Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\frac {24 i b^3 e^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {48 b^3 e^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {24 b^2 e^{3/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{3/2}}-\frac {6 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x}+\frac {24 i b^3 e^{3/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{3/2}}-\frac {2 b e^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^{2/3}},x\right )}{d} \] Output:
24*I*b^3*e^(3/2)*n^3*arctan(e^(1/2)*x^(1/3)/d^(1/2))^2/d^(3/2)+48*b^3*e^(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)))/d^(3/2)+24*b^2*e^(3/2)*n^2*arctan(e^(1/2)*x^(1/3)/d^(1/2))*(a+b*ln (c*(d+e*x^(2/3))^n))/d^(3/2)-6*b*e*n*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d/x^(1/ 3)-(a+b*ln(c*(d+e*x^(2/3))^n))^3/x+24*I*b^3*e^(3/2)*n^3*polylog(2,1-2*d^(1 /2)/(d^(1/2)+I*e^(1/2)*x^(1/3)))/d^(3/2)-2*b*e^2*n*Defer(Int)((a+b*ln(c*(d +e*x^(2/3))^n))^2/(d+e*x^(2/3))/x^(2/3),x)/d
Leaf count is larger than twice the leaf count of optimal. \(1158\) vs. \(2(319)=638\).
Time = 10.31 (sec) , antiderivative size = 1158, normalized size of antiderivative = 48.25 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^2,x]
Output:
(-6*b*e*n*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(d* x^(1/3)) - (6*b*e^(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)/d^(3/2) - (3*b*n*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])^3/x + (3*b^2*e*n ^2*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])*((-16*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)]])/d^(3/2) - (8*Log[d + e*x^(2/3)])/d - (2*Log[d + e*x^(2/3)]^2)/(e *x^(2/3)) - (8*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]))/(-d)^(3/2) - (2*Sqrt[-((e*x^(2/3))/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] + Log[1 + (e*x^(2/3))/d]^2 - 4*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/d))/(2*x^(1/3)) + (b^3*n^3*(48*Sqrt[-d^ 2]*e*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(2/3)*HypergeometricPFQ[{1/2, 1/2 , 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] - 12*d*Sqrt[-d^2]*(-((e*x ^(2/3))/d))^(3/2)*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 24*Sqrt[d]*(e*x^ (2/3))^(3/2)*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*Log[d + e*x^(2/3)] + 24*Sqr t[-d^2]*e*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*x^(2/3)*HypergeometricPFQ[{1/2 , 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3)] - 6*Sqrt[-d ^2]*e*x^(2/3)*Log[d + e*x^(2/3)]^2 + 6*Sqrt[-d]*(d + e*x^(2/3))^(3/2)*(...
Not integrable
Time = 1.20 (sec) , antiderivative size = 24, 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 = {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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx\) |
| \(\Big \downarrow \) 2908 |
| \(\displaystyle 3 \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^{4/3}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2907 |
| \(\displaystyle 3 \left (2 b e n \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^{2/3}}d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x}\right )\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle 3 \left (2 b e n \int \left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d x^{2/3}}-\frac {e \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \left (d+e x^{2/3}\right )}\right )d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x}+2 b e n \left (-\frac {e \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d+e x^{2/3}}d\sqrt [3]{x}}{d}+\frac {4 b \sqrt {e} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{3/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d \sqrt [3]{x}}+\frac {4 i b^2 \sqrt {e} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {8 b^2 \sqrt {e} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{3/2}}+\frac {4 i b^2 \sqrt {e} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{d^{3/2}}\right )\right )\) |
Input:
Int[(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^2,x]
Output:
$Aborted
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]
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]
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]
Not integrable
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{3}}{x^{2}}d x\]
Input:
int((a+b*ln(c*(d+e*x^(2/3))^n))^3/x^2,x)
Output:
int((a+b*ln(c*(d+e*x^(2/3))^n))^3/x^2,x)
Not integrable
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(2/3))^n))^3/x^2,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^2, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e*x**(2/3))**n))**3/x**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*log(c*(d+e*x^(2/3))^n))^3/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
Not integrable
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(2/3))^n))^3/x^2,x, algorithm="giac")
Output:
integrate((b*log((e*x^(2/3) + d)^n*c) + a)^3/x^2, x)
Not integrable
Time = 25.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3}{x^2} \,d x \] Input:
int((a + b*log(c*(d + e*x^(2/3))^n))^3/x^2,x)
Output:
int((a + b*log(c*(d + e*x^(2/3))^n))^3/x^2, x)
Not integrable
Time = 0.18 (sec) , antiderivative size = 393, normalized size of antiderivative = 16.38 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^2} \, dx=\frac {-18 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) a^{2} b e n x -24 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) a \,b^{2} e \,n^{2} x -16 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} e}{\sqrt {e}\, \sqrt {d}}\right ) b^{3} e \,n^{3} x -18 x^{\frac {2}{3}} a^{2} b d e n -24 x^{\frac {2}{3}} a \,b^{2} d e \,n^{2}-16 x^{\frac {2}{3}} b^{3} d e \,n^{3}-6 \left (\int \frac {{\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2}}{x^{\frac {8}{3}} e +d \,x^{2}}d x \right ) b^{3} d^{3} n x -12 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}{x^{\frac {8}{3}} e +d \,x^{2}}d x \right ) a \,b^{2} d^{3} n x -8 \left (\int \frac {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}{x^{\frac {8}{3}} e +d \,x^{2}}d x \right ) b^{3} d^{3} n^{2} x -3 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{3} b^{3} d^{2}-9 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2} a \,b^{2} d^{2}-6 {\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right )}^{2} b^{3} d^{2} n -9 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a^{2} b \,d^{2}-12 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) a \,b^{2} d^{2} n -8 \,\mathrm {log}\left (\left (x^{\frac {2}{3}} e +d \right )^{n} c \right ) b^{3} d^{2} n^{2}-3 a^{3} d^{2}}{3 d^{2} x} \] Input:
int((a+b*log(c*(d+e*x^(2/3))^n))^3/x^2,x)
Output:
( - 18*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*a**2*b*e*n*x - 24*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*a*b**2*e*n**2*x - 16*sqrt(e)*sqrt(d)*atan((x**(1/3)*e)/(sqrt(e)*sqrt(d)))*b**3*e*n**3*x - 1 8*x**(2/3)*a**2*b*d*e*n - 24*x**(2/3)*a*b**2*d*e*n**2 - 16*x**(2/3)*b**3*d *e*n**3 - 6*int(log((x**(2/3)*e + d)**n*c)**2/(x**(2/3)*e*x**2 + d*x**2),x )*b**3*d**3*n*x - 12*int(log((x**(2/3)*e + d)**n*c)/(x**(2/3)*e*x**2 + d*x **2),x)*a*b**2*d**3*n*x - 8*int(log((x**(2/3)*e + d)**n*c)/(x**(2/3)*e*x** 2 + d*x**2),x)*b**3*d**3*n**2*x - 3*log((x**(2/3)*e + d)**n*c)**3*b**3*d** 2 - 9*log((x**(2/3)*e + d)**n*c)**2*a*b**2*d**2 - 6*log((x**(2/3)*e + d)** n*c)**2*b**3*d**2*n - 9*log((x**(2/3)*e + d)**n*c)*a**2*b*d**2 - 12*log((x **(2/3)*e + d)**n*c)*a*b**2*d**2*n - 8*log((x**(2/3)*e + d)**n*c)*b**3*d** 2*n**2 - 3*a**3*d**2)/(3*d**2*x)