\(\int (a+b \log (c (d+\frac {e}{x^{2/3}})^n))^3 \, dx\) [529]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}+\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} 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)^{3/2}}-\frac {12 b^3 e^{3/2} 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)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {12 b^3 e^{3/2} n^3 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {2 b e^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{\left (e+d x^{2/3}\right ) x^{2/3}},x\right )}{d} \]

[Out]

6*b*e*n*x^(1/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d+x*(a+b*ln(c*(d+e/x^(2/3))^n))^3+12*b^2*e^(3/2)*n^2*(a+b*ln(c*(
d+e/x^(2/3))^n))*ln(-x^(1/3)*(-d)^(1/2)+e^(1/2))/(-d)^(3/2)-12*b^3*e^(3/2)*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)^(3/2)+24*b^3*e^(3/2)*n^3*ln(x^(1/3)*(-d)^(1/2)/e^(1/2))*ln(-x^(1/
3)*(-d)^(1/2)+e^(1/2))/(-d)^(3/2)-6*b^3*e^(3/2)*n^3*ln(-x^(1/3)*(-d)^(1/2)+e^(1/2))^2/(-d)^(3/2)-12*b^2*e^(3/2
)*n^2*(a+b*ln(c*(d+e/x^(2/3))^n))*ln(x^(1/3)*(-d)^(1/2)+e^(1/2))/(-d)^(3/2)+12*b^3*e^(3/2)*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)^(3/2)-24*b^3*e^(3/2)*n^3*ln(-x^(1/3)*(-d)^(1/2)/e^
(1/2))*ln(x^(1/3)*(-d)^(1/2)+e^(1/2))/(-d)^(3/2)+6*b^3*e^(3/2)*n^3*ln(x^(1/3)*(-d)^(1/2)+e^(1/2))^2/(-d)^(3/2)
+24*b^3*e^(3/2)*n^3*polylog(2,1-x^(1/3)*(-d)^(1/2)/e^(1/2))/(-d)^(3/2)-12*b^3*e^(3/2)*n^3*polylog(2,1/2-1/2*x^
(1/3)*(-d)^(1/2)/e^(1/2))/(-d)^(3/2)+12*b^3*e^(3/2)*n^3*polylog(2,1/2+1/2*x^(1/3)*(-d)^(1/2)/e^(1/2))/(-d)^(3/
2)-24*b^3*e^(3/2)*n^3*polylog(2,1+x^(1/3)*(-d)^(1/2)/e^(1/2))/(-d)^(3/2)-2*b*e^2*n*Unintegrable((a+b*ln(c*(d+e
/x^(2/3))^n))^2/(e+d*x^(2/3))/x^(2/3),x)/d

Rubi [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

[In]

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

[Out]

(6*b*e*n*x^(1/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/d + x*(a + b*Log[c*(d + e/x^(2/3))^n])^3 + (12*b^2*e^(3/2
)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)])/(-d)^(3/2) - (6*b^3*e^(3/2)*n^3*Log[Sq
rt[e] - Sqrt[-d]*x^(1/3)]^2)/(-d)^(3/2) - (12*b^2*e^(3/2)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + S
qrt[-d]*x^(1/3)])/(-d)^(3/2) + (6*b^3*e^(3/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2)/(-d)^(3/2) + (12*b^3*e^(3
/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(3/2) - (12*b^3*e^(3/2
)*n^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(3/2) - (24*b^3*e^(3/2)*n^
3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])])/(-d)^(3/2) + (24*b^3*e^(3/2)*n^3*Log[Sqr
t[e] - Sqrt[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(3/2) + (24*b^3*e^(3/2)*n^3*PolyLog[2, 1 - (Sqr
t[-d]*x^(1/3))/Sqrt[e]])/(-d)^(3/2) - (12*b^3*e^(3/2)*n^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-
d)^(3/2) + (12*b^3*e^(3/2)*n^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(3/2) - (24*b^3*e^(3/2)*n^
3*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(3/2) - (6*b*e^2*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

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+(6 b e n) \text {Subst}\left (\int \frac {\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 ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+(6 b e n) \text {Subst}\left (\int \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d}-\frac {e \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {(6 b e n) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (24 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\left (d+\frac {e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{d} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (24 b^2 e^2 n^2\right ) \text {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^2 e^{3/2} n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^2 e^{3/2} n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (24 b^3 e^{5/2} n^3\right ) \text {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)^{3/2}}-\frac {\left (24 b^3 e^{5/2} n^3\right ) \text {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)^{3/2}} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (24 b^3 e^{5/2} n^3\right ) \text {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)^{3/2}}-\frac {\left (24 b^3 e^{5/2} n^3\right ) \text {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)^{3/2}} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {x \log \left (\sqrt {e}-\sqrt {-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {x \log \left (\sqrt {e}+\sqrt {-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {-d}} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {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 )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {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 )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {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}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {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} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}+\frac {12 b^3 e^{3/2} 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)^{3/2}}-\frac {12 b^3 e^{3/2} 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)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {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}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {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} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}+\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} 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)^{3/2}}-\frac {12 b^3 e^{3/2} 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)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {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)^{3/2}}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {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)^{3/2}} \\ & = \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} 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)^{3/2}}-\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} 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)^{3/2}}+\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} 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)^{3/2}}-\frac {12 b^3 e^{3/2} 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)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {12 b^3 e^{3/2} n^3 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \text {Li}_2\left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{d} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} \,d x } \]

[In]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((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)

Mupad [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3 \,d x \]

[In]

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

[Out]

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