3.531 \(\int \frac {(a+b \log (c (d+\frac {e}{x^{2/3}})^n))^3}{x^4} \, dx\)

Optimal. Leaf size=784 \[ \frac {2 b d^5 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^{2/3} \left (d x^{2/3}+e\right )},x\right )}{3 e^4}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}+\frac {4504 i b^3 d^{9/2} n^3 \text {Li}_2\left (\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}-1\right )}{315 e^{9/2}}+\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{315 e^{9/2}}+\frac {3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{99225 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{315 e^{9/2}}+\frac {3475504 b^3 d^4 n^3}{99225 e^4 \sqrt [3]{x}}-\frac {637984 b^3 d^3 n^3}{297675 e^3 x}+\frac {221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac {3088 b^3 d n^3}{27783 e x^{7/3}}+\frac {16 b^3 n^3}{729 x^3} \]

[Out]

16/729*b^3*n^3/x^3-3088/27783*b^3*d*n^3/e/x^(7/3)+221344/496125*b^3*d^2*n^3/e^2/x^(5/3)-637984/297675*b^3*d^3*
n^3/e^3/x+3475504/99225*b^3*d^4*n^3/e^4/x^(1/3)+3475504/99225*b^3*d^(9/2)*n^3*arctan(x^(1/3)*d^(1/2)/e^(1/2))/
e^(9/2)+4504/315*I*b^3*d^(9/2)*n^3*polylog(2,-1+2*e^(1/2)/(-I*x^(1/3)*d^(1/2)+e^(1/2)))/e^(9/2)-8/81*b^2*n^2*(
a+b*ln(c*(d+e/x^(2/3))^n))/x^3+128/441*b^2*d*n^2*(a+b*ln(c*(d+e/x^(2/3))^n))/e/x^(7/3)-1144/1575*b^2*d^2*n^2*(
a+b*ln(c*(d+e/x^(2/3))^n))/e^2/x^(5/3)+1984/945*b^2*d^3*n^2*(a+b*ln(c*(d+e/x^(2/3))^n))/e^3/x-4504/315*b^2*d^4
*n^2*(a+b*ln(c*(d+e/x^(2/3))^n))/e^4/x^(1/3)-4504/315*b^2*d^(9/2)*n^2*arctan(x^(1/3)*d^(1/2)/e^(1/2))*(a+b*ln(
c*(d+e/x^(2/3))^n))/e^(9/2)+2/9*b*n*(a+b*ln(c*(d+e/x^(2/3))^n))^2/x^3-2/7*b*d*n*(a+b*ln(c*(d+e/x^(2/3))^n))^2/
e/x^(7/3)+2/5*b*d^2*n*(a+b*ln(c*(d+e/x^(2/3))^n))^2/e^2/x^(5/3)-2/3*b*d^3*n*(a+b*ln(c*(d+e/x^(2/3))^n))^2/e^3/
x+2*b*d^4*n*(a+b*ln(c*(d+e/x^(2/3))^n))^2/e^4/x^(1/3)-1/3*(a+b*ln(c*(d+e/x^(2/3))^n))^3/x^3-9008/315*b^3*d^(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)))/e^(9/2)+4504/315*I*b^3*d^(
9/2)*n^3*arctan(x^(1/3)*d^(1/2)/e^(1/2))^2/e^(9/2)+2/3*b*d^5*n*Unintegrable((a+b*ln(c*(d+e/x^(2/3))^n))^2/(e+d
*x^(2/3))/x^(2/3),x)/e^4

________________________________________________________________________________________

Rubi [A]  time = 3.62, 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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(16*b^3*n^3)/(729*x^3) - (3088*b^3*d*n^3)/(27783*e*x^(7/3)) + (221344*b^3*d^2*n^3)/(496125*e^2*x^(5/3)) - (637
984*b^3*d^3*n^3)/(297675*e^3*x) + (3475504*b^3*d^4*n^3)/(99225*e^4*x^(1/3)) + (3475504*b^3*d^(9/2)*n^3*ArcTan[
(Sqrt[d]*x^(1/3))/Sqrt[e]])/(99225*e^(9/2)) + (((4504*I)/315)*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]
]^2)/e^(9/2) - (9008*b^3*d^(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))])/(315*e^(9/2)) - (8*b^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(81*x^3) + (128*b^2*d*n^2*(a + b*Log
[c*(d + e/x^(2/3))^n]))/(441*e*x^(7/3)) - (1144*b^2*d^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(1575*e^2*x^(5/3
)) + (1984*b^2*d^3*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(945*e^3*x) - (4504*b^2*d^4*n^2*(a + b*Log[c*(d + e/x
^(2/3))^n]))/(315*e^4*x^(1/3)) - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x
^(2/3))^n]))/(315*e^(9/2)) + (2*b*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(9*x^3) - (2*b*d*n*(a + b*Log[c*(d + e
/x^(2/3))^n])^2)/(7*e*x^(7/3)) + (2*b*d^2*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*e^2*x^(5/3)) - (2*b*d^3*n*(
a + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*e^3*x) + (2*b*d^4*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(e^4*x^(1/3)) -
(a + b*Log[c*(d + e/x^(2/3))^n])^3/(3*x^3) + (((4504*I)/315)*b^3*d^(9/2)*n^3*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt
[e] - I*Sqrt[d]*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/(e + d*
x^2), x], x, x^(1/3)])/e^4

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x^4} \, dx &=3 \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^3}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}-(2 b e n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{\left (d+\frac {e}{x^2}\right ) x^{12}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}-(2 b e n) \operatorname {Subst}\left (\int \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e x^{10}}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e^2 x^8}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e^3 x^6}-\frac {d^3 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e^4 x^4}+\frac {d^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e^5 x^2}-\frac {d^5 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e^5 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}-(2 b n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{x^{10}} \, dx,x,\sqrt [3]{x}\right )-\frac {\left (2 b d^4 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{x^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+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (2 b d^3 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{x^4} \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac {\left (2 b d^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{x^6} \, dx,x,\sqrt [3]{x}\right )}{e^2}+\frac {(2 b d n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{x^8} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b d^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 )}{e^4}-\frac {1}{7} \left (8 b^2 d 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^{10}} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^4 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^4} \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac {\left (8 b^2 d^3 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^6} \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac {\left (8 b^2 d^2 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^8} \, dx,x,\sqrt [3]{x}\right )}{5 e}+\frac {1}{9} \left (8 b^2 e 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^{12}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b d^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 )}{e^4}-\frac {1}{7} \left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e x^8}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^2 x^6}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^3 x^4}-\frac {d^3 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^4 x^2}+\frac {d^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^4 \left (e+d 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+\frac {e}{x^2}\right )^n\right )}{e x^2}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e \left (e+d 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 {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e x^4}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^2 x^2}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^2 \left (e+d 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 {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e x^6}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^2 x^4}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^3 x^2}-\frac {d^3 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^3 \left (e+d 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 {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{e x^{10}}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^2 x^8}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^3 x^6}-\frac {d^3 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^4 x^4}+\frac {d^4 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^5 x^2}-\frac {d^5 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )}{e^5 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b d^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 )}{e^4}+\frac {1}{9} \left (8 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^{10}} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^2} \, 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+\frac {e}{x^2}\right )^n\right )}{e+d 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+\frac {e}{x^2}\right )^n\right )}{e+d 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+\frac {e}{x^2}\right )^n\right )}{e+d 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+\frac {e}{x^2}\right )^n\right )}{e+d 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+\frac {e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{7 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{5 e^2}-\frac {\left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^8} \, dx,x,\sqrt [3]{x}\right )}{9 e}-\frac {\left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{x^8} \, dx,x,\sqrt [3]{x}\right )}{7 e}\\ &=-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b d^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 )}{e^4}+\frac {1}{63} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^{10}} \, dx,x,\sqrt [3]{x}\right )+\frac {1}{49} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^{10}} \, dx,x,\sqrt [3]{x}\right )-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac {\left (16 b^3 d^5 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 )}{9 e^3}-\frac {\left (16 b^3 d^5 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 e^3}-\frac {\left (16 b^3 d^5 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 e^3}-\frac {\left (16 b^3 d^5 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 e^3}-\frac {\left (16 b^3 d^5 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 )}{e^3}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^8} \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^8} \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^8} \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac {1}{81} \left (16 b^3 e n^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^2}\right ) x^{12}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b d^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 )}{e^4}+\frac {1}{63} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac {1}{49} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )-\frac {\left (16 b^3 d^{9/2} 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 )}{9 e^{7/2}}-\frac {\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac {\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac {\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac {\left (16 b^3 d^{9/2} 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 )}{e^{7/2}}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d 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 {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac {1}{81} \left (16 b^3 e n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {16 b^3 n^3}{729 x^3}-\frac {256 b^3 d n^3}{3087 e x^{7/3}}+\frac {2288 b^3 d^2 n^3}{7875 e^2 x^{5/3}}-\frac {3968 b^3 d^3 n^3}{2835 e^3 x}+\frac {9008 b^3 d^4 n^3}{315 e^4 \sqrt [3]{x}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b d^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 )}{e^4}+\frac {1}{81} \left (16 b^3 d n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d 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}{e+d 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}{e+d 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}{e+d 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (16 b^3 d^{9/2} 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 )}{9 e^{7/2}}-\frac {\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac {\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac {\left (16 b^3 d^{9/2} 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 e^{7/2}}-\frac {\left (16 b^3 d^{9/2} 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 )}{e^{7/2}}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{27 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{25 e^2}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{63 e}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{49 e}\\ &=\frac {16 b^3 n^3}{729 x^3}-\frac {3088 b^3 d n^3}{27783 e x^{7/3}}+\frac {7472 b^3 d^2 n^3}{18375 e^2 x^{5/3}}-\frac {26704 b^3 d^3 n^3}{14175 e^3 x}+\frac {30992 b^3 d^4 n^3}{945 e^4 \sqrt [3]{x}}+\frac {9008 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{315 e^{9/2}}+\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{315 e^{9/2}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b d^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 )}{e^4}-\frac {\left (16 i b^3 d^{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 )}{9 e^{9/2}}-\frac {\left (16 i b^3 d^{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 e^{9/2}}-\frac {\left (16 i b^3 d^{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 e^{9/2}}-\frac {\left (16 i b^3 d^{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 e^{9/2}}-\frac {\left (16 i b^3 d^{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 )}{e^{9/2}}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d 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}{e+d 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{25 e^3}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{63 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{49 e^2}-\frac {\left (16 b^3 d^2 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{81 e}\\ &=\frac {16 b^3 n^3}{729 x^3}-\frac {3088 b^3 d n^3}{27783 e x^{7/3}}+\frac {221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac {206128 b^3 d^3 n^3}{99225 e^3 x}+\frac {161824 b^3 d^4 n^3}{4725 e^4 \sqrt [3]{x}}+\frac {30992 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{945 e^{9/2}}+\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{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 )}{315 e^{9/2}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {\left (2 b d^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 )}{e^4}+\frac {\left (16 b^3 d^5 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 )}{9 e^5}+\frac {\left (16 b^3 d^5 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 e^5}+\frac {\left (16 b^3 d^5 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 e^5}+\frac {\left (16 b^3 d^5 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 e^5}+\frac {\left (16 b^3 d^5 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 )}{e^5}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d 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}{e+d 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{25 e^4}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{63 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{49 e^3}+\frac {\left (16 b^3 d^3 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{81 e^2}\\ &=\frac {16 b^3 n^3}{729 x^3}-\frac {3088 b^3 d n^3}{27783 e x^{7/3}}+\frac {221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac {637984 b^3 d^3 n^3}{297675 e^3 x}+\frac {1151968 b^3 d^4 n^3}{33075 e^4 \sqrt [3]{x}}+\frac {161824 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{4725 e^{9/2}}+\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{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 )}{315 e^{9/2}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {4504 i b^3 d^{9/2} n^3 \text {Li}_2\left (-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\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+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{49 e^4}-\frac {\left (16 b^3 d^4 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{81 e^3}\\ &=\frac {16 b^3 n^3}{729 x^3}-\frac {3088 b^3 d n^3}{27783 e x^{7/3}}+\frac {221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac {637984 b^3 d^3 n^3}{297675 e^3 x}+\frac {3475504 b^3 d^4 n^3}{99225 e^4 \sqrt [3]{x}}+\frac {1151968 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{33075 e^{9/2}}+\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{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 )}{315 e^{9/2}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {4504 i b^3 d^{9/2} n^3 \text {Li}_2\left (-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\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+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{81 e^4}\\ &=\frac {16 b^3 n^3}{729 x^3}-\frac {3088 b^3 d n^3}{27783 e x^{7/3}}+\frac {221344 b^3 d^2 n^3}{496125 e^2 x^{5/3}}-\frac {637984 b^3 d^3 n^3}{297675 e^3 x}+\frac {3475504 b^3 d^4 n^3}{99225 e^4 \sqrt [3]{x}}+\frac {3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{99225 e^{9/2}}+\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{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 )}{315 e^{9/2}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{81 x^3}+\frac {128 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{441 e x^{7/3}}-\frac {1144 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{1575 e^2 x^{5/3}}+\frac {1984 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{945 e^3 x}-\frac {4504 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{315 e^4 \sqrt [3]{x}}-\frac {4504 b^2 d^{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 )}{315 e^{9/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{9 x^3}-\frac {2 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{7 e x^{7/3}}+\frac {2 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{5 e^2 x^{5/3}}-\frac {2 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{3 e^3 x}+\frac {2 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^4 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{3 x^3}+\frac {4504 i b^3 d^{9/2} n^3 \text {Li}_2\left (-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\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+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ \end {align*}

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Mathematica [A]  time = 8.71, size = 2726, normalized size = 3.48 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*n^3*(32*e^4*Sqrt[-(e/(d*x^(2/3)))] - 32*d^4*x^(8/3) - 1584*d^3*e*x^2*HypergeometricPFQ[{-7/2, 1, 1, 1}, {
2, 2, 2}, 1 + e/(d*x^(2/3))] - 1584*d^4*x^(8/3)*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3)
)] + 4536*d^3*e*x^2*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + 4536*d^4*x^(8/3)*Hyperg
eometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3))] - 3780*d^3*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1},
{2, 2, 2}, 1 + e/(d*x^(2/3))] - 3780*d^4*x^(8/3)*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/
3))] + 864*d^3*e*x^2*HypergeometricPFQ[{-7/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] + 864*d^4*x^(8/3)*
HypergeometricPFQ[{-7/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] - 3024*d^3*e*x^2*HypergeometricPFQ[{-5/
2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] - 3024*d^4*x^(8/3)*HypergeometricPFQ[{-5/2, 1, 1, 1, 1}, {2,
2, 2, 2}, 1 + e/(d*x^(2/3))] + 3780*d^3*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(
2/3))] + 3780*d^4*x^(8/3)*HypergeometricPFQ[{-3/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] - 1890*d^3*e*
x^2*HypergeometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] - 1890*d^4*x^(8/3)*HypergeometricPF
Q[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] - (288*e^4*Log[d + e/x^(2/3)])/Sqrt[-(e/(d*x^(2/3)))] +
48*e^4*Sqrt[-(e/(d*x^(2/3)))]*Log[d + e/x^(2/3)] + 240*d^4*x^(8/3)*Log[d + e/x^(2/3)] + 3780*d^3*e*x^2*Hyperg
eometricPFQ[{-3/2, 1, 1}, {2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)] + 3780*d^4*x^(8/3)*HypergeometricPFQ[{
-3/2, 1, 1}, {2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)] - 864*d^3*e*x^2*HypergeometricPFQ[{-7/2, 1, 1, 1},
{2, 2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)] - 864*d^4*x^(8/3)*HypergeometricPFQ[{-7/2, 1, 1, 1}, {2, 2, 2
}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)] + 3024*d^3*e*x^2*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, 1 + e/
(d*x^(2/3))]*Log[d + e/x^(2/3)] + 3024*d^4*x^(8/3)*HypergeometricPFQ[{-5/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2
/3))]*Log[d + e/x^(2/3)] - 3780*d^3*e*x^2*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3))]*Log
[d + e/x^(2/3)] - 3780*d^4*x^(8/3)*HypergeometricPFQ[{-3/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/
x^(2/3)] + 1890*d^3*e*x^2*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)]
+ 1890*d^4*x^(8/3)*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)] + (252*
e^4*Log[d + e/x^(2/3)]^2)/(-(e/(d*x^(2/3))))^(3/2) - (36*e^4*Log[d + e/x^(2/3)]^2)/Sqrt[-(e/(d*x^(2/3)))] + 68
*e^4*Sqrt[-(e/(d*x^(2/3)))]*Log[d + e/x^(2/3)]^2 - 284*d^4*x^(8/3)*Log[d + e/x^(2/3)]^2 + 1890*d^3*e*x^2*Hyper
geometricPFQ[{-3/2, 1, 1}, {2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)]^2 + 1890*d^4*x^(8/3)*HypergeometricPF
Q[{-3/2, 1, 1}, {2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)]^2 - 945*d^3*e*x^2*HypergeometricPFQ[{-1/2, 1, 1}
, {2, 2}, 1 + e/(d*x^(2/3))]*Log[d + e/x^(2/3)]^2 - 945*d^4*x^(8/3)*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1
+ e/(d*x^(2/3))]*Log[d + e/x^(2/3)]^2 - 70*e^4*Sqrt[-(e/(d*x^(2/3)))]*Log[d + e/x^(2/3)]^3 + 70*d^4*x^(8/3)*Lo
g[d + e/x^(2/3)]^3 - 1512*d^3*(e + d*x^(2/3))*x^2*HypergeometricPFQ[{-5/2, 1, 1}, {2, 2}, 1 + e/(d*x^(2/3))]*(
1 + 3*Log[d + e/x^(2/3)] + Log[d + e/x^(2/3)]^2) + 144*d^3*(e + d*x^(2/3))*x^2*HypergeometricPFQ[{-7/2, 1, 1},
{2, 2}, 1 + e/(d*x^(2/3))]*(6 + 11*Log[d + e/x^(2/3)] + 3*Log[d + e/x^(2/3)]^2)))/(210*e^4*Sqrt[-(e/(d*x^(2/3
)))]*x^3) - (2*b*d*n*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*e*x^(7/3)) + (2*b*d^2*n*(
a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*e^2*x^(5/3)) - (2*b*d^3*n*(a - b*n*Log[d + e/x^
(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*e^3*x) + (2*b*d^4*n*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^
(2/3))^n])^2)/(e^4*x^(1/3)) + (2*b*d^(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)/e^(9/2) - (b*n*Log[d + e/x^(2/3)]*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x
^(2/3))^n])^2)/x^3 - ((a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2*(3*a - 2*b*n - 3*b*n*Log[d +
e/x^(2/3)] + 3*b*Log[c*(d + e/x^(2/3))^n]))/(9*x^3) + (b^2*n^2*(-a + b*n*Log[d + e/x^(2/3)] - b*Log[c*(d + e/
x^(2/3))^n])*(9800*e^(9/2) - 28800*d*e^(7/2)*x^(2/3) + 72072*d^2*e^(5/2)*x^(4/3) - 208320*d^3*e^(3/2)*x^2 + 14
18760*d^4*Sqrt[e]*x^(8/3) - 1418760*d^(9/2)*x^3*ArcTan[Sqrt[e]/(Sqrt[d]*x^(1/3))] - 44100*e^(9/2)*Log[d + e/x^
(2/3)] + 56700*d*e^(7/2)*x^(2/3)*Log[d + e/x^(2/3)] - 79380*d^2*e^(5/2)*x^(4/3)*Log[d + e/x^(2/3)] + 132300*d^
3*e^(3/2)*x^2*Log[d + e/x^(2/3)] - 396900*d^4*Sqrt[e]*x^(8/3)*Log[d + e/x^(2/3)] + 99225*e^(9/2)*Log[d + e/x^(
2/3)]^2 - 198450*(-d)^(9/2)*x^3*Log[d + e/x^(2/3)]*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 99225*(-d)^(9/2)*x^3*Log[
Sqrt[e] - Sqrt[-d]*x^(1/3)]^2 + 198450*(-d)^(9/2)*x^3*Log[d + e/x^(2/3)]*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] - 992
25*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2 - 198450*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Lo
g[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] + 198450*(-d)^(9/2)*x^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt
[-d]*x^(1/3))/Sqrt[e])/2] + 396900*(-d)^(9/2)*x^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqr
t[e])] - 396900*(-d)^(9/2)*x^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]] - 396900*(-d)^(
9/2)*x^3*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] + 198450*(-d)^(9/2)*x^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3)
)/(2*Sqrt[e])] - 198450*(-d)^(9/2)*x^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] + 396900*(-d)^(9/2)*x^3*
PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]]))/(99225*e^(9/2)*x^3)

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right ) + a^{3}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^3/x^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^3/x^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a \right )^{3}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*(d+e/x^(2/3))^n)+a)^3/x^4,x)

[Out]

int((b*ln(c*(d+e/x^(2/3))^n)+a)^3/x^4,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{3} n^{3} \log \left (d x^{\frac {2}{3}} + e\right )^{3}}{3 \, x^{3}} - \int -\frac {{\left (2 \, b^{3} d n x + 9 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x - 18 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + 9 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} n^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} - 24 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{3} + 9 \, {\left (4 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x - 4 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {1}{3}}\right )} n \log \left (d x^{\frac {2}{3}} + e\right ) + 36 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} + 3 \, {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d\right )} x - 18 \, {\left ({\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + 3 \, {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x^{\frac {1}{3}}}{3 \, {\left (d x^{5} + e x^{\frac {13}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^3/x^4,x, algorithm="maxima")

[Out]

-1/3*b^3*n^3*log(d*x^(2/3) + e)^3/x^3 - integrate(-1/3*((2*b^3*d*n*x + 9*(b^3*d*log(c) + a*b^2*d)*x - 18*(b^3*
d*x + b^3*e*x^(1/3))*log(x^(1/3*n)) + 9*(b^3*e*log(c) + a*b^2*e)*x^(1/3))*n^2*log(d*x^(2/3) + e)^2 - 24*(b^3*d
*x + b^3*e*x^(1/3))*log(x^(1/3*n))^3 + 9*(4*(b^3*d*x + b^3*e*x^(1/3))*log(x^(1/3*n))^2 + (b^3*d*log(c)^2 + 2*a
*b^2*d*log(c) + a^2*b*d)*x - 4*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*x^(1/3))*log(x^(1/3*n))
+ (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(1/3))*n*log(d*x^(2/3) + e) + 36*((b^3*d*log(c) + a*b^2*d)*x
+ (b^3*e*log(c) + a*b^2*e)*x^(1/3))*log(x^(1/3*n))^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 - 18*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^
2*b*e)*x^(1/3))*log(x^(1/3*n)) + 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^(1/3))/(
d*x^5 + e*x^(13/3)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a + b*log(c*(d + e/x^(2/3))^n))^3/x^4, 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((a+b*ln(c*(d+e/x**(2/3))**n))**3/x**4,x)

[Out]

Timed out

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