Integrand size = 24, antiderivative size = 24 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\frac {568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac {16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac {16 b^3 e^3 n^3 x}{105 d^3}+\frac {1376 b^3 e^{9/2} n^3 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{105 d^{9/2}}+\frac {1408 i b^3 e^{9/2} n^3 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{105 d^{9/2}}-\frac {2816 b^3 e^{9/2} n^3 \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 )}{105 d^{9/2}}+\frac {568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac {32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac {8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac {1408 b^2 e^{9/2} n^2 \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 )}{105 d^{9/2}}-\frac {2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac {2 b e^3 n x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac {2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac {2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {1408 i b^3 e^{9/2} n^3 \operatorname {PolyLog}\left (2,-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac {2 b e^5 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{\left (e+d x^{2/3}\right ) x^{2/3}},x\right )}{3 d^4} \] Output:
568/105*a*b^2*e^4*n^2*x^(1/3)/d^4-16/7*b^3*e^4*n^3*x^(1/3)/d^4+16/105*b^3* e^3*n^3*x/d^3+1376/105*b^3*e^(9/2)*n^3*arctan(d^(1/2)*x^(1/3)/e^(1/2))/d^( 9/2)+1408/105*I*b^3*e^(9/2)*n^3*arctan(d^(1/2)*x^(1/3)/e^(1/2))^2/d^(9/2)- 2816/105*b^3*e^(9/2)*n^3*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^(9/2)+568/105*b^3*e^4*n^2*x^(1/3)*ln(c*(d+e/x ^(2/3))^n)/d^4-32/35*b^2*e^3*n^2*x*(a+b*ln(c*(d+e/x^(2/3))^n))/d^3+8/35*b^ 2*e^2*n^2*x^(5/3)*(a+b*ln(c*(d+e/x^(2/3))^n))/d^2-1408/105*b^2*e^(9/2)*n^2 *arctan(d^(1/2)*x^(1/3)/e^(1/2))*(a+b*ln(c*(d+e/x^(2/3))^n))/d^(9/2)-2*b*e ^4*n*x^(1/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d^4+2/3*b*e^3*n*x*(a+b*ln(c*(d+ e/x^(2/3))^n))^2/d^3-2/5*b*e^2*n*x^(5/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d^2 +2/7*b*e*n*x^(7/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d+1/3*x^3*(a+b*ln(c*(d+e/ x^(2/3))^n))^3+1408/105*I*b^3*e^(9/2)*n^3*polylog(2,-1+2*e^(1/2)/(e^(1/2)- I*d^(1/2)*x^(1/3)))/d^(9/2)+2/3*b*e^5*n*Defer(Int)((a+b*ln(c*(d+e/x^(2/3)) ^n))^2/(e+d*x^(2/3))/x^(2/3),x)/d^4
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5975\) vs. \(2(646)=1292\).
Time = 29.29 (sec) , antiderivative size = 5975, normalized size of antiderivative = 248.96 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Result too large to show} \] Input:
Integrate[x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3,x]
Output:
Result too large to show
Not integrable
Time = 3.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {2908, 2907, 2005, 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.
| \(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2908 |
| \(\displaystyle 3 \int x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2907 |
| \(\displaystyle 3 \left (\frac {2}{3} b e n \int \frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d+\frac {e}{x^{2/3}}}d\sqrt [3]{x}+\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle 3 \left (\frac {2}{3} b e n \int \frac {x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^{2/3} d+e}d\sqrt [3]{x}+\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle 3 \left (\frac {2}{3} b e n \int \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 e^4}{d^4 \left (x^{2/3} d+e\right )}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 e^3}{d^4}+\frac {x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 e^2}{d^3}-\frac {x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 e}{d^2}+\frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}\right )d\sqrt [3]{x}+\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {2}{3} b e n \left (\frac {e^4 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^{2/3} d+e}d\sqrt [3]{x}}{d^4}-\frac {704 b e^{7/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{105 d^{9/2}}-\frac {e^3 \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac {e^2 x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac {16 b e^2 n x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}-\frac {e x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac {4 b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}+\frac {x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac {284 a b e^3 n \sqrt [3]{x}}{105 d^4}+\frac {704 i b^2 e^{7/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{105 d^{9/2}}+\frac {688 b^2 e^{7/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{105 d^{9/2}}-\frac {1408 b^2 e^{7/2} n^2 \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac {284 b^2 e^3 n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{105 d^4}+\frac {704 i b^2 e^{7/2} n^2 \operatorname {PolyLog}\left (2,\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right )}{105 d^{9/2}}-\frac {8 b^2 e^3 n^2 \sqrt [3]{x}}{7 d^4}+\frac {8 b^2 e^2 n^2 x}{105 d^3}\right )\right )\) |
Input:
Int[x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3,x]
Output:
$Aborted
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
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 = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int x^{2} {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{3}d x\]
Input:
int(x^2*(a+b*ln(c*(d+e/x^(2/3))^n))^3,x)
Output:
int(x^2*(a+b*ln(c*(d+e/x^(2/3))^n))^3,x)
Not integrable
Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.88 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(2/3))^n))^3,x, algorithm="fricas")
Output:
integral(b^3*x^2*log(c*((d*x + e*x^(1/3))/x)^n)^3 + 3*a*b^2*x^2*log(c*((d* x + e*x^(1/3))/x)^n)^2 + 3*a^2*b*x^2*log(c*((d*x + e*x^(1/3))/x)^n) + a^3* x^2, x)
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Timed out} \] Input:
integrate(x**2*(a+b*ln(c*(d+e/x**(2/3))**n))**3,x)
Output:
Timed out
Exception generated. \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(2/3))^n))^3,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 = 0.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{2} \,d x } \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(2/3))^n))^3,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/x^(2/3))^n) + a)^3*x^2, x)
Not integrable
Time = 25.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3 \,d x \] Input:
int(x^2*(a + b*log(c*(d + e/x^(2/3))^n))^3,x)
Output:
int(x^2*(a + b*log(c*(d + e/x^(2/3))^n))^3, x)
Not integrable
Time = 0.20 (sec) , antiderivative size = 935, normalized size of antiderivative = 38.96 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:
int(x^2*(a+b*log(c*(d+e/x^(2/3))^n))^3,x)
Output:
(630*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*a**2*b*e**4*n - 4224*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*a*b**2*e**4*n**2 + 4128*sqrt(e)*sqrt(d)*atan((x**(1/3)*d)/(sqrt(e)*sqrt(d)))*b**3*e**4*n** 3 - 126*x**(2/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**3*d**3*e* *2*n*x - 252*x**(2/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a*b**2*d** 3*e**2*n*x + 72*x**(2/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**3*d* *3*e**2*n**2*x - 126*x**(2/3)*a**2*b*d**3*e**2*n*x + 72*x**(2/3)*a*b**2*d* *3*e**2*n**2*x - 105*x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**3 *b**3*d*e**4 + 90*x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b* *3*d**4*e*n*x**2 - 630*x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))* *2*b**3*d*e**4*n + 180*x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))* a*b**2*d**4*e*n*x**2 - 1260*x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n) /3))*a*b**2*d*e**4*n + 1704*x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n) /3))*b**3*d*e**4*n**2 + 90*x**(1/3)*a**2*b*d**4*e*n*x**2 - 630*x**(1/3)*a* *2*b*d*e**4*n + 1704*x**(1/3)*a*b**2*d*e**4*n**2 - 720*x**(1/3)*b**3*d*e** 4*n**3 + 35*int(log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**3/(x**(2/3)*e + x**(1/3)*d*x),x)*b**3*d*e**5 + 420*int(log(((x**(2/3)*d + e)**n*c)/x**((2 *n)/3))/(x**(2/3)*e + x**(1/3)*d*x),x)*a*b**2*d*e**5*n - 1408*int(log(((x* *(2/3)*d + e)**n*c)/x**((2*n)/3))/(x**(2/3)*e + x**(1/3)*d*x),x)*b**3*d*e* *5*n**2 + 35*int((x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**3...