d^4 \left (d+e x^{2/3}\right )}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) e^3}{d^4 x^{2/3}}+\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) e^2}{d^3 x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) e}{d^2 x^2}+\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{d x^{8/3}}\right )d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{9 x^3}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{9 x^3}+\frac {4}{9} b e n \left (\frac {e^{7/2} \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{9/2}}+\frac {e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^4 \sqrt [3]{x}}-\frac {e^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^3 x}+\frac {e \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^2 x^{5/3}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{7 d x^{7/3}}+\frac {i b e^{7/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{9/2}}-\frac {352 b e^{7/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{105 d^{9/2}}+\frac {2 b e^{7/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{9/2}}+\frac {i b e^{7/2} n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{9/2}}-\frac {142 b e^3 n}{105 d^4 \sqrt [3]{x}}+\frac {8 b e^2 n}{35 d^3 x}-\frac {2 b e n}{35 d^2 x^{5/3}}\right )\right )\)

Input:

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

Output:

3*(-1/9*(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^3 + (4*b*e*n*((-2*b*e*n)/(35* 
d^2*x^(5/3)) + (8*b*e^2*n)/(35*d^3*x) - (142*b*e^3*n)/(105*d^4*x^(1/3)) - 
(352*b*e^(7/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(105*d^(9/2)) + (I*b*e 
^(7/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/d^(9/2) + (2*b*e^(7/2)*n*Arc 
Tan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3 
))])/d^(9/2) - (a + b*Log[c*(d + e*x^(2/3))^n])/(7*d*x^(7/3)) + (e*(a + b* 
Log[c*(d + e*x^(2/3))^n]))/(5*d^2*x^(5/3)) - (e^2*(a + b*Log[c*(d + e*x^(2 
/3))^n]))/(3*d^3*x) + (e^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(d^4*x^(1/3)) 
 + (e^(7/2)*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3)) 
^n]))/d^(9/2) + (I*b*e^(7/2)*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqr 
t[e]*x^(1/3))])/d^(9/2)))/9)
 

Defintions of rubi rules used

rule 2009

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

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 2908

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

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 [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{2}}{x^{4}}d x\]

Input:

int((a+b*ln(c*(d+e*x^(2/3))^n))^2/x^4,x)
 

Output:

int((a+b*ln(c*(d+e*x^(2/3))^n))^2/x^4,x)
 

Fricas [F]

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

integrate((a+b*log(c*(d+e*x^(2/3))^n))^2/x^4,x, algorithm="fricas")
 

Output:

integral((b^2*log((e*x^(2/3) + d)^n*c)^2 + 2*a*b*log((e*x^(2/3) + d)^n*c) 
+ a^2)/x^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*(d+e*x**(2/3))**n))**2/x**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*log(c*(d+e*x^(2/3))^n))^2/x^4,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]

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

integrate((a+b*log(c*(d+e*x^(2/3))^n))^2/x^4,x, algorithm="giac")
 

Output:

integrate((b*log((e*x^(2/3) + d)^n*c) + a)^2/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x^4} \,d x \] Input:

int((a + b*log(c*(d + e*x^(2/3))^n))^2/x^4,x)
 

Output:

int((a + b*log(c*(d + e*x^(2/3))^n))^2/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\frac {3780 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) a b \,e^{4} n \,x^{3}+840 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) b^{2} e^{4} n^{2} x^{3}-540 x^{\frac {2}{3}} a b \,d^{4} e n +3780 x^{\frac {8}{3}} a b d \,e^{4} n -120 x^{\frac {2}{3}} b^{2} d^{4} e \,n^{2}+840 x^{\frac {8}{3}} b^{2} d \,e^{4} n^{2}+756 x^{\frac {4}{3}} a b \,d^{3} e^{2} n +168 x^{\frac {4}{3}} b^{2} d^{3} e^{2} n^{2}-1260 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}{x^{\frac {14}{3}} e +d \,x^{4}}d x \right ) b^{2} d^{6} n \,x^{3}-945 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2} b^{2} d^{5}-1890 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a b \,d^{5}-420 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{2} d^{5} n -945 a^{2} d^{5}-1260 a b \,d^{2} e^{3} n \,x^{2}-280 b^{2} d^{2} e^{3} n^{2} x^{2}}{2835 d^{5} x^{3}} \] Input:

int((a+b*log(c*(d+e*x^(2/3))^n))^2/x^4,x)
 

Output:

(3780*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*a*b*e**4*n*x**3 
 + 840*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*b**2*e**4*n**2 
*x**3 - 540*x**(2/3)*a*b*d**4*e*n + 3780*x**(2/3)*a*b*d*e**4*n*x**2 - 120* 
x**(2/3)*b**2*d**4*e*n**2 + 840*x**(2/3)*b**2*d*e**4*n**2*x**2 + 756*x**(1 
/3)*a*b*d**3*e**2*n*x + 168*x**(1/3)*b**2*d**3*e**2*n**2*x - 1260*int(log( 
(x**(2/3)*e + d)**n*c)/(x**(2/3)*e*x**4 + d*x**4),x)*b**2*d**6*n*x**3 - 94 
5*log((x**(2/3)*e + d)**n*c)**2*b**2*d**5 - 1890*log((x**(2/3)*e + d)**n*c 
)*a*b*d**5 - 420*log((x**(2/3)*e + d)**n*c)*b**2*d**5*n - 945*a**2*d**5 - 
1260*a*b*d**2*e**3*n*x**2 - 280*b**2*d**2*e**3*n**2*x**2)/(2835*d**5*x**3)