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

Optimal. Leaf size=485 \[ -\frac{2 b d^2 n \text{Unintegrable}\left (\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{2/3} \left (d+e x^{2/3}\right )},x\right )}{e}+\frac{32 i b^3 d^{3/2} n^3 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+\frac{8}{3} b^2 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{32 a b^2 d n^2 \sqrt [3]{x}}{e}-2 b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{6 b d n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac{32 b^3 d n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}+\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{e^{3/2}}-\frac{208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{3/2}}+\frac{64 b^3 d^{3/2} n^3 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{e^{3/2}}+\frac{208 b^3 d n^3 \sqrt [3]{x}}{3 e}-\frac{16}{9} b^3 n^3 x \]

[Out]

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

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Rubi [A]  time = 1.07922, 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 = {} \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.24265, size = 598, normalized size = 1.23 \[ \frac{3 b^2 n^2 x \left (-a-b \log \left (c \left (d+e x^{2/3}\right )^n\right )+b n \log \left (d+e x^{2/3}\right )\right ) \left (3 \left (d+e x^{2/3}\right ) \, _4F_3\left (-\frac{1}{2},1,1,1;2,2,2;\frac{x^{2/3} e}{d}+1\right )+\log \left (d+e x^{2/3}\right ) \left (\left (d-d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}\right ) \log \left (d+e x^{2/3}\right )-3 \left (d+e x^{2/3}\right ) \, _3F_2\left (-\frac{1}{2},1,1;2,2;\frac{x^{2/3} e}{d}+1\right )\right )\right )}{d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}}-\frac{b^3 n^3 x \left (\log \left (d+e x^{2/3}\right ) \left (18 \left (d+e x^{2/3}\right ) \, _4F_3\left (-\frac{1}{2},1,1,1;2,2,2;\frac{x^{2/3} e}{d}+1\right )+\log \left (d+e x^{2/3}\right ) \left (2 \left (d-d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}\right ) \log \left (d+e x^{2/3}\right )-9 \left (d+e x^{2/3}\right ) \, _3F_2\left (-\frac{1}{2},1,1;2,2;\frac{x^{2/3} e}{d}+1\right )\right )\right )-18 \left (d+e x^{2/3}\right ) \, _5F_4\left (-\frac{1}{2},1,1,1,1;2,2,2,2;\frac{x^{2/3} e}{d}+1\right )\right )}{2 d \left (-\frac{e x^{2/3}}{d}\right )^{3/2}}-\frac{6 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2}{e^{3/2}}+3 b n x \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+\frac{6 b d n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2}{e}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )-2 b n\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^3*n^3*x*(-18*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1, 1}, {2, 2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[
d + e*x^(2/3)]*(18*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d +
e*x^(2/3)]*(-9*(d + e*x^(2/3))*HypergeometricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + (e*x^(2/3))/d] + 2*(d - d*(-((e*x^(
2/3))/d))^(3/2))*Log[d + e*x^(2/3)]))))/(2*d*(-((e*x^(2/3))/d))^(3/2)) + (3*b^2*n^2*x*(3*(d + e*x^(2/3))*Hyper
geometricPFQ[{-1/2, 1, 1, 1}, {2, 2, 2}, 1 + (e*x^(2/3))/d] + Log[d + e*x^(2/3)]*(-3*(d + e*x^(2/3))*Hypergeom
etricPFQ[{-1/2, 1, 1}, {2, 2}, 1 + (e*x^(2/3))/d] + (d - d*(-((e*x^(2/3))/d))^(3/2))*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*(-((e*x^(2/3))/d))^(3/2)) + (6*b*d*n*x^(1/3)*(a - b
*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e - (6*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(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*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])^2*
(a - 2*b*n - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])

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Maple [A]  time = 0.334, 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 ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\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((e*x^(2/3) + d)^n*c)^3 + 3*a*b^2*log((e*x^(2/3) + d)^n*c)^2 + 3*a^2*b*log((e*x^(2/3) + d)^n*c
) + 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 ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\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((e*x^(2/3) + d)^n*c) + a)^3, x)