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

Optimal. Leaf size=737 \[ -\frac{2 b e^2 n \text{Unintegrable}\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{24 b^3 e^{3/2} n^3 \text{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 \text{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 \text{PolyLog}\left (2,\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 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )}{(-d)^{3/2}}+\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{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*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) - (2*b*e^2*n*Unintegrable[(a + b*Log[c*(d + e/x^(2/3)
)^n])^2/((e + d*x^(2/3))*x^(2/3)), x])/d

________________________________________________________________________________________

Rubi [A]  time = 1.29714, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 5.68845, size = 824, normalized size = 1.12 \[ \frac{b^3 \sqrt [3]{x} \left (\sqrt{d} \left (6 e+d x^{2/3} \log \left (d+\frac{e}{x^{2/3}}\right )\right ) \log ^2\left (d+\frac{e}{x^{2/3}}\right )-6 e \sqrt{\frac{e}{x^{2/3} d+e}} \left (8 \sqrt{d} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{d}{d+\frac{e}{x^{2/3}}}\right )+\log \left (d+\frac{e}{x^{2/3}}\right ) \left (4 \sqrt{d} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{d}{d+\frac{e}{x^{2/3}}}\right )+\sqrt{d+\frac{e}{x^{2/3}}} \sin ^{-1}\left (\frac{\sqrt{d}}{\sqrt{d+\frac{e}{x^{2/3}}}}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )\right )+6 \sqrt{d} e \left (\frac{4 \sqrt{\frac{e}{x^{2/3}}} \tanh ^{-1}\left (\frac{\sqrt{\frac{e}{x^{2/3}}}}{\sqrt{-d}}\right ) \left (\log \left (d+\frac{e}{x^{2/3}}\right )-\log \left (\frac{e}{d x^{2/3}}+1\right )\right )}{\sqrt{-d}}-\sqrt{-\frac{e}{d x^{2/3}}} \left (2 \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{e}{d x^{2/3}}}+1\right )\right )-4 \log \left (\frac{e}{d x^{2/3}}+1\right ) \log \left (\frac{1}{2} \left (\sqrt{-\frac{e}{d x^{2/3}}}+1\right )\right )+\log ^2\left (\frac{e}{d x^{2/3}}+1\right )-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{e}{d x^{2/3}}}\right )\right )\right )\right ) n^3}{d^{3/2}}+\frac{3 b^2 \left (-3 \left (x^{2/3} d+e\right ) \, _4F_3\left (1,1,1,\frac{5}{2};2,2,2;\frac{e}{d x^{2/3}}+1\right ) e^2-d x^{2/3} \log \left (d+\frac{e}{x^{2/3}}\right ) \left (4 \log \left (\frac{1}{2} \left (\sqrt{-\frac{e}{d x^{2/3}}}+1\right )\right ) e^2+4 \left (e-\frac{e}{\sqrt{-\frac{e}{d x^{2/3}}}}\right ) e+\left (d^2 \sqrt{-\frac{e}{d x^{2/3}}} x^{4/3}-e^2\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )\right ) \left (-a+b n \log \left (d+\frac{e}{x^{2/3}}\right )-b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) n^2}{d^3 \sqrt{-\frac{e}{d x^{2/3}}} x}-\frac{6 b e^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{d^{3/2}}+3 b x \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n+\frac{6 b e \sqrt [3]{x} \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{d}+x \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^2*n^2*(-3*e^2*(e + d*x^(2/3))*HypergeometricPFQ[{1, 1, 1, 5/2}, {2, 2, 2}, 1 + e/(d*x^(2/3))] - d*x^(2/3)
*Log[d + e/x^(2/3)]*(4*e*(e - e/Sqrt[-(e/(d*x^(2/3)))]) + 4*e^2*Log[(1 + Sqrt[-(e/(d*x^(2/3)))])/2] + (-e^2 +
d^2*Sqrt[-(e/(d*x^(2/3)))]*x^(4/3))*Log[d + e/x^(2/3)]))*(-a + b*n*Log[d + e/x^(2/3)] - b*Log[c*(d + e/x^(2/3)
)^n]))/(d^3*Sqrt[-(e/(d*x^(2/3)))]*x) + (6*b*e*n*x^(1/3)*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))
^n])^2)/d - (6*b*e^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2
/3))^n])^2)/d^(3/2) + 3*b*n*x*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*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^3 + (b^3*n^3*x^(1/3)*(Sqrt[d]*Log[d + e/x^(2/3)]^
2*(6*e + d*x^(2/3)*Log[d + e/x^(2/3)]) - 6*e*Sqrt[e/(e + d*x^(2/3))]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1
/2, 1/2}, {3/2, 3/2, 3/2}, d/(d + e/x^(2/3))] + Log[d + e/x^(2/3)]*(4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2
}, {3/2, 3/2}, d/(d + e/x^(2/3))] + Sqrt[d + e/x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e/x^(2/3)]]*Log[d + e/x^(2/3)]
)) + 6*Sqrt[d]*e*((4*Sqrt[e/x^(2/3)]*ArcTanh[Sqrt[e/x^(2/3)]/Sqrt[-d]]*(Log[d + e/x^(2/3)] - Log[1 + e/(d*x^(2
/3))]))/Sqrt[-d] - Sqrt[-(e/(d*x^(2/3)))]*(2*Log[(1 + Sqrt[-(e/(d*x^(2/3)))])/2]^2 - 4*Log[(1 + Sqrt[-(e/(d*x^
(2/3)))])/2]*Log[1 + e/(d*x^(2/3))] + Log[1 + e/(d*x^(2/3))]^2 - 4*PolyLog[2, 1/2 - Sqrt[-(e/(d*x^(2/3)))]/2])
)))/d^(3/2)

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Maple [A]  time = 0.345, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right ) + a\right )}^{3}\,{d x} \end{align*}

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)