\(\int x^2 (a+b \log (c (d+\frac {e}{x^{2/3}})^n))^3 \, dx\) [528]

Optimal result

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 {568 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 {1136 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 {568 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 {4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac {2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac {4 b^3 e^{9/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )}{(-d)^{9/2}}-\frac {4 b^3 e^{9/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{(-d)^{9/2}}-\frac {8 b^3 e^{9/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}+\frac {8 b^3 e^{9/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}+\frac {568 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 {8 b^3 e^{9/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}-\frac {4 b^3 e^{9/2} n^3 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )}{(-d)^{9/2}}+\frac {4 b^3 e^{9/2} n^3 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{(-d)^{9/2}}-\frac {8 b^3 e^{9/2} n^3 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-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} \]

[Out]

568/105*I*b^3*e^(9/2)*n^3*arctan(x^(1/3)*d^(1/2)/e^(1/2))^2/d^(9/2)+16/105*b^3*e^3*n^3*x/d^3-16/7*b^3*e^4*n^3*
x^(1/3)/d^4+2*b^3*e^(9/2)*n^3*ln(x^(1/3)*(-d)^(1/2)+e^(1/2))^2/(-d)^(9/2)+8*b^3*e^(9/2)*n^3*polylog(2,1-x^(1/3
)*(-d)^(1/2)/e^(1/2))/(-d)^(9/2)-4*b^3*e^(9/2)*n^3*polylog(2,1/2-1/2*x^(1/3)*(-d)^(1/2)/e^(1/2))/(-d)^(9/2)+4*
b^3*e^(9/2)*n^3*polylog(2,1/2+1/2*x^(1/3)*(-d)^(1/2)/e^(1/2))/(-d)^(9/2)-8*b^3*e^(9/2)*n^3*polylog(2,1+x^(1/3)
*(-d)^(1/2)/e^(1/2))/(-d)^(9/2)+2/3*b*e^5*n*Unintegrable((a+b*ln(c*(d+e/x^(2/3))^n))^2/(e+d*x^(2/3))/x^(2/3),x
)/d^4+1376/105*b^3*e^(9/2)*n^3*arctan(x^(1/3)*d^(1/2)/e^(1/2))/d^(9/2)-2*b^3*e^(9/2)*n^3*ln(-x^(1/3)*(-d)^(1/2
)+e^(1/2))^2/(-d)^(9/2)+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+4*b^2*e^(9/2)*n^2*(a+b*ln(c*(d+e/x^(2/3))^n)
)*ln(-x^(1/3)*(-d)^(1/2)+e^(1/2))/(-d)^(9/2)-4*b^3*e^(9/2)*n^3*ln(1/2+1/2*x^(1/3)*(-d)^(1/2)/e^(1/2))*ln(-x^(1
/3)*(-d)^(1/2)+e^(1/2))/(-d)^(9/2)+8*b^3*e^(9/2)*n^3*ln(x^(1/3)*(-d)^(1/2)/e^(1/2))*ln(-x^(1/3)*(-d)^(1/2)+e^(
1/2))/(-d)^(9/2)-4*b^2*e^(9/2)*n^2*(a+b*ln(c*(d+e/x^(2/3))^n))*ln(x^(1/3)*(-d)^(1/2)+e^(1/2))/(-d)^(9/2)+4*b^3
*e^(9/2)*n^3*ln(1/2-1/2*x^(1/3)*(-d)^(1/2)/e^(1/2))*ln(x^(1/3)*(-d)^(1/2)+e^(1/2))/(-d)^(9/2)-8*b^3*e^(9/2)*n^
3*ln(-x^(1/3)*(-d)^(1/2)/e^(1/2))*ln(x^(1/3)*(-d)^(1/2)+e^(1/2))/(-d)^(9/2)-1136/105*b^3*e^(9/2)*n^3*arctan(x^
(1/3)*d^(1/2)/e^(1/2))*ln(2-2*e^(1/2)/(-I*x^(1/3)*d^(1/2)+e^(1/2)))/d^(9/2)+568/105*I*b^3*e^(9/2)*n^3*polylog(
2,-1+2*e^(1/2)/(-I*x^(1/3)*d^(1/2)+e^(1/2)))/d^(9/2)+1/3*x^3*(a+b*ln(c*(d+e/x^(2/3))^n))^3+568/105*a*b^2*e^4*n
^2*x^(1/3)/d^4+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^2-568/105*b^2*e^(9/2)*n^2*arctan(x^(1/3)*d^(
1/2)/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

Rubi [N/A]

Not integrable

Time = 2.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

[In]

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

[Out]

(568*a*b^2*e^4*n^2*x^(1/3))/(105*d^4) - (16*b^3*e^4*n^3*x^(1/3))/(7*d^4) + (16*b^3*e^3*n^3*x)/(105*d^3) + (137
6*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(105*d^(9/2)) + (((568*I)/105)*b^3*e^(9/2)*n^3*ArcTan[(Sq
rt[d]*x^(1/3))/Sqrt[e]]^2)/d^(9/2) - (1136*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e
])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(105*d^(9/2)) + (568*b^3*e^4*n^2*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/(105*d^4
) - (32*b^2*e^3*n^2*x*(a + b*Log[c*(d + e/x^(2/3))^n]))/(35*d^3) + (8*b^2*e^2*n^2*x^(5/3)*(a + b*Log[c*(d + e/
x^(2/3))^n]))/(35*d^2) - (568*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n
]))/(105*d^(9/2)) - (2*b*e^4*n*x^(1/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/d^4 + (2*b*e^3*n*x*(a + b*Log[c*(d
+ e/x^(2/3))^n])^2)/(3*d^3) - (2*b*e^2*n*x^(5/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*d^2) + (2*b*e*n*x^(7/3
)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*d) + (x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/3 + (4*b^2*e^(9/2)*n^2*
(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)])/(-d)^(9/2) - (2*b^3*e^(9/2)*n^3*Log[Sqrt[e]
- Sqrt[-d]*x^(1/3)]^2)/(-d)^(9/2) - (4*b^2*e^(9/2)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]
*x^(1/3)])/(-d)^(9/2) + (2*b^3*e^(9/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2)/(-d)^(9/2) + (4*b^3*e^(9/2)*n^3*
Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(9/2) - (4*b^3*e^(9/2)*n^3*Log
[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(9/2) - (8*b^3*e^(9/2)*n^3*Log[Sqrt
[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])])/(-d)^(9/2) + (8*b^3*e^(9/2)*n^3*Log[Sqrt[e] - Sqrt
[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) + (((568*I)/105)*b^3*e^(9/2)*n^3*PolyLog[2, -1 + (2*
Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2) + (8*b^3*e^(9/2)*n^3*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e
]])/(-d)^(9/2) - (4*b^3*e^(9/2)*n^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(9/2) + (4*b^3*e^(9
/2)*n^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(9/2) - (8*b^3*e^(9/2)*n^3*PolyLog[2, 1 + (Sqrt[-
d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) + (2*b*e^5*n*Defer[Subst][Defer[Int][(a + b*Log[c*(d + e/x^2)^n])^2/(e + d*x^
2), x], x, x^(1/3)])/d^4

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+(2 b e n) \text {Subst}\left (\int \frac {x^6 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+(2 b e n) \text {Subst}\left (\int \left (-\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d^4}+\frac {e^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d^3}-\frac {e x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d^2}+\frac {x^6 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d}+\frac {e^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d^4 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {(2 b e n) \text {Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^2}+\frac {\left (2 b e^3 n\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^3}-\frac {\left (2 b e^4 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4} \\ & = -\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 {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (8 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac {\left (8 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac {\left (8 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\left (d+\frac {e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4} \\ & = -\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 {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (8 b^2 e^2 n^2\right ) \text {Subst}\left (\int \left (\frac {e^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d^3}-\frac {e x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d^2}+\frac {x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d}-\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d^3 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac {\left (8 b^2 e^3 n^2\right ) \text {Subst}\left (\int \left (-\frac {e \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d^2}+\frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d}+\frac {e^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac {\left (8 b^2 e^4 n^2\right ) \text {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{d}-\frac {e \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^4} \\ & = -\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 {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (8 b^2 e^2 n^2\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^2}-\frac {\left (8 b^2 e^3 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^3}-\frac {\left (8 b^2 e^3 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^3}+\frac {\left (8 b^2 e^4 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac {\left (8 b^2 e^4 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac {\left (8 b^2 e^4 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac {\left (4 b^2 e^{9/2} n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (4 b^2 e^{9/2} n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5975\) vs. \(2(1278)=2556\).

Time = 23.09 (sec) , antiderivative size = 5975, normalized size of antiderivative = 248.96 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.88 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3 \,d x \]

[In]

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

[Out]

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