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

Optimal. Leaf size=794 \[ \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}{x^{2/3} \left (d+e x^{2/3}\right )},x\right )}{3 e^4}-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\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 {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 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\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 {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\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 {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 {4504 i b^3 d^{9/2} n^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{315 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}+\frac {3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{315 e^{9/2}}-\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 \]

[Out]

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+637984/297675*b^3*d^3*n^3*x/e^3-22134
4/496125*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*polylog(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

________________________________________________________________________________________

Rubi [A]  time = 3.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(4504*a*b^2*d^4*n^2*x^(1/3))/(315*e^4) - (3475504*b^3*d^4*n^3*x^(1/3))/(99225*e^4) + (637984*b^3*d^3*n^3*x)/(2
97675*e^3) - (221344*b^3*d^2*n^3*x^(5/3))/(496125*e^2) + (3088*b^3*d*n^3*x^(7/3))/(27783*e) - (16*b^3*n^3*x^3)
/729 + (3475504*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(99225*e^(9/2)) - (((4504*I)/315)*b^3*d^(9/
2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(9/2) - (9008*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*
Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(315*e^(9/2)) + (4504*b^3*d^4*n^2*x^(1/3)*Log[c*(d + e*x^(2/3)
)^n])/(315*e^4) - (1984*b^2*d^3*n^2*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/(945*e^3) + (1144*b^2*d^2*n^2*x^(5/3)*
(a + b*Log[c*(d + e*x^(2/3))^n]))/(1575*e^2) - (128*b^2*d*n^2*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(441*e
) + (8*b^2*n^2*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/81 - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[
d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(315*e^(9/2)) - (2*b*d^4*n*x^(1/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e
^4 + (2*b*d^3*n*x*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3)
)^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(7*e) - (2*b*n*x^3*(a + b*Log[c*(d + e
*x^(2/3))^n])^2)/9 + (x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/3 - (((4504*I)/315)*b^3*d^(9/2)*n^3*PolyLog[2, 1
- (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(9/2) + (2*b*d^5*n*Defer[Subst][Defer[Int][(a + b*Log[c*(d +
e*x^2)^n])^2/(d + e*x^2), x], x, x^(1/3)])/e^4

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx &=3 \operatorname {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+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+e x^{2/3}\right )^n\right )\right )^3-(2 b e n) \operatorname {Subst}\left (\int \frac {x^{10} \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-(2 b e n) \operatorname {Subst}\left (\int \left (\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^5}-\frac {d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^4}+\frac {d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^3}-\frac {d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^2}+\frac {x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-(2 b n) \operatorname {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )-\frac {\left (2 b d^4 n\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (2 b d^3 n\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac {\left (2 b d^2 n\right ) \operatorname {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^2}+\frac {(2 b d n) \operatorname {Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=-\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 {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {1}{7} \left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e}+\frac {1}{9} \left (8 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {x^{10} \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=-\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 {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {1}{7} \left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \left (-\frac {d^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac {d^2 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{e}-\frac {d \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \left (-\frac {d \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac {d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \left (\frac {d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac {d^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e}+\frac {1}{9} \left (8 b^2 e n^2\right ) \operatorname {Subst}\left (\int \left (\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5}-\frac {d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac {d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\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 {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {1}{9} \left (8 b^2 n^2\right ) \operatorname {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^3}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^2}-\frac {\left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e}-\frac {\left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e}\\ &=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{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 \tan ^{-1}\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 {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {1}{63} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac {1}{49} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{e^3}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac {1}{81} \left (16 b^3 e n^3\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}+\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 \tan ^{-1}\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 {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {1}{63} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^4}{e^2}+\frac {x^6}{e}+\frac {d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac {1}{49} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^4}{e^2}+\frac {x^6}{e}+\frac {d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac {\left (16 b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^{7/2}}+\frac {\left (16 b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^{7/2}}+\frac {\left (16 b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^{7/2}}+\frac {\left (16 b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^{7/2}}+\frac {\left (16 b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^{7/2}}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^3}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x^2}{e^2}+\frac {x^4}{e}-\frac {d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x^2}{e^2}+\frac {x^4}{e}-\frac {d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x^2}{e^2}+\frac {x^4}{e}-\frac {d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac {1}{81} \left (16 b^3 e n^3\right ) \operatorname {Subst}\left (\int \left (\frac {d^4}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^4}{e^3}-\frac {d x^6}{e^2}+\frac {x^8}{e}-\frac {d^5}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\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 {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{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 \tan ^{-1}\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 {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{81 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{63 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{49 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{45 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{25 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^4}+2 \frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ &=\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 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \tan ^{-1}\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 \tan ^{-1}\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 {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ &=\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 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \tan ^{-1}\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 \tan ^{-1}\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 {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (16 i b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{9 e^{9/2}}-\frac {\left (16 i b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{7 e^{9/2}}-\frac {\left (16 i b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{5 e^{9/2}}-\frac {\left (16 i b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{3 e^{9/2}}-\frac {\left (16 i b^3 d^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{e^{9/2}}\\ &=\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 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \tan ^{-1}\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 \tan ^{-1}\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 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{315 e^{9/2}}+\frac {\left (2 b d^5 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ \end {align*}

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Mathematica [A]  time = 9.25, size = 3146, normalized size = 3.96 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*n^3*x^(1/3)*(32*d^4 - 32*d^4*Sqrt[1 - (d + e*x^(2/3))/d] + 128*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(
2/3)) - 192*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2 + 128*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/
3))^3 - 32*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4 + 1584*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1
, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] - 4536*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}
, (d + e*x^(2/3))/d] + 3780*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/
d] - 864*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] + 3024*d^3
*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] - 3780*d^3*(d + e*x^(2
/3))*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] + 1890*d^3*(d + e*x^(2/3))*Hyperge
ometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, (d + e*x^(2/3))/d] - 240*d^4*Log[d + e*x^(2/3)] + 240*d^4*Sqrt[1
- (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 672*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3
)] + 576*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)] - 96*d*Sqrt[1 - (d + e*x^(2/3))/
d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)] - 48*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]
- 3780*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 864
*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 302
4*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 37
80*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 1
890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] +
284*d^4*Log[d + e*x^(2/3)]^2 - 284*d^4*Sqrt[1 - (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 + 668*d^3*Sqrt[1 - (d
+ e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)]^2 - 552*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*L
og[d + e*x^(2/3)]^2 + 236*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)]^2 - 68*Sqrt[1 - (
d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^2 - 1890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1
, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 + 945*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}
, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 - 70*d^4*Log[d + e*x^(2/3)]^3 + 70*d^4*Sqrt[1 - (d + e*x^(2/
3))/d]*Log[d + e*x^(2/3)]^3 - 280*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)]^3 + 420*d
^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)]^3 - 280*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d +
e*x^(2/3))^3*Log[d + e*x^(2/3)]^3 + 70*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^3 + 1
512*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*(1 + 3*Log[d + e*x^(2/3)] +
Log[d + e*x^(2/3)]^2) - 144*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*(6
+ 11*Log[d + e*x^(2/3)] + 3*Log[d + e*x^(2/3)]^2)))/(210*e^4*Sqrt[1 - (d + e*x^(2/3))/d]) + (b^2*n^2*x^(1/3)*
(-120*d^4 + 120*d^4*Sqrt[1 - (d + e*x^(2/3))/d] - 336*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3)) + 288*d^
2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2 - 48*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3 - 24*Sqrt
[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4 - 1890*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (
d + e*x^(2/3))/d] + 432*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] -
1512*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] + 1890*d^3*(d + e*x
^(2/3))*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] - 945*d^3*(d + e*x^(2/3))*Hypergeomet
ricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, (d + e*x^(2/3))/d] + 284*d^4*Log[d + e*x^(2/3)] - 284*d^4*Sqrt[1 - (d + e*x
^(2/3))/d]*Log[d + e*x^(2/3)] + 668*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))*Log[d + e*x^(2/3)] - 552*d
^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)] + 236*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e
*x^(2/3))^3*Log[d + e*x^(2/3)] - 68*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)] - 1890*d^
3*(d + e*x^(2/3))*HypergeometricPFQ[{-3/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] + 945*d^3*(d +
e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)] - 105*d^4*Log[d + e*
x^(2/3)]^2 + 105*d^4*Sqrt[1 - (d + e*x^(2/3))/d]*Log[d + e*x^(2/3)]^2 - 420*d^3*Sqrt[1 - (d + e*x^(2/3))/d]*(d
+ e*x^(2/3))*Log[d + e*x^(2/3)]^2 + 630*d^2*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^2*Log[d + e*x^(2/3)]^
2 - 420*d*Sqrt[1 - (d + e*x^(2/3))/d]*(d + e*x^(2/3))^3*Log[d + e*x^(2/3)]^2 + 105*Sqrt[1 - (d + e*x^(2/3))/d]
*(d + e*x^(2/3))^4*Log[d + e*x^(2/3)]^2 + 756*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2}, (d +
e*x^(2/3))/d]*(3 + 2*Log[d + e*x^(2/3)]) - 72*d^3*(d + e*x^(2/3))*HypergeometricPFQ[{-7/2, 1, 1}, {2, 2}, (d
+ e*x^(2/3))/d]*(11 + 6*Log[d + e*x^(2/3)]))*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])))/(10
5*e^4*Sqrt[1 - (d + e*x^(2/3))/d]) - (2*b*d^4*n*x^(1/3)*(a + b*(-(n*Log[d + e*x^(2/3)]) + 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)]) + 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)]) + 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)]) + 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)]) + 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)]) + Log[c*(d + e*x^(2/3))^n]))^2 + (x^3*(a + b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d +
e*x^(2/3))^n]))^2*(3*a - 2*b*n + 3*b*(-(n*Log[d + e*x^(2/3)]) + Log[c*(d + e*x^(2/3))^n])))/9

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} x^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b x^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a^{3} x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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((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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{3} x^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, b^{3} n^{3} x^{3} \log \left (e x^{\frac {2}{3}} + d\right )^{3} + \int -\frac {{\left (2 \, b^{3} e n x^{3} - 9 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{3} - 9 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{\frac {7}{3}}\right )} n^{2} \log \left (e x^{\frac {2}{3}} + d\right )^{2} - 3 \, {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x^{3} - 3 \, {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d\right )} x^{\frac {7}{3}} - 9 \, {\left ({\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{3} + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x^{\frac {7}{3}}\right )} n \log \left (e x^{\frac {2}{3}} + d\right )}{3 \, {\left (e x + d x^{\frac {1}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^3*n^3*x^3*log(e*x^(2/3) + d)^3 + integrate(-1/3*((2*b^3*e*n*x^3 - 9*(b^3*e*log(c) + a*b^2*e)*x^3 - 9*(b^
3*d*log(c) + a*b^2*d)*x^(7/3))*n^2*log(e*x^(2/3) + d)^2 - 3*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*l
og(c) + a^3*e)*x^3 - 3*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x^(7/3) - 9*((b^3*e*lo
g(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^3 + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^(7/3))*n*log(e*x^(2
/3) + d))/(e*x + d*x^(1/3)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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