∫ x2(a + b log(c(d + e _

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

Optimal. Leaf size=1278 \[ \frac {2 b n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{\left (x^{2/3} d+e\right ) x^{2/3}},x\right ) e^5}{3 d^4}+\frac {568 i b^3 n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2 e^{9/2}}{105 d^{9/2}}-\frac {2 b^3 n^3 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) e^{9/2}}{(-d)^{9/2}}+\frac {2 b^3 n^3 \log ^2\left (\sqrt [3]{x} \sqrt {-d}+\sqrt {e}\right ) e^{9/2}}{(-d)^{9/2}}+\frac {1376 b^3 n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) e^{9/2}}{105 d^{9/2}}-\frac {1136 b^3 n^3 \tan ^{-1}\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 ) e^{9/2}}{105 d^{9/2}}-\frac {568 b^2 n^2 \tan ^{-1}\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 ) e^{9/2}}{105 d^{9/2}}+\frac {4 b^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 ) e^{9/2}}{(-d)^{9/2}}-\frac {4 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt [3]{x} \sqrt {-d}+\sqrt {e}\right ) e^{9/2}}{(-d)^{9/2}}+\frac {4 b^3 n^3 \log \left (\sqrt [3]{x} \sqrt {-d}+\sqrt {e}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right ) e^{9/2}}{(-d)^{9/2}}-\frac {4 b^3 n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {1}{2} \left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )\right ) e^{9/2}}{(-d)^{9/2}}-\frac {8 b^3 n^3 \log \left (\sqrt [3]{x} \sqrt {-d}+\sqrt {e}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right ) e^{9/2}}{(-d)^{9/2}}+\frac {8 b^3 n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right ) e^{9/2}}{(-d)^{9/2}}+\frac {568 i b^3 n^3 \text {Li}_2\left (\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right ) e^{9/2}}{105 d^{9/2}}+\frac {8 b^3 n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right ) e^{9/2}}{(-d)^{9/2}}-\frac {4 b^3 n^3 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right ) e^{9/2}}{(-d)^{9/2}}+\frac {4 b^3 n^3 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )\right ) e^{9/2}}{(-d)^{9/2}}-\frac {8 b^3 n^3 \text {Li}_2\left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right ) e^{9/2}}{(-d)^{9/2}}-\frac {2 b n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 e^4}{d^4}+\frac {568 b^3 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) e^4}{105 d^4}-\frac {16 b^3 n^3 \sqrt [3]{x} e^4}{7 d^4}+\frac {568 a b^2 n^2 \sqrt [3]{x} e^4}{105 d^4}+\frac {2 b n x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 e^3}{3 d^3}+\frac {16 b^3 n^3 x e^3}{105 d^3}-\frac {32 b^2 n^2 x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) e^3}{35 d^3}-\frac {2 b n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 e^2}{5 d^2}+\frac {8 b^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) e^2}{35 d^2}+\frac {2 b n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 e}{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 \]

[Out]

568/105*a*b^2*e^4*n^2*x^(1/3)/d^4+1/3*x^3*(a+b*ln(c*(d+e/x^(2/3))^n))^3-16/7*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(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*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+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*arcta

n(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+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)+568/105*I*b^3*e^(9/2)*n^3*arctan(x^(1/3)*d^(1/2)/e^(1/2))^2/d^(9/2)

________________________________________________________________________________________

Rubi [A] time = 3.17, 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 = {} \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

(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*} \int x^2 \left (a+b \log \left (c \left (d+\frac {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+\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) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}\\ &=\frac {568 a b^2 e^4 n^2 \sqrt [3]{x}}{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 \tan ^{-1}\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 {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 {\left (2 b e^5 n\right ) \operatorname {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^3 e^4 n^2\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+\frac {e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac {\left (8 b^3 e^4 n^2\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+\frac {e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac {\left (8 b^3 e^4 n^2\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+\frac {e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac {\left (16 b^3 e^3 n^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{35 d^2}-\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}-\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}+\frac {\left (8 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {\left (8 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {\left (16 b^3 e^6 n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{\sqrt {d} \sqrt {e} \left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}-\frac {\left (16 b^3 e^6 n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{\sqrt {d} \sqrt {e} \left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}-\frac {\left (16 b^3 e^6 n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{\sqrt {d} \sqrt {e} \left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=\frac {568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}+\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 \tan ^{-1}\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 {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 {\left (2 b e^5 n\right ) \operatorname {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 (16 b^3 e^3 n^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^2}-\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}-\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac {\left (8 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \left (\frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{e x}-\frac {d x \log \left (\sqrt {e}-\sqrt {-d} x\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {\left (8 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \left (\frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{e x}-\frac {d x \log \left (\sqrt {e}+\sqrt {-d} x\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {\left (16 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{7 d^{9/2}}-\frac {\left (16 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{5 d^{9/2}}-\frac {\left (16 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}\\ &=\frac {568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac {64 b^3 e^4 n^3 \sqrt [3]{x}}{35 d^4}+\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 \tan ^{-1}\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 {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 {\left (2 b e^5 n\right ) \operatorname {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 (16 b^3 e^3 n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {e}{d^2}+\frac {x^2}{d}+\frac {e^2}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{35 d^2}+\frac {\left (8 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {\left (8 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac {\left (8 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {x \log \left (\sqrt {e}-\sqrt {-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac {\left (8 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {x \log \left (\sqrt {e}+\sqrt {-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac {\left (16 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{7 d^{9/2}}-\frac {\left (16 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 d^{9/2}}-\frac {\left (16 b^3 e^{11/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}\\ &=\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 {1328 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{105 d^{9/2}}+\frac {568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{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 \tan ^{-1}\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 {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 {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 {\left (2 b e^5 n\right ) \operatorname {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^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt {-d} \log \left (\sqrt {e}-\sqrt {-d} x\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \log \left (\sqrt {e}-\sqrt {-d} x\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac {\left (8 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt {-d} \log \left (\sqrt {e}+\sqrt {-d} x\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \log \left (\sqrt {e}+\sqrt {-d} x\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac {\left (16 i b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{x \left (i+\frac {\sqrt {d} x}{\sqrt {e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{7 d^{9/2}}-\frac {\left (16 i b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{x \left (i+\frac {\sqrt {d} x}{\sqrt {e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{5 d^{9/2}}-\frac {\left (16 i b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {e}}\right )}{x \left (i+\frac {\sqrt {d} x}{\sqrt {e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}+\frac {\left (8 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {\sqrt {-d} x}{\sqrt {e}}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (8 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {\sqrt {-d} x}{\sqrt {e}}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (16 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^4}\\ &=\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 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{105 d^{9/2}}+\frac {568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{105 d^{9/2}}-\frac {1136 b^3 e^{9/2} n^3 \tan ^{-1}\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 \tan ^{-1}\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 {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 {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 {8 b^3 e^{9/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}-\frac {8 b^3 e^{9/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}+\frac {\left (2 b e^5 n\right ) \operatorname {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 (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-\frac {i \sqrt {d} x}{\sqrt {e}}}\right )}{1+\frac {d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-\frac {i \sqrt {d} x}{\sqrt {e}}}\right )}{1+\frac {d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac {\left (16 b^3 e^4 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-\frac {i \sqrt {d} x}{\sqrt {e}}}\right )}{1+\frac {d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac {\left (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac {\left (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=\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 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{105 d^{9/2}}+\frac {568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{105 d^{9/2}}-\frac {1136 b^3 e^{9/2} n^3 \tan ^{-1}\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 \tan ^{-1}\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 {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 {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 \text {Li}_2\left (-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 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}-\frac {8 b^3 e^{9/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}+\frac {\left (2 b e^5 n\right ) \operatorname {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 (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac {\left (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {\left (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {\sqrt {e}-\sqrt {-d} x}{2 \sqrt {e}}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac {\left (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {\sqrt {e}+\sqrt {-d} x}{2 \sqrt {e}}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=\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 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{105 d^{9/2}}+\frac {568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{105 d^{9/2}}-\frac {1136 b^3 e^{9/2} n^3 \tan ^{-1}\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 \tan ^{-1}\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 \text {Li}_2\left (-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 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}-\frac {8 b^3 e^{9/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}+\frac {\left (2 b e^5 n\right ) \operatorname {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 (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {e}}\right )}{x} \, dx,x,\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac {\left (4 b^3 e^{9/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {e}}\right )}{x} \, dx,x,\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}\\ &=\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 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{105 d^{9/2}}+\frac {568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{105 d^{9/2}}-\frac {1136 b^3 e^{9/2} n^3 \tan ^{-1}\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 \tan ^{-1}\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 \text {Li}_2\left (-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 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}-\frac {4 b^3 e^{9/2} n^3 \text {Li}_2\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 \text {Li}_2\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 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{9/2}}+\frac {\left (2 b e^5 n\right ) \operatorname {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}\\ \end {align*}

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Mathematica [A] time = 4.95, size = 764, normalized size = 0.60 \[ -\frac {b^2 n^2 \left (-a-b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+b n \log \left (d+\frac {e}{x^{2/3}}\right )\right ) \left (\log \left (d+\frac {e}{x^{2/3}}\right ) \left (9 e^5 \left (d x^{2/3}+e\right ) \, _3F_2\left (1,1,\frac {11}{2};2,2;\frac {e}{d x^{2/3}}+1\right )+d x^{2/3} \left (d^5 x^{10/3} \sqrt {-\frac {e}{d x^{2/3}}}+e^5\right ) \log \left (d+\frac {e}{x^{2/3}}\right )\right )-9 e^5 \left (d x^{2/3}+e\right ) \, _4F_3\left (1,1,1,\frac {11}{2};2,2,2;\frac {e}{d x^{2/3}}+1\right )\right )}{d^6 x \sqrt {-\frac {e}{d x^{2/3}}}}+\frac {b^3 n^3 \left (\log \left (d+\frac {e}{x^{2/3}}\right ) \left (\log \left (d+\frac {e}{x^{2/3}}\right ) \left (27 e^5 \left (d x^{2/3}+e\right ) \, _3F_2\left (1,1,\frac {11}{2};2,2;\frac {e}{d x^{2/3}}+1\right )+2 d x^{2/3} \left (d^5 x^{10/3} \sqrt {-\frac {e}{d x^{2/3}}}+e^5\right ) \log \left (d+\frac {e}{x^{2/3}}\right )\right )-54 e^5 \left (d x^{2/3}+e\right ) \, _4F_3\left (1,1,1,\frac {11}{2};2,2,2;\frac {e}{d x^{2/3}}+1\right )\right )+54 e^5 \left (d x^{2/3}+e\right ) \, _5F_4\left (1,1,1,1,\frac {11}{2};2,2,2,2;\frac {e}{d x^{2/3}}+1\right )\right )}{6 d^6 x \sqrt {-\frac {e}{d x^{2/3}}}}+\frac {2 b e^{9/2} n \tan ^{-1}\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 )-b n \log \left (d+\frac {e}{x^{2/3}}\right )\right )^2}{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 )-b n \log \left (d+\frac {e}{x^{2/3}}\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 )-b n \log \left (d+\frac {e}{x^{2/3}}\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 )-b n \log \left (d+\frac {e}{x^{2/3}}\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 )-b n \log \left (d+\frac {e}{x^{2/3}}\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 )-b n \log \left (d+\frac {e}{x^{2/3}}\right )\right )^3+b n x^3 \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac {e}{x^{2/3}}\right )\right )^2 \]

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*(54*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 1, 11/2}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] + Log[

d + e/x^(2/3)]*(-54*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + Log

[d + e/x^(2/3)]*(27*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 11/2}, {2, 2}, 1 + e/(d*x^(2/3))] + 2*d*x^(2/

3)*(e^5 + d^5*Sqrt[-(e/(d*x^(2/3)))]*x^(10/3))*Log[d + e/x^(2/3)]))))/(6*d^6*Sqrt[-(e/(d*x^(2/3)))]*x) - (b^2*

n^2*(-9*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 11/2}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + Log[d + e/x^(2/

3)]*(9*e^5*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 11/2}, {2, 2}, 1 + e/(d*x^(2/3))] + d*x^(2/3)*(e^5 + d^5*S

qrt[-(e/(d*x^(2/3)))]*x^(10/3))*Log[d + e/x^(2/3)]))*(-a + b*n*Log[d + e/x^(2/3)] - b*Log[c*(d + e/x^(2/3))^n]

))/(d^6*Sqrt[-(e/(d*x^(2/3)))]*x) - (2*b*e^4*n*x^(1/3)*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n

])^2)/d^4 + (2*b*e^3*n*x*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*d^3) - (2*b*e^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*d^2) + (2*b*e*n*x^(7/3)*(a - b*n*Log[d +

e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*d) + (2*b*e^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a - b*n*

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

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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{2} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} x^{2} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b x^{2} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right ) + a^{3} x^{2}, x\right ) \]

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(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)

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

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

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maple [A] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a \right )^{3} x^{2}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, b^{3} n^{3} x^{3} \log \left (d x^{\frac {2}{3}} + e\right )^{3} - \int \frac {{\left (2 \, b^{3} d n x^{3} - 9 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{3} - 9 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {7}{3}} + 18 \, {\left (b^{3} d x^{3} + b^{3} e x^{\frac {7}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )\right )} n^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} - 3 \, {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d\right )} x^{3} + 24 \, {\left (b^{3} d x^{3} + b^{3} e x^{\frac {7}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{3} - 3 \, {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x^{\frac {7}{3}} - 9 \, {\left ({\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x^{3} + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {7}{3}} + 4 \, {\left (b^{3} d x^{3} + b^{3} e x^{\frac {7}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} - 4 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{3} + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {7}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )\right )} n \log \left (d x^{\frac {2}{3}} + e\right ) - 36 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{3} + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {7}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} + 18 \, {\left ({\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x^{3} + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {7}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )}{3 \, {\left (d x + e x^{\frac {1}{3}}\right )}}\,{d x} \]

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(d*x^(2/3) + e)^3 - integrate(1/3*((2*b^3*d*n*x^3 - 9*(b^3*d*log(c) + a*b^2*d)*x^3 - 9*(b^3

*e*log(c) + a*b^2*e)*x^(7/3) + 18*(b^3*d*x^3 + b^3*e*x^(7/3))*log(x^(1/3*n)))*n^2*log(d*x^(2/3) + e)^2 - 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^3 + 24*(b^3*d*x^3 + b^3*e*x^(7/3))*log(x^(1/3*

n))^3 - 3*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*x^(7/3) - 9*((b^3*d*log(c)^2 + 2*a*

b^2*d*log(c) + a^2*b*d)*x^3 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(7/3) + 4*(b^3*d*x^3 + b^3*e*x^(

7/3))*log(x^(1/3*n))^2 - 4*((b^3*d*log(c) + a*b^2*d)*x^3 + (b^3*e*log(c) + a*b^2*e)*x^(7/3))*log(x^(1/3*n)))*n

*log(d*x^(2/3) + e) - 36*((b^3*d*log(c) + a*b^2*d)*x^3 + (b^3*e*log(c) + a*b^2*e)*x^(7/3))*log(x^(1/3*n))^2 +

18*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^3 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(7/3))

*log(x^(1/3*n)))/(d*x + e*x^(1/3)), x)

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

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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