Integrand size = 22, antiderivative size = 77 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx=\frac {2 b n}{3 x}-\frac {2 b d n}{e \sqrt [3]{x}}-\frac {2 b d^{3/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \] Output:
2/3*b*n/x-2*b*d*n/e/x^(1/3)-2*b*d^(3/2)*n*arctan(d^(1/2)*x^(1/3)/e^(1/2))/ e^(3/2)-(a+b*ln(c*(d+e/x^(2/3))^n))/x
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx=-\frac {a}{x}+\frac {2 b n}{3 x}-\frac {2 b d n}{e \sqrt [3]{x}}+\frac {2 b d^{3/2} n \arctan \left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x} \] Input:
Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])/x^2,x]
Output:
-(a/x) + (2*b*n)/(3*x) - (2*b*d*n)/(e*x^(1/3)) + (2*b*d^(3/2)*n*ArcTan[Sqr t[e]/(Sqrt[d]*x^(1/3))])/e^(3/2) - (b*Log[c*(d + e/x^(2/3))^n])/x
Time = 0.40 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2905, 795, 864, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle -\frac {2}{3} b e n \int \frac {1}{\left (d+\frac {e}{x^{2/3}}\right ) x^{8/3}}dx-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}\) |
\(\Big \downarrow \) 795 |
\(\displaystyle -\frac {2}{3} b e n \int \frac {1}{\left (x^{2/3} d+e\right ) x^2}dx-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}\) |
\(\Big \downarrow \) 864 |
\(\displaystyle -2 b e n \int \frac {1}{\left (x^{2/3} d+e\right ) x^{4/3}}d\sqrt [3]{x}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -2 b e n \left (-\frac {d \int \frac {1}{\left (x^{2/3} d+e\right ) x^{2/3}}d\sqrt [3]{x}}{e}-\frac {1}{3 e x}\right )-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -2 b e n \left (-\frac {d \left (-\frac {d \int \frac {1}{x^{2/3} d+e}d\sqrt [3]{x}}{e}-\frac {1}{e \sqrt [3]{x}}\right )}{e}-\frac {1}{3 e x}\right )-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x}-2 b e n \left (-\frac {d \left (-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {1}{e \sqrt [3]{x}}\right )}{e}-\frac {1}{3 e x}\right )\) |
Input:
Int[(a + b*Log[c*(d + e/x^(2/3))^n])/x^2,x]
Output:
-2*b*e*n*(-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) - (a + b*Log[c*(d + e/x^(2/3))^n])/x
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]
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]
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]
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]
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]
\[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )}{x^{2}}d x\]
Input:
int((a+b*ln(c*(d+e/x^(2/3))^n))/x^2,x)
Output:
int((a+b*ln(c*(d+e/x^(2/3))^n))/x^2,x)
Time = 0.11 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.05 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx=\left [\frac {3 \, b d n x \sqrt {-\frac {d}{e}} \log \left (\frac {d^{3} x^{2} + 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - e^{3} - 2 \, {\left (d^{2} e x \sqrt {-\frac {d}{e}} - d e^{2}\right )} x^{\frac {2}{3}} - 2 \, {\left (d^{2} e x + e^{3} \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{d^{3} x^{2} + e^{3}}\right ) - 3 \, b e n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) - 6 \, b d n x^{\frac {2}{3}} + 2 \, b e n - 3 \, b e \log \left (c\right ) - 3 \, a e}{3 \, e x}, -\frac {6 \, b d n x \sqrt {\frac {d}{e}} \arctan \left (x^{\frac {1}{3}} \sqrt {\frac {d}{e}}\right ) + 3 \, b e n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) + 6 \, b d n x^{\frac {2}{3}} - 2 \, b e n + 3 \, b e \log \left (c\right ) + 3 \, a e}{3 \, e x}\right ] \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))/x^2,x, algorithm="fricas")
Output:
[1/3*(3*b*d*n*x*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)) - 3*b*e*n*log((d*x + e*x^(1/3))/x) - 6*b*d*n*x^(2/3) + 2*b*e*n - 3*b*e*log(c) - 3*a*e)/(e*x), -1/3*(6*b*d*n*x*sqrt(d/e)*arctan(x^ (1/3)*sqrt(d/e)) + 3*b*e*n*log((d*x + e*x^(1/3))/x) + 6*b*d*n*x^(2/3) - 2* b*e*n + 3*b*e*log(c) + 3*a*e)/(e*x)]
Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e/x**(2/3))**n))/x**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))/x^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
Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx=-\frac {1}{3} \, {\left (2 \, e {\left (\frac {3 \, d^{2} \arctan \left (\frac {d x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {3 \, d x^{\frac {2}{3}} - e}{e^{2} x}\right )} + \frac {3 \, \log \left (d + \frac {e}{x^{\frac {2}{3}}}\right )}{x}\right )} b n - \frac {b \log \left (c\right )}{x} - \frac {a}{x} \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))/x^2,x, algorithm="giac")
Output:
-1/3*(2*e*(3*d^2*arctan(d*x^(1/3)/sqrt(d*e))/(sqrt(d*e)*e^2) + (3*d*x^(2/3 ) - e)/(e^2*x)) + 3*log(d + e/x^(2/3))/x)*b*n - b*log(c)/x - a/x
Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{x^2} \,d x \] Input:
int((a + b*log(c*(d + e/x^(2/3))^n))/x^2,x)
Output:
int((a + b*log(c*(d + e/x^(2/3))^n))/x^2, x)
Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^2} \, dx=\frac {-6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} d}{\sqrt {e}\, \sqrt {d}}\right ) b d n x -6 x^{\frac {2}{3}} b d e n -3 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b \,e^{2}-3 a \,e^{2}+2 b \,e^{2} n}{3 e^{2} x} \] Input:
int((a+b*log(c*(d+e/x^(2/3))^n))/x^2,x)
Output:
( - 6*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*b*d*n*x - 6*x** (2/3)*b*d*e*n - 3*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b*e**2 - 3*a*e **2 + 2*b*e**2*n)/(3*e**2*x)