\(\int x^2 (a+b \log (c (d+e x^{2/3})^n))^3 \, dx\) [485]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac {3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac {637984 b^3 d^3 n^3 x}{297675 e^3}-\frac {221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac {3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac {16}{729} b^3 n^3 x^3+\frac {3475504 b^3 d^{9/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} 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 )}{315 e^{9/2}}+\frac {4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} 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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac {4504 i b^3 d^{9/2} n^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{315 e^{9/2}}+\frac {2 b d^5 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 )}{3 e^4} \]

[Out]

4504/315*a*b^2*d^4*n^2*x^(1/3)/e^4-3475504/99225*b^3*d^4*n^3*x^(1/3)/e^4+637984/297675*b^3*d^3*n^3*x/e^3-22134
4/496125*b^3*d^2*n^3*x^(5/3)/e^2+3088/27783*b^3*d*n^3*x^(7/3)/e-16/729*b^3*n^3*x^3+3475504/99225*b^3*d^(9/2)*n
^3*arctan(x^(1/3)*e^(1/2)/d^(1/2))/e^(9/2)-4504/315*I*b^3*d^(9/2)*n^3*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x^(1/3)
*e^(1/2)))/e^(9/2)+4504/315*b^3*d^4*n^2*x^(1/3)*ln(c*(d+e*x^(2/3))^n)/e^4-1984/945*b^2*d^3*n^2*x*(a+b*ln(c*(d+
e*x^(2/3))^n))/e^3+1144/1575*b^2*d^2*n^2*x^(5/3)*(a+b*ln(c*(d+e*x^(2/3))^n))/e^2-128/441*b^2*d*n^2*x^(7/3)*(a+
b*ln(c*(d+e*x^(2/3))^n))/e+8/81*b^2*n^2*x^3*(a+b*ln(c*(d+e*x^(2/3))^n))-4504/315*b^2*d^(9/2)*n^2*arctan(x^(1/3
)*e^(1/2)/d^(1/2))*(a+b*ln(c*(d+e*x^(2/3))^n))/e^(9/2)-2*b*d^4*n*x^(1/3)*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^4+2/3
*b*d^3*n*x*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^3-2/5*b*d^2*n*x^(5/3)*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^2+2/7*b*d*n*x
^(7/3)*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e-2/9*b*n*x^3*(a+b*ln(c*(d+e*x^(2/3))^n))^2+1/3*x^3*(a+b*ln(c*(d+e*x^(2/3
))^n))^3-9008/315*b^3*d^(9/2)*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^
(9/2)-4504/315*I*b^3*d^(9/2)*n^3*arctan(x^(1/3)*e^(1/2)/d^(1/2))^2/e^(9/2)+2/3*b*d^5*n*Unintegrable((a+b*ln(c*
(d+e*x^(2/3))^n))^2/(d+e*x^(2/3))/x^(2/3),x)/e^4

Rubi [N/A]

Not integrable

Time = 1.93 (sec) , antiderivative size = 24, 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 = {} \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \]

[In]

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

[Out]

(4504*a*b^2*d^4*n^2*x^(1/3))/(315*e^4) - (3475504*b^3*d^4*n^3*x^(1/3))/(99225*e^4) + (637984*b^3*d^3*n^3*x)/(2
97675*e^3) - (221344*b^3*d^2*n^3*x^(5/3))/(496125*e^2) + (3088*b^3*d*n^3*x^(7/3))/(27783*e) - (16*b^3*n^3*x^3)
/729 + (3475504*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(99225*e^(9/2)) - (((4504*I)/315)*b^3*d^(9/
2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(9/2) - (9008*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*
Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(315*e^(9/2)) + (4504*b^3*d^4*n^2*x^(1/3)*Log[c*(d + e*x^(2/3)
)^n])/(315*e^4) - (1984*b^2*d^3*n^2*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/(945*e^3) + (1144*b^2*d^2*n^2*x^(5/3)*
(a + b*Log[c*(d + e*x^(2/3))^n]))/(1575*e^2) - (128*b^2*d*n^2*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(441*e
) + (8*b^2*n^2*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/81 - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[
d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(315*e^(9/2)) - (2*b*d^4*n*x^(1/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e
^4 + (2*b*d^3*n*x*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3)
)^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(7*e) - (2*b*n*x^3*(a + b*Log[c*(d + e
*x^(2/3))^n])^2)/9 + (x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/3 - (((4504*I)/315)*b^3*d^(9/2)*n^3*PolyLog[2, 1
- (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(9/2) + (2*b*d^5*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^4

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-(2 b e n) \text {Subst}\left (\int \frac {x^{10} \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 ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-(2 b e n) \text {Subst}\left (\int \left (\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^5}-\frac {d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^4}+\frac {d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^3}-\frac {d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^2}+\frac {x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-(2 b n) \text {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )-\frac {\left (2 b d^4 n\right ) \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^4}+\frac {\left (2 b d^5 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^4}+\frac {\left (2 b d^3 n\right ) \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 )}{e^3}-\frac {\left (2 b d^2 n\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^2}+\frac {(2 b d n) \text {Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e} \\ & = -\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 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^4}-\frac {1}{7} \left (8 b^2 d n^2\right ) \text {Subst}\left (\int \frac {x^8 \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 {\left (8 b^2 d^4 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 )}{e^3}-\frac {\left (8 b^2 d^3 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 )}{3 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {x^6 \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 )}{5 e}+\frac {1}{9} \left (8 b^2 e n^2\right ) \text {Subst}\left (\int \frac {x^{10} \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 {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 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^4}-\frac {1}{7} \left (8 b^2 d n^2\right ) \text {Subst}\left (\int \left (-\frac {d^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac {d^2 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^4 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 )}{e^3}-\frac {\left (8 b^2 d^3 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 )}{3 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \left (\frac {d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac {d^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e}+\frac {1}{9} \left (8 b^2 e n^2\right ) \text {Subst}\left (\int \left (\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5}-\frac {d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac {d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.80 (sec) , antiderivative size = 1552, normalized size of antiderivative = 64.67 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=-\frac {2 b d^4 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^4}+\frac {2 b d^3 n 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}{3 e^3}-\frac {2 b d^2 n x^{5/3} \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}{5 e^2}+\frac {2 b d n x^{7/3} \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}{7 e}+\frac {2 b d^{9/2} 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^{9/2}}+b n x^3 \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+\frac {1}{9} x^3 \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 (3 a-2 b n-3 b n \log \left (d+e x^{2/3}\right )+3 b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {b^3 n^3 \left (1094783760 d^{9/2} \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 )-e x^{2/3} \left (-16 \left (68423985 d^4-4186770 d^3 e x^{2/3}+871542 d^2 e^2 x^{4/3}-217125 d e^3 x^2+42875 e^4 x^{8/3}\right )+2520 \left (177345 d^4-26040 d^3 e x^{2/3}+9009 d^2 e^2 x^{4/3}-3600 d e^3 x^2+1225 e^4 x^{8/3}\right ) \log \left (d+e x^{2/3}\right )-198450 \left (315 d^4-105 d^3 e x^{2/3}+63 d^2 e^2 x^{4/3}-45 d e^3 x^2+35 e^4 x^{8/3}\right ) \log ^2\left (d+e x^{2/3}\right )[phone] e^4 x^{8/3} \log ^3\left (d+e x^{2/3}\right )\right )[phone] d^{9/2} \sqrt {\frac {e x^{2/3}}{d+e x^{2/3}}} \left (8 \sqrt {d} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^{2/3}}\right )+\log \left (d+e x^{2/3}\right ) \left (4 \sqrt {d} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^{2/3}}\right )+\sqrt {d+e x^{2/3}} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^{2/3}}}\right ) \log \left (d+e x^{2/3}\right )\right )\right )[phone] (-d)^{9/2} \left (4 \sqrt {e x^{2/3}} \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 {-d} \sqrt {-\frac {e x^{2/3}}{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 )\right )\right )}{31255875 e^5 \sqrt [3]{x}}+\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 {1418760 d^{9/2} \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}}}}+1225 \left (d+e x^{2/3}\right )^4 \left (8-36 \log \left (d+e x^{2/3}\right )+81 \log ^2\left (d+e x^{2/3}\right )\right )-100 d \left (d+e x^{2/3}\right )^3 \left (680-2331 \log \left (d+e x^{2/3}\right )+3969 \log ^2\left (d+e x^{2/3}\right )\right )+d^4 \left (1737752-709380 \log \left (d+e x^{2/3}\right )+99225 \log ^2\left (d+e x^{2/3}\right )\right )-4 d^3 \left (d+e x^{2/3}\right ) \left (119516-159390 \log \left (d+e x^{2/3}\right )+99225 \log ^2\left (d+e x^{2/3}\right )\right )+6 d^2 \left (d+e x^{2/3}\right )^2 \left (36212-85680 \log \left (d+e x^{2/3}\right )+99225 \log ^2\left (d+e x^{2/3}\right )\right )+\frac {396900 (-d)^{9/2} \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 {99225 d^4 \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 )}{99225 e^4} \]

[In]

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

[Out]

(-2*b*d^4*n*x^(1/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/e^4 + (2*b*d^3*n*x*(a - b*n*L
og[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a - b*n*Log[d + e*x^(2/3)] +
b*Log[c*(d + e*x^(2/3))^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3)
)^n])^2)/(7*e) + (2*b*d^(9/2)*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^(9/2) + b*n*x^3*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n]
)^2 + (x^3*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*(3*a - 2*b*n - 3*b*n*Log[d + e*x^(2/3)]
+ 3*b*Log[c*(d + e*x^(2/3))^n]))/9 - (b^3*n^3*(1094783760*d^(9/2)*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)]] - e*x^(2/3)*(-16*(68423985*d^4 - 4186770*d^3*e*x^(2/3) + 871542
*d^2*e^2*x^(4/3) - 217125*d*e^3*x^2 + 42875*e^4*x^(8/3)) + 2520*(177345*d^4 - 26040*d^3*e*x^(2/3) + 9009*d^2*e
^2*x^(4/3) - 3600*d*e^3*x^2 + 1225*e^4*x^(8/3))*Log[d + e*x^(2/3)] - 198450*(315*d^4 - 105*d^3*e*x^(2/3) + 63*
d^2*e^2*x^(4/3) - 45*d*e^3*x^2 + 35*e^4*x^(8/3))*Log[d + e*x^(2/3)]^2 + 10418625*e^4*x^(8/3)*Log[d + e*x^(2/3)
]^3) + 62511750*d^(9/2)*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]*(8*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {
3/2, 3/2, 3/2}, d/(d + e*x^(2/3))] + Log[d + e*x^(2/3)]*(4*Sqrt[d]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/
2}, d/(d + e*x^(2/3))] + Sqrt[d + e*x^(2/3)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]]*Log[d + e*x^(2/3)])) + 111727
350*(-d)^(9/2)*(4*Sqrt[e*x^(2/3)]*ArcTanh[Sqrt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/
d]) - Sqrt[-d]*Sqrt[-((e*x^(2/3))/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*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))))/
(31255875*e^5*x^(1/3)) + (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])*((1418760*
d^(9/2)*ArcSin[Sqrt[d]/Sqrt[d + e*x^(2/3)]])/(Sqrt[d + e*x^(2/3)]*Sqrt[(e*x^(2/3))/(d + e*x^(2/3))]) + 1225*(d
+ e*x^(2/3))^4*(8 - 36*Log[d + e*x^(2/3)] + 81*Log[d + e*x^(2/3)]^2) - 100*d*(d + e*x^(2/3))^3*(680 - 2331*Lo
g[d + e*x^(2/3)] + 3969*Log[d + e*x^(2/3)]^2) + d^4*(1737752 - 709380*Log[d + e*x^(2/3)] + 99225*Log[d + e*x^(
2/3)]^2) - 4*d^3*(d + e*x^(2/3))*(119516 - 159390*Log[d + e*x^(2/3)] + 99225*Log[d + e*x^(2/3)]^2) + 6*d^2*(d
+ e*x^(2/3))^2*(36212 - 85680*Log[d + e*x^(2/3)] + 99225*Log[d + e*x^(2/3)]^2) + (396900*(-d)^(9/2)*ArcTanh[Sq
rt[e*x^(2/3)]/Sqrt[-d]]*(Log[d + e*x^(2/3)] - Log[1 + (e*x^(2/3))/d]))/Sqrt[e*x^(2/3)] - (99225*d^4*(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*PolyLog[2, 1/2 - Sqrt[-((e*x^(2/3))/d)]/2]))/Sqrt[-((e*x^(2/3))/d)]))/(99225*e^4)

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x^2*(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 = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

integrate(x^2*(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^2, x)

Mupad [N/A]

Not integrable

Time = 1.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3 \,d x \]

[In]

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

[Out]

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