\(\int \frac {a+b \log (c (d+\frac {e}{x^{2/3}})^n)}{x^4} \, dx\) [515]

Optimal result

Integrand size = 22, antiderivative size = 132 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}+\frac {2 b d^{9/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3} \] Output:

2/27*b*n/x^3-2/21*b*d*n/e/x^(7/3)+2/15*b*d^2*n/e^2/x^(5/3)-2/9*b*d^3*n/e^3 
/x+2/3*b*d^4*n/e^4/x^(1/3)+2/3*b*d^(9/2)*n*arctan(d^(1/2)*x^(1/3)/e^(1/2)) 
/e^(9/2)-1/3*(a+b*ln(c*(d+e/x^(2/3))^n))/x^3
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {2}{9} b e n \left (-\frac {1}{3 e x^3}+\frac {3 d}{7 e^2 x^{7/3}}-\frac {3 d^2}{5 e^3 x^{5/3}}+\frac {d^3}{e^4 x}-\frac {3 d^4}{e^5 \sqrt [3]{x}}+\frac {3 d^{9/2} \arctan \left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )}{e^{11/2}}\right )-\frac {b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3} \] Input:

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

Output:

-1/3*a/x^3 - (2*b*e*n*(-1/3*1/(e*x^3) + (3*d)/(7*e^2*x^(7/3)) - (3*d^2)/(5 
*e^3*x^(5/3)) + d^3/(e^4*x) - (3*d^4)/(e^5*x^(1/3)) + (3*d^(9/2)*ArcTan[Sq 
rt[e]/(Sqrt[d]*x^(1/3))])/e^(11/2)))/9 - (b*Log[c*(d + e/x^(2/3))^n])/(3*x 
^3)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2905, 795, 864, 264, 264, 264, 264, 264, 218}

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.

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

\(\Big \downarrow \) 2905

\(\displaystyle -\frac {2}{9} b e n \int \frac {1}{\left (d+\frac {e}{x^{2/3}}\right ) x^{14/3}}dx-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\)

\(\Big \downarrow \) 795

\(\displaystyle -\frac {2}{9} b e n \int \frac {1}{\left (x^{2/3} d+e\right ) x^4}dx-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\)

\(\Big \downarrow \) 864

\(\displaystyle -\frac {2}{3} b e n \int \frac {1}{\left (x^{2/3} d+e\right ) x^{10/3}}d\sqrt [3]{x}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {2}{3} b e n \left (-\frac {d \int \frac {1}{\left (x^{2/3} d+e\right ) x^{8/3}}d\sqrt [3]{x}}{e}-\frac {1}{9 e x^3}\right )-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {2}{3} b e n \left (-\frac {d \left (-\frac {d \int \frac {1}{\left (x^{2/3} d+e\right ) x^2}d\sqrt [3]{x}}{e}-\frac {1}{7 e x^{7/3}}\right )}{e}-\frac {1}{9 e x^3}\right )-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {2}{3} b e n \left (-\frac {d \left (-\frac {d \left (-\frac {d \int \frac {1}{\left (x^{2/3} d+e\right ) x^{4/3}}d\sqrt [3]{x}}{e}-\frac {1}{5 e x^{5/3}}\right )}{e}-\frac {1}{7 e x^{7/3}}\right )}{e}-\frac {1}{9 e x^3}\right )-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {2}{3} b e n \left (-\frac {d \left (-\frac {d \left (-\frac {d \left (-\frac {d \int \frac {1}{\left (x^{2/3} d+e\right ) x^{2/3}}d\sqrt [3]{x}}{e}-\frac {1}{3 e x}\right )}{e}-\frac {1}{5 e x^{5/3}}\right )}{e}-\frac {1}{7 e x^{7/3}}\right )}{e}-\frac {1}{9 e x^3}\right )-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {2}{3} b e n \left (-\frac {d \left (-\frac {d \left (-\frac {d \left (-\frac {d \left (-\frac {d \int \frac {1}{x^{2/3} d+e}d\sqrt [3]{x}}{e}-\frac {1}{e \sqrt [3]{x}}\right )}{e}-\frac {1}{3 e x}\right )}{e}-\frac {1}{5 e x^{5/3}}\right )}{e}-\frac {1}{7 e x^{7/3}}\right )}{e}-\frac {1}{9 e x^3}\right )-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {2}{3} b e n \left (-\frac {d \left (-\frac {d \left (-\frac {d \left (-\frac {d \left (-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {1}{e \sqrt [3]{x}}\right )}{e}-\frac {1}{3 e x}\right )}{e}-\frac {1}{5 e x^{5/3}}\right )}{e}-\frac {1}{7 e x^{7/3}}\right )}{e}-\frac {1}{9 e x^3}\right )\)

Input:

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

Output:

(-2*b*e*n*(-1/9*1/(e*x^3) - (d*(-1/7*1/(e*x^(7/3)) - (d*(-1/5*1/(e*x^(5/3) 
) - (d*(-1/3*1/(e*x) - (d*(-(1/(e*x^(1/3))) - (Sqrt[d]*ArcTan[(Sqrt[d]*x^( 
1/3))/Sqrt[e]])/e^(3/2)))/e))/e))/e))/e))/3 - (a + b*Log[c*(d + e/x^(2/3)) 
^n])/(3*x^3)
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 795

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* 
(b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
 

rule 864

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi 
nator[n]}, Simp[k   Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x 
^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
 

rule 2905

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.57 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\left [\frac {315 \, b d^{4} n x^{3} \sqrt {-\frac {d}{e}} \log \left (\frac {d^{3} x^{2} - 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - e^{3} + 2 \, {\left (d^{2} e x \sqrt {-\frac {d}{e}} + d e^{2}\right )} x^{\frac {2}{3}} - 2 \, {\left (d^{2} e x - e^{3} \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{d^{3} x^{2} + e^{3}}\right ) - 210 \, b d^{3} e n x^{2} + 126 \, b d^{2} e^{2} n x^{\frac {4}{3}} - 315 \, b e^{4} n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) + 70 \, b e^{4} n - 315 \, b e^{4} \log \left (c\right ) - 315 \, a e^{4} + 90 \, {\left (7 \, b d^{4} n x^{2} - b d e^{3} n\right )} x^{\frac {2}{3}}}{945 \, e^{4} x^{3}}, \frac {630 \, b d^{4} n x^{3} \sqrt {\frac {d}{e}} \arctan \left (x^{\frac {1}{3}} \sqrt {\frac {d}{e}}\right ) - 210 \, b d^{3} e n x^{2} + 126 \, b d^{2} e^{2} n x^{\frac {4}{3}} - 315 \, b e^{4} n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) + 70 \, b e^{4} n - 315 \, b e^{4} \log \left (c\right ) - 315 \, a e^{4} + 90 \, {\left (7 \, b d^{4} n x^{2} - b d e^{3} n\right )} x^{\frac {2}{3}}}{945 \, e^{4} x^{3}}\right ] \] Input:

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

Output:

[1/945*(315*b*d^4*n*x^3*sqrt(-d/e)*log((d^3*x^2 - 2*d*e^2*x*sqrt(-d/e) - e 
^3 + 2*(d^2*e*x*sqrt(-d/e) + d*e^2)*x^(2/3) - 2*(d^2*e*x - e^3*sqrt(-d/e)) 
*x^(1/3))/(d^3*x^2 + e^3)) - 210*b*d^3*e*n*x^2 + 126*b*d^2*e^2*n*x^(4/3) - 
 315*b*e^4*n*log((d*x + e*x^(1/3))/x) + 70*b*e^4*n - 315*b*e^4*log(c) - 31 
5*a*e^4 + 90*(7*b*d^4*n*x^2 - b*d*e^3*n)*x^(2/3))/(e^4*x^3), 1/945*(630*b* 
d^4*n*x^3*sqrt(d/e)*arctan(x^(1/3)*sqrt(d/e)) - 210*b*d^3*e*n*x^2 + 126*b* 
d^2*e^2*n*x^(4/3) - 315*b*e^4*n*log((d*x + e*x^(1/3))/x) + 70*b*e^4*n - 31 
5*b*e^4*log(c) - 315*a*e^4 + 90*(7*b*d^4*n*x^2 - b*d*e^3*n)*x^(2/3))/(e^4* 
x^3)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((a+b*log(c*(d+e/x^(2/3))^n))/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 [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\frac {1}{945} \, {\left (2 \, e {\left (\frac {315 \, d^{5} \arctan \left (\frac {d x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{5}} + \frac {315 \, d^{4} x^{\frac {8}{3}} - 105 \, d^{3} e x^{2} + 63 \, d^{2} e^{2} x^{\frac {4}{3}} - 45 \, d e^{3} x^{\frac {2}{3}} + 35 \, e^{4}}{e^{5} x^{3}}\right )} - \frac {315 \, \log \left (d + \frac {e}{x^{\frac {2}{3}}}\right )}{x^{3}}\right )} b n - \frac {b \log \left (c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \] Input:

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

Output:

1/945*(2*e*(315*d^5*arctan(d*x^(1/3)/sqrt(d*e))/(sqrt(d*e)*e^5) + (315*d^4 
*x^(8/3) - 105*d^3*e*x^2 + 63*d^2*e^2*x^(4/3) - 45*d*e^3*x^(2/3) + 35*e^4) 
/(e^5*x^3)) - 315*log(d + e/x^(2/3))/x^3)*b*n - 1/3*b*log(c)/x^3 - 1/3*a/x 
^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\frac {630 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) b \,d^{4} n \,x^{3}+630 x^{\frac {8}{3}} b \,d^{4} e n -90 x^{\frac {2}{3}} b d \,e^{4} n +126 x^{\frac {4}{3}} b \,d^{2} e^{3} n -315 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b \,e^{5}-315 a \,e^{5}-210 b \,d^{3} e^{2} n \,x^{2}+70 b \,e^{5} n}{945 e^{5} x^{3}} \] Input:

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

Output:

(630*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*b*d**4*n*x**3 + 
630*x**(2/3)*b*d**4*e*n*x**2 - 90*x**(2/3)*b*d*e**4*n + 126*x**(1/3)*b*d** 
2*e**3*n*x - 315*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b*e**5 - 315*a* 
e**5 - 210*b*d**3*e**2*n*x**2 + 70*b*e**5*n)/(945*e**5*x**3)