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

Optimal. Leaf size=486 \[ -\frac {2 b d^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{2/3} \left (d+e x^{2/3}\right )},x\right )}{e}+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{e^{3/2}}+\frac {8}{3} b^2 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {32 a b^2 d n^2 \sqrt [3]{x}}{e}-2 b n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {6 b d n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e}+x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac {32 b^3 d n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e}+\frac {32 i b^3 d^{3/2} n^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac {32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{e^{3/2}}-\frac {208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {64 b^3 d^{3/2} n^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{e^{3/2}}+\frac {208 b^3 d n^3 \sqrt [3]{x}}{3 e}-\frac {16}{9} b^3 n^3 x \]

[Out]

-32*a*b^2*d*n^2*x^(1/3)/e+208/3*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 [A]  time = 1.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

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 = 1.28, 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]

-1/2*(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)]))))/(d*(-((e*x^(2/3))/d))^(3/2)) + (3*b^2*n^2*x*(3*(d + e*x^(2/3))*Hyp
ergeometricPFQ[{-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))*Hyperge
ometricPFQ[{-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])

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a^{3}, x\right ) \]

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ b^{3} n^{3} x \log \left (e x^{\frac {2}{3}} + d\right )^{3} - {\left (2 \, e n {\left (\frac {3 \, d^{2} \arctan \left (\frac {e x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {e x - 3 \, d x^{\frac {1}{3}}}{e^{2}}\right )} - 3 \, x \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )\right )} a^{2} b + a^{3} x + \int -\frac {{\left (2 \, b^{3} e n x - 3 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x - 3 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{\frac {1}{3}}\right )} n^{2} \log \left (e x^{\frac {2}{3}} + d\right )^{2} - 3 \, {\left ({\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c)\right )} x + {\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c)\right )} x^{\frac {1}{3}}\right )} n \log \left (e x^{\frac {2}{3}} + d\right ) - {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2}\right )} x - {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2}\right )} x^{\frac {1}{3}}}{e x + d x^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________