Integrand size = 24, antiderivative size = 24 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac {3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac {637984 b^3 d^3 n^3 x}{297675 e^3}-\frac {221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac {3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac {16}{729} b^3 n^3 x^3+\frac {3475504 b^3 d^{9/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} 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 )}{315 e^{9/2}}+\frac {4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} 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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac {4504 i b^3 d^{9/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{315 e^{9/2}}+\frac {2 b d^5 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 )}{3 e^4} \]
4504/315*a*b^2*d^4*n^2*x^(1/3)/e^4-3475504/99225*b^3*d^4*n^3*x^(1/3)/e^4+6 37984/297675*b^3*d^3*n^3*x/e^3-221344/496125*b^3*d^2*n^3*x^(5/3)/e^2+3088/ 27783*b^3*d*n^3*x^(7/3)/e-16/729*b^3*n^3*x^3+3475504/99225*b^3*d^(9/2)*n^3 *arctan(x^(1/3)*e^(1/2)/d^(1/2))/e^(9/2)-4504/315*I*b^3*d^(9/2)*n^3*polylo g(2,1-2*d^(1/2)/(d^(1/2)+I*x^(1/3)*e^(1/2)))/e^(9/2)+4504/315*b^3*d^4*n^2* x^(1/3)*ln(c*(d+e*x^(2/3))^n)/e^4-1984/945*b^2*d^3*n^2*x*(a+b*ln(c*(d+e*x^ (2/3))^n))/e^3+1144/1575*b^2*d^2*n^2*x^(5/3)*(a+b*ln(c*(d+e*x^(2/3))^n))/e ^2-128/441*b^2*d*n^2*x^(7/3)*(a+b*ln(c*(d+e*x^(2/3))^n))/e+8/81*b^2*n^2*x^ 3*(a+b*ln(c*(d+e*x^(2/3))^n))-4504/315*b^2*d^(9/2)*n^2*arctan(x^(1/3)*e^(1 /2)/d^(1/2))*(a+b*ln(c*(d+e*x^(2/3))^n))/e^(9/2)-2*b*d^4*n*x^(1/3)*(a+b*ln (c*(d+e*x^(2/3))^n))^2/e^4+2/3*b*d^3*n*x*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^3 -2/5*b*d^2*n*x^(5/3)*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^2+2/7*b*d*n*x^(7/3)*( a+b*ln(c*(d+e*x^(2/3))^n))^2/e-2/9*b*n*x^3*(a+b*ln(c*(d+e*x^(2/3))^n))^2+1 /3*x^3*(a+b*ln(c*(d+e*x^(2/3))^n))^3-9008/315*b^3*d^(9/2)*n^3*arctan(x^(1/ 3)*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*x^(1/3)*e^(1/2)))/e^(9/2)-4504 /315*I*b^3*d^(9/2)*n^3*arctan(x^(1/3)*e^(1/2)/d^(1/2))^2/e^(9/2)+2/3*b*d^5 *n*Unintegrable((a+b*ln(c*(d+e*x^(2/3))^n))^2/(d+e*x^(2/3))/x^(2/3),x)/e^4
Time = 7.80 (sec) , antiderivative size = 1552, normalized size of antiderivative = 64.67 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \]
(-2*b*d^4*n*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^ n])^2)/e^4 + (2*b*d^3*n*x*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^( 2/3))^n])^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a - b*n*Log[d + e*x^(2/3)] + b* Log[c*(d + e*x^(2/3))^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a - b*n*Log[d + e *x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(7*e) + (2*b*d^(9/2)*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)/e^(9/2) + b*n*x^3*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^3*(a - b*n*Log[d + e*x^(2/3)] + b *Log[c*(d + e*x^(2/3))^n])^2*(3*a - 2*b*n - 3*b*n*Log[d + e*x^(2/3)] + 3*b *Log[c*(d + e*x^(2/3))^n]))/9 - (b^3*n^3*(1094783760*d^(9/2)*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) ]] - e*x^(2/3)*(-16*(68423985*d^4 - 4186770*d^3*e*x^(2/3) + 871542*d^2*e^2 *x^(4/3) - 217125*d*e^3*x^2 + 42875*e^4*x^(8/3)) + 2520*(177345*d^4 - 2604 0*d^3*e*x^(2/3) + 9009*d^2*e^2*x^(4/3) - 3600*d*e^3*x^2 + 1225*e^4*x^(8/3) )*Log[d + e*x^(2/3)] - 198450*(315*d^4 - 105*d^3*e*x^(2/3) + 63*d^2*e^2*x^ (4/3) - 45*d*e^3*x^2 + 35*e^4*x^(8/3))*Log[d + e*x^(2/3)]^2 + 10418625*e^4 *x^(8/3)*Log[d + e*x^(2/3)]^3) + 62511750*d^(9/2)*Sqrt[(e*x^(2/3))/(d + e* x^(2/3))]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/ 2}, d/(d + e*x^(2/3))] + Log[d + e*x^(2/3)]*(4*Sqrt[d]*HypergeometricPFQ[{ 1/2, 1/2, 1/2}, {3/2, 3/2}, d/(d + e*x^(2/3))] + Sqrt[d + e*x^(2/3)]*Ar...
Not integrable
Time = 2.37 (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.
| \(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2908 |
| \(\displaystyle 3 \int x^{8/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2907 |
| \(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac {2}{3} b e n \int \frac {x^{10/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{d+e x^{2/3}}d\sqrt [3]{x}\right )\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac {2}{3} b e n \int \left (-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^5}{e^5 \left (d+e x^{2/3}\right )}+\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^4}{e^5}-\frac {x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^3}{e^4}+\frac {x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d^2}{e^3}-\frac {x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 d}{e^2}+\frac {x^{8/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e}\right )d\sqrt [3]{x}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac {2}{3} b e n \left (-\frac {d^5 \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}}{e^5}+\frac {2252 b d^{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 )}{315 e^{11/2}}+\frac {d^4 \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^5}-\frac {d^3 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^4}+\frac {992 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^4}+\frac {d^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^3}-\frac {572 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^3}-\frac {d x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e^2}+\frac {64 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e^2}+\frac {x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{9 e}-\frac {4 b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{81 e}-\frac {2252 a b d^4 n \sqrt [3]{x}}{315 e^5}+\frac {2252 i b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{11/2}}-\frac {1737752 b^2 d^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{11/2}}+\frac {4504 b^2 d^{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 )}{315 e^{11/2}}-\frac {2252 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^5}+\frac {2252 i b^2 d^{9/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{315 e^{11/2}}+\frac {1737752 b^2 d^4 n^2 \sqrt [3]{x}}{99225 e^5}-\frac {318992 b^2 d^3 n^2 x}{297675 e^4}+\frac {110672 b^2 d^2 n^2 x^{5/3}}{496125 e^3}-\frac {1544 b^2 d n^2 x^{7/3}}{27783 e^2}+\frac {8 b^2 n^2 x^3}{729 e}\right )\right )\) |
3.5.85.3.1 Defintions of rubi rules used
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 = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{3}d x\]
Not integrable
Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3} x^{2} \,d x } \]
integral(b^3*x^2*log((e*x^(2/3) + d)^n*c)^3 + 3*a*b^2*x^2*log((e*x^(2/3) + d)^n*c)^2 + 3*a^2*b*x^2*log((e*x^(2/3) + d)^n*c) + a^3*x^2, x)
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Timed out} \]
Exception generated. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Exception raised: ValueError} \]
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 = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3} x^{2} \,d x } \]
Not integrable
Time = 1.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3 \,d x \]