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

Optimal. Leaf size=482 \[ -\frac{2 b d^2 n \text{Unintegrable}\left (\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x^{2/3} \left (d x^{2/3}+e\right )},x\right )}{e}-\frac{32 i b^3 d^{3/2} n^3 \text{PolyLog}\left (2,-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}-\frac{208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}+\frac{64 b^3 d^{3/2} n^3 \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{208 b^3 d n^3}{3 e \sqrt [3]{x}}+\frac{16 b^3 n^3}{9 x} \]

[Out]

(16*b^3*n^3)/(9*x) - (208*b^3*d*n^3)/(3*e*x^(1/3)) - (208*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(
3*e^(3/2)) - ((32*I)*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/e^(3/2) + (64*b^3*d^(3/2)*n^3*ArcTan
[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) - (8*b^2*n^2*(a + b*Lo
g[c*(d + e/x^(2/3))^n]))/(3*x) + (32*b^2*d*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(e*x^(1/3)) + (32*b^2*d^(3/2)
*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^(3/2) + (2*b*n*(a + b*Log[c*(d + e/
x^(2/3))^n])^2)/x - (6*b*d*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(e*x^(1/3)) - (a + b*Log[c*(d + e/x^(2/3))^n]
)^3/x - ((32*I)*b^3*d^(3/2)*n^3*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) - (2*b*d^2
*n*Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^n])^2/((e + d*x^(2/3))*x^(2/3)), x])/e

________________________________________________________________________________________

Rubi [A]  time = 1.38325, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(16*b^3*n^3)/(9*x) - (208*b^3*d*n^3)/(3*e*x^(1/3)) - (208*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(
3*e^(3/2)) - ((32*I)*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/e^(3/2) + (64*b^3*d^(3/2)*n^3*ArcTan
[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) - (8*b^2*n^2*(a + b*Lo
g[c*(d + e/x^(2/3))^n]))/(3*x) + (32*b^2*d*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(e*x^(1/3)) + (32*b^2*d^(3/2)
*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^(3/2) + (2*b*n*(a + b*Log[c*(d + e/
x^(2/3))^n])^2)/x - (6*b*d*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(e*x^(1/3)) - (a + b*Log[c*(d + e/x^(2/3))^n]
)^3/x - ((32*I)*b^3*d^(3/2)*n^3*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) - (6*b*d^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)])/e

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^2} \, 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^4} \, 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}{x}-(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}{\left (d+\frac{e}{x^2}\right ) x^6} \, 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}{x}-(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}{e x^4}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^2 x^2}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^2 \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}{x}-(6 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^4} \, dx,x,\sqrt [3]{x}\right )+\frac{(6 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^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (6 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}{e+d x^2} \, 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}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\left (24 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^4} \, dx,x,\sqrt [3]{x}\right )+\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^6} \, 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}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\left (24 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^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 )+\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^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 )\\ &=\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\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^4} \, dx,x,\sqrt [3]{x}\right )-\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^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (24 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^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\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 )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{\left (24 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 )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )+\left (48 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )+\left (16 b^3 d^2 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 )+\left (48 b^3 d^2 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 )-\frac{1}{3} \left (16 b^3 e n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^6} \, 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 )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\left (48 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (16 b^3 d^{3/2} 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 )}{\sqrt{e}}+\frac{\left (48 b^3 d^{3/2} 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 )}{\sqrt{e}}-\frac{1}{3} \left (16 b^3 e n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{16 b^3 n^3}{9 x}-\frac{64 b^3 d n^3}{e \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 )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{1}{3} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (48 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{\left (16 b^3 d^{3/2} 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 )}{\sqrt{e}}+\frac{\left (48 b^3 d^{3/2} 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 )}{\sqrt{e}}\\ &=\frac{16 b^3 n^3}{9 x}-\frac{208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac{64 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{\left (16 i b^3 d^{3/2} 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^{3/2}}+\frac{\left (48 i b^3 d^{3/2} 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^{3/2}}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e}\\ &=\frac{16 b^3 n^3}{9 x}-\frac{208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac{208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}+\frac{64 b^3 d^{3/2} 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 )}{e^{3/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (16 b^3 d^2 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^2}-\frac{\left (48 b^3 d^2 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^2}\\ &=\frac{16 b^3 n^3}{9 x}-\frac{208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac{208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}+\frac{64 b^3 d^{3/2} 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 )}{e^{3/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{32 i b^3 d^{3/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{\left (6 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}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ \end{align*}

Mathematica [A]  time = 2.23848, size = 1097, normalized size = 2.28 \[ \frac{b^3 \left (18 \left (x^{2/3} d+e\right ) \, _5F_4\left (-\frac{1}{2},1,1,1,1;2,2,2,2;\frac{e}{d x^{2/3}}+1\right )-\log \left (d+\frac{e}{x^{2/3}}\right ) \left (18 \left (x^{2/3} d+e\right ) \, _4F_3\left (-\frac{1}{2},1,1,1;2,2,2;\frac{e}{d x^{2/3}}+1\right )+\log \left (d+\frac{e}{x^{2/3}}\right ) \left (2 \left (x^{2/3} d+e \sqrt{-\frac{e}{d x^{2/3}}}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )-9 \left (x^{2/3} d+e\right ) \, _3F_2\left (-\frac{1}{2},1,1;2,2;\frac{e}{d x^{2/3}}+1\right )\right )\right )\right ) n^3}{2 e \sqrt{-\frac{e}{d x^{2/3}}} x}+\frac{b^2 \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 ) \left (9 e^{3/2} \log ^2\left (d+\frac{e}{x^{2/3}}\right )-12 e^{3/2} \log \left (d+\frac{e}{x^{2/3}}\right )+18 \sqrt{-d} d x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )+18 (-d)^{3/2} x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )+36 d \sqrt{e} x^{2/3} \log \left (d+\frac{e}{x^{2/3}}\right )+9 (-d)^{3/2} x \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )+9 \sqrt{-d} d x \log ^2\left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right )+8 e^{3/2}+96 d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} \sqrt [3]{x}}\right )+18 \sqrt{-d} d x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+18 (-d)^{3/2} x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right )+36 (-d)^{3/2} x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+36 \sqrt{-d} d x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+36 \sqrt{-d} d x \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+18 (-d)^{3/2} x \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+18 \sqrt{-d} d x \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right )+36 (-d)^{3/2} x \text{PolyLog}\left (2,\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )-96 d \sqrt{e} x^{2/3}\right ) n^2}{3 e^{3/2} x}-\frac{6 b d^{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}{e^{3/2}}-\frac{3 b \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}{x}-\frac{6 b d \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}{e \sqrt [3]{x}}-\frac{\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 \left (a-2 b n-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 )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*n^3*(18*(e + d*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + e/(d*x^(2/3))] - Log[d +
e/x^(2/3)]*(18*(e + d*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + e/(d*x^(2/3))] + Log[d + e/x^
(2/3)]*(-9*(e + d*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + e/(d*x^(2/3))] + 2*(e*Sqrt[-(e/(d*x^(2/
3)))] + d*x^(2/3))*Log[d + e/x^(2/3)]))))/(2*e*Sqrt[-(e/(d*x^(2/3)))]*x) - (6*b*d*n*(a - b*n*Log[d + e/x^(2/3)
] + b*Log[c*(d + e/x^(2/3))^n])^2)/(e*x^(1/3)) - (6*b*d^(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)/e^(3/2) - (3*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 - ((a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2*(a - 2*b*
n - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n]))/x + (b^2*n^2*(-a + b*n*Log[d + e/x^(2/3)] - b*Log[c*
(d + e/x^(2/3))^n])*(8*e^(3/2) - 96*d*Sqrt[e]*x^(2/3) + 96*d^(3/2)*x*ArcTan[Sqrt[e]/(Sqrt[d]*x^(1/3))] - 12*e^
(3/2)*Log[d + e/x^(2/3)] + 36*d*Sqrt[e]*x^(2/3)*Log[d + e/x^(2/3)] + 9*e^(3/2)*Log[d + e/x^(2/3)]^2 + 18*Sqrt[
-d]*d*x*Log[d + e/x^(2/3)]*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 9*(-d)^(3/2)*x*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]^2
+ 18*(-d)^(3/2)*x*Log[d + e/x^(2/3)]*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 9*Sqrt[-d]*d*x*Log[Sqrt[e] + Sqrt[-d]*x
^(1/3)]^2 + 18*Sqrt[-d]*d*x*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] + 18*(-d
)^(3/2)*x*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] + 36*(-d)^(3/2)*x*Log[Sqrt[e
] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])] + 36*Sqrt[-d]*d*x*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log
[(Sqrt[-d]*x^(1/3))/Sqrt[e]] + 36*Sqrt[-d]*d*x*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] + 18*(-d)^(3/2)*x*Po
lyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] + 18*Sqrt[-d]*d*x*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2
] + 36*(-d)^(3/2)*x*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]]))/(3*e^(3/2)*x)

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Maple [A]  time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^2,x)

[Out]

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

[Out]

Timed out

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

[Out]

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