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

Optimal. Leaf size=738 \[ -\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}{x^{2/3} \left (d x^{2/3}+e\right )},x\right )}{d}+\frac {12 b^2 e^{3/2} n^2 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \log \left (\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 )}{(-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 {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 (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\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 {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \log \left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\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 (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}+1\right )\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {-d} \sqrt [3]{x}+\sqrt {e}\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}} \]

[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 [A]  time = 1.30, 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 = {} \[ \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(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*} \int \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^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) \operatorname {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) \operatorname {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) \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}-\frac {\left (6 b e^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}{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 ) \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}+\frac {\left (24 b^2 e^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^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 ) \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}+\frac {\left (24 b^2 e^2 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}\\ &=\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 ) \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}+\frac {\left (12 b^2 e^{3/2} 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}+\frac {\left (12 b^2 e^{3/2} 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}\\ &=\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 ) \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}+\frac {\left (24 b^3 e^{5/2} 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)^{3/2}}-\frac {\left (24 b^3 e^{5/2} 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)^{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 ) \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}+\frac {\left (24 b^3 e^{5/2} 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)^{3/2}}-\frac {\left (24 b^3 e^{5/2} 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)^{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 ) \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}+\frac {\left (24 b^3 e^{3/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {\left (24 b^3 e^{3/2} 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 )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} 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 )}{\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 ) \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}+\frac {\left (24 b^3 e^{3/2} 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 )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} 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 )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} 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}-\frac {\left (24 b^3 e^{3/2} 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}\\ &=\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 ) \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}-\frac {\left (12 b^3 e^{3/2} 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}+\frac {\left (12 b^3 e^{3/2} 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}+\frac {\left (12 b^3 e^{3/2} 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}-\frac {\left (12 b^3 e^{3/2} 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}\\ &=\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 ) \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}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \operatorname {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 ) \operatorname {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 ) \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}+\frac {\left (12 b^3 e^{3/2} 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}\\ &=\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 ) \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}+\frac {\left (12 b^3 e^{3/2} 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)^{3/2}}-\frac {\left (12 b^3 e^{3/2} 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)^{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 ) \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}\\ \end {align*}

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Mathematica [A]  time = 3.22, size = 475, normalized size = 0.64 \[ \frac {-9 b^2 e n^2 \left (d x^{2/3}+e\right ) \sqrt {-\frac {e}{d x^{2/3}}} \, _4F_3\left (1,1,1,\frac {5}{2};2,2,2;\frac {e}{d x^{2/3}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+9 b^3 e n^3 \left (d x^{2/3}+e\right ) \sqrt {-\frac {e}{d x^{2/3}}} \, _5F_4\left (1,1,1,1,\frac {5}{2};2,2,2,2;\frac {e}{d x^{2/3}}+1\right )+d x^{2/3} \left (3 b^2 e n^2 \sqrt {-\frac {e}{d x^{2/3}}} \log ^2\left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+2 b n \log \left (\frac {1}{2} \left (\sqrt {-\frac {e}{d x^{2/3}}}+1\right )\right )+2 b n\right )-12 b^2 e n^2 \sqrt {-\frac {e}{d x^{2/3}}} \left (\log \left (\frac {1}{2} \left (\sqrt {-\frac {e}{d x^{2/3}}}+1\right )\right )+1\right ) \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )+\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \left (a d x^{2/3}+b d x^{2/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+6 b e n\right )-2 b^3 e n^3 \sqrt {-\frac {e}{d x^{2/3}}} \log ^3\left (d+\frac {e}{x^{2/3}}\right )\right )-6 b \sqrt {d} e^{3/2} n \sqrt [3]{x} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac {e}{x^{2/3}}\right )\right )^2}{d^2 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*b^3*e*n^3*(e + d*x^(2/3))*Sqrt[-(e/(d*x^(2/3)))]*HypergeometricPFQ[{1, 1, 1, 1, 5/2}, {2, 2, 2, 2}, 1 + e/(
d*x^(2/3))] - 9*b^2*e*n^2*(e + d*x^(2/3))*Sqrt[-(e/(d*x^(2/3)))]*HypergeometricPFQ[{1, 1, 1, 5/2}, {2, 2, 2},
1 + e/(d*x^(2/3))]*(a + b*Log[c*(d + e/x^(2/3))^n]) - 6*b*Sqrt[d]*e^(3/2)*n*x^(1/3)*ArcTan[(Sqrt[d]*x^(1/3))/S
qrt[e]]*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2 + d*x^(2/3)*(-2*b^3*e*n^3*Sqrt[-(e/(d*x^(2
/3)))]*Log[d + e/x^(2/3)]^3 - 12*b^2*e*n^2*Sqrt[-(e/(d*x^(2/3)))]*(1 + Log[(1 + Sqrt[-(e/(d*x^(2/3)))])/2])*Lo
g[d + e/x^(2/3)]*(a + b*Log[c*(d + e/x^(2/3))^n]) + 3*b^2*e*n^2*Sqrt[-(e/(d*x^(2/3)))]*Log[d + e/x^(2/3)]^2*(a
+ 2*b*n + 2*b*n*Log[(1 + Sqrt[-(e/(d*x^(2/3)))])/2] + b*Log[c*(d + e/x^(2/3))^n]) + (a + b*Log[c*(d + e/x^(2/
3))^n])^2*(6*b*e*n + a*d*x^(2/3) + b*d*x^(2/3)*Log[c*(d + e/x^(2/3))^n])))/(d^2*x^(1/3))

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right ) + a^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a \right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((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 \[ b^{3} n^{3} x \log \left (d x^{\frac {2}{3}} + e\right )^{3} - 3 \, {\left (2 \, e n {\left (\frac {e \arctan \left (\frac {d x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} d} - \frac {x^{\frac {1}{3}}}{d}\right )} - x \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )\right )} a^{2} b + a^{3} x - \int \frac {{\left (2 \, b^{3} d n x - 3 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + 6 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) - 3 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} n^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 8 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{3} - 3 \, {\left (4 \, {\left (b^{3} d x + b^{3} e x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c)\right )} x - 4 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c)\right )} x^{\frac {1}{3}}\right )} n \log \left (d x^{\frac {2}{3}} + e\right ) - 12 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} - {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2}\right )} x + 6 \, {\left ({\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c)\right )} x + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c)\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right ) - {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2}\right )} x^{\frac {1}{3}}}{d x + e x^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*n^3*x*log(d*x^(2/3) + e)^3 - 3*(2*e*n*(e*arctan(d*x^(1/3)/sqrt(d*e))/(sqrt(d*e)*d) - x^(1/3)/d) - x*log(c*
(d + e/x^(2/3))^n))*a^2*b + a^3*x - integrate(((2*b^3*d*n*x - 3*(b^3*d*log(c) + a*b^2*d)*x + 6*(b^3*d*x + b^3*
e*x^(1/3))*log(x^(1/3*n)) - 3*(b^3*e*log(c) + a*b^2*e)*x^(1/3))*n^2*log(d*x^(2/3) + e)^2 + 8*(b^3*d*x + b^3*e*
x^(1/3))*log(x^(1/3*n))^3 - 3*(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))*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))*x^(1/3))*n*log(d*x^(2/3) + e) - 12*((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 - (b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2)*x + 6*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c))*x +
(b^3*e*log(c)^2 + 2*a*b^2*e*log(c))*x^(1/3))*log(x^(1/3*n)) - (b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2)*x^(1/3))/(
d*x + e*x^(1/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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)

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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)

[Out]

Timed out

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