\(\int (a+b \log (c (d+e x^{2/3})^n))^3 \, dx\) [486]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=-\frac {32 a b^2 d n^2 \sqrt [3]{x}}{e}+\frac {208 b^3 d n^3 \sqrt [3]{x}}{3 e}-\frac {16}{9} b^3 n^3 x-\frac {208 b^3 d^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {32 i b^3 d^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}+\frac {64 b^3 d^{3/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {32 b^3 d n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}+\frac {8}{3} b^2 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {32 b^2 d^{3/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+\frac {6 b d n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e}-2 b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {32 i b^3 d^{3/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {2 b d^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^{2/3}},x\right )}{e} \]

[Out]

-32*a*b^2*d*n^2*x^(1/3)/e+208/3*b^3*d*n^3*x^(1/3)/e-16/9*b^3*n^3*x-208/3*b^3*d^(3/2)*n^3*arctan(x^(1/3)*e^(1/2
)/d^(1/2))/e^(3/2)+32*I*b^3*d^(3/2)*n^3*arctan(x^(1/3)*e^(1/2)/d^(1/2))^2/e^(3/2)-32*b^3*d*n^2*x^(1/3)*ln(c*(d
+e*x^(2/3))^n)/e+8/3*b^2*n^2*x*(a+b*ln(c*(d+e*x^(2/3))^n))+32*b^2*d^(3/2)*n^2*arctan(x^(1/3)*e^(1/2)/d^(1/2))*
(a+b*ln(c*(d+e*x^(2/3))^n))/e^(3/2)+6*b*d*n*x^(1/3)*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e-2*b*n*x*(a+b*ln(c*(d+e*x^(
2/3))^n))^2+x*(a+b*ln(c*(d+e*x^(2/3))^n))^3+64*b^3*d^(3/2)*n^3*arctan(x^(1/3)*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d
^(1/2)+I*x^(1/3)*e^(1/2)))/e^(3/2)+32*I*b^3*d^(3/2)*n^3*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x^(1/3)*e^(1/2)))/e^(
3/2)-2*b*d^2*n*Unintegrable((a+b*ln(c*(d+e*x^(2/3))^n))^2/(d+e*x^(2/3))/x^(2/3),x)/e

Rubi [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

[In]

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

[Out]

(-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*} \text {integral}& = 3 \text {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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\left (48 b^3 d n^3\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1299\) vs. \(2(486)=972\).

Time = 5.89 (sec) , antiderivative size = 1299, normalized size of antiderivative = 64.95 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\frac {6 b d n \sqrt [3]{x} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e}-\frac {6 b d^{3/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^{3/2}}+3 b n x \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+x \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (a-2 b n-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {b^2 n^2 \sqrt [3]{x} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (-\frac {96 d^{3/2} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right )}{\sqrt {d+e x^{2/3}} \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}}}-d \left (104-48 \log \left (d+e x^{2/3}\right )+9 \log ^2\left (d+e x^{2/3}\right )\right )+\left (d+e x^{2/3}\right ) \left (8-12 \log \left (d+e x^{2/3}\right )+9 \log ^2\left (d+e x^{2/3}\right )\right )+\frac {36 (-d)^{3/2} \text {arctanh}\left (\frac {\sqrt {e x^{2/3}}}{\sqrt {-d}}\right ) \left (\log \left (d+e x^{2/3}\right )-\log \left (1+\frac {e x^{2/3}}{d}\right )\right )}{\sqrt {e x^{2/3}}}+\frac {9 d \left (2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right )-4 \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right ) \log \left (1+\frac {e x^{2/3}}{d}\right )+\log ^2\left (1+\frac {e x^{2/3}}{d}\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {e x^{2/3}}{d}}\right )\right )}{\sqrt {-\frac {e x^{2/3}}{d}}}\right )}{3 e}+\frac {b^3 n^3 \left (624 d e x^{2/3}-16 e^2 x^{4/3}+624 d^{3/2} \sqrt {d+e x^{2/3}} \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right )+432 d^2 \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} \, _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+e x^{2/3}}\right )+144 d^2 \sqrt {-\frac {e x^{2/3}}{d}} \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right )-288 d e x^{2/3} \log \left (d+e x^{2/3}\right )+24 e^2 x^{4/3} \log \left (d+e x^{2/3}\right )+288 \sqrt {-d} d \sqrt {e x^{2/3}} \text {arctanh}\left (\frac {\sqrt {e x^{2/3}}}{\sqrt {-d}}\right ) \log \left (d+e x^{2/3}\right )+216 d^2 \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^{2/3}}\right ) \log \left (d+e x^{2/3}\right )+54 d e x^{2/3} \log ^2\left (d+e x^{2/3}\right )-18 e^2 x^{4/3} \log ^2\left (d+e x^{2/3}\right )+54 d^{3/2} \sqrt {d+e x^{2/3}} \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right ) \log ^2\left (d+e x^{2/3}\right )+9 e^2 x^{4/3} \log ^3\left (d+e x^{2/3}\right )+288 (-d)^{3/2} \sqrt {e x^{2/3}} \text {arctanh}\left (\frac {\sqrt {e x^{2/3}}}{\sqrt {-d}}\right ) \log \left (1+\frac {e x^{2/3}}{d}\right )-288 d^2 \sqrt {-\frac {e x^{2/3}}{d}} \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {e x^{2/3}}{d}}\right )\right ) \log \left (1+\frac {e x^{2/3}}{d}\right )+72 d^2 \sqrt {-\frac {e x^{2/3}}{d}} \log ^2\left (1+\frac {e x^{2/3}}{d}\right )-288 d^2 \sqrt {-\frac {e x^{2/3}}{d}} \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {e x^{2/3}}{d}}\right )\right )}{9 e^2 \sqrt [3]{x}} \]

[In]

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

[Out]

(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]) + (b^2*n^2*x^(1/3)*(a
- b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])*((-96*d^(3/2)*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/(Sq
rt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]) - d*(104 - 48*Log[d + e*x^(2/3)] + 9*Log[d + e*x^(2/3)]^2
) + (d + e*x^(2/3))*(8 - 12*Log[d + e*x^(2/3)] + 9*Log[d + e*x^(2/3)]^2) + (36*(-d)^(3/2)*ArcTanh[Sqrt[e*x^(2/
3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]))/Sqrt[e*x^(2/3)] + (9*d*(2*Log[(1 + Sqrt[-((e*x^(2
/3))/d)])/2]^2 - 4*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*Log[1 + (e*x^(2/3))/d] + Log[1 + (e*x^(2/3))/d]^2 - 4*P
olyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/Sqrt[-((e*x^(2/3))/d)]))/(3*e) + (b^3*n^3*(624*d*e*x^(2/3) - 16*e^
2*x^(4/3) + 624*d^(3/2)*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3
)]] + 432*d^2*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, d/(d
+ e*x^(2/3))] + 144*d^2*Sqrt[-((e*x^(2/3))/d)]*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]^2 - 288*d*e*x^(2/3)*Log[d +
e*x^(2/3)] + 24*e^2*x^(4/3)*Log[d + e*x^(2/3)] + 288*Sqrt[-d]*d*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[
-d]]*Log[d + e*x^(2/3)] + 216*d^2*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3
/2}, d/(d + e*x^(2/3))]*Log[d + e*x^(2/3)] + 54*d*e*x^(2/3)*Log[d + e*x^(2/3)]^2 - 18*e^2*x^(4/3)*Log[d + e*x^
(2/3)]^2 + 54*d^(3/2)*Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]
]*Log[d + e*x^(2/3)]^2 + 9*e^2*x^(4/3)*Log[d + e*x^(2/3)]^3 + 288*(-d)^(3/2)*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^
(2/3)]/Sqrt[-d]]*Log[1 + (e*x^(2/3))/d] - 288*d^2*Sqrt[-((e*x^(2/3))/d)]*Log[(1 + Sqrt[-((e*x^(2/3))/d)])/2]*L
og[1 + (e*x^(2/3))/d] + 72*d^2*Sqrt[-((e*x^(2/3))/d)]*Log[1 + (e*x^(2/3))/d]^2 - 288*d^2*Sqrt[-((e*x^(2/3))/d)
]*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/(9*e^2*x^(1/3))

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.10 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

integral(b^3*log((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)

Sympy [N/A]

Not integrable

Time = 61.66 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int \left (a + b \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}\right )^{3}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3 \,d x \]

[In]

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

[Out]

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