3.6.30 \(\int \frac {(a+b \log (c (d+\frac {e}{x^{2/3}})^n))^3}{x^2} \, dx\) [530]

Optimal. Leaf size=483 \[ \frac {16 b^3 n^3}{9 x}-\frac {208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac {208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{3/2}}-\frac {32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )^2}{e^{3/2}}+\frac {64 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \log \left (2-\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac {32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac {32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac {2 b n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {6 b d n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac {32 i b^3 d^{3/2} n^3 \text {Li}_2\left (-1+\frac {2 \sqrt {e}}{\sqrt {e}-i \sqrt {d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac {2 b d^2 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{\left (e+d x^{2/3}\right ) x^{2/3}},x\right )}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.87, 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 = {} \begin {gather*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(1097\) vs. \(2(483)=966\).
time = 1.30, size = 1097, normalized size = 2.27 \begin {gather*} \frac {b^3 n^3 \left (18 \left (e+d x^{2/3}\right ) \, _5F_4\left (-\frac {1}{2},1,1,1,1;2,2,2,2;1+\frac {e}{d x^{2/3}}\right )-\log \left (d+\frac {e}{x^{2/3}}\right ) \left (18 \left (e+d x^{2/3}\right ) \, _4F_3\left (-\frac {1}{2},1,1,1;2,2,2;1+\frac {e}{d x^{2/3}}\right )+\log \left (d+\frac {e}{x^{2/3}}\right ) \left (-9 \left (e+d x^{2/3}\right ) \, _3F_2\left (-\frac {1}{2},1,1;2,2;1+\frac {e}{d x^{2/3}}\right )+2 \left (e \sqrt {-\frac {e}{d x^{2/3}}}+d x^{2/3}\right ) \log \left (d+\frac {e}{x^{2/3}}\right )\right )\right )\right )}{2 e \sqrt {-\frac {e}{d x^{2/3}}} x}-\frac {6 b d n \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac {6 b d^{3/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right ) \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{e^{3/2}}-\frac {3 b n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac {\left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \left (a-2 b n-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{x}+\frac {b^2 n^2 \left (-a+b n \log \left (d+\frac {e}{x^{2/3}}\right )-b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \left (8 e^{3/2}-96 d \sqrt {e} x^{2/3}+96 d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )-12 e^{3/2} \log \left (d+\frac {e}{x^{2/3}}\right )+36 d \sqrt {e} x^{2/3} \log \left (d+\frac {e}{x^{2/3}}\right )+9 e^{3/2} \log ^2\left (d+\frac {e}{x^{2/3}}\right )+18 \sqrt {-d} d x \log \left (d+\frac {e}{x^{2/3}}\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+9 (-d)^{3/2} x \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+18 (-d)^{3/2} x \log \left (d+\frac {e}{x^{2/3}}\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+9 \sqrt {-d} d x \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+18 \sqrt {-d} d x \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+18 (-d)^{3/2} x \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )+36 (-d)^{3/2} x \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+36 \sqrt {-d} d x \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+36 \sqrt {-d} d x \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+18 (-d)^{3/2} x \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+18 \sqrt {-d} d x \text {Li}_2\left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )+36 (-d)^{3/2} x \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{3 e^{3/2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )^{3}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^3*n^3*log(d*x^(2/3) + e)^3/x - integrate(-((2*b^3*d*n*x + 3*(b^3*log(c) + a*b^2)*x^(1/3)*e + 3*(b^3*d*log(c
) + a*b^2*d)*x - 6*(b^3*d*x + b^3*x^(1/3)*e)*log(x^(1/3*n)))*n^2*log(d*x^(2/3) + e)^2 - 8*(b^3*d*x + b^3*x^(1/
3)*e)*log(x^(1/3*n))^3 + 3*(4*(b^3*d*x + b^3*x^(1/3)*e)*log(x^(1/3*n))^2 + (b^3*log(c)^2 + 2*a*b^2*log(c) + a^
2*b)*x^(1/3)*e + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x - 4*((b^3*log(c) + a*b^2)*x^(1/3)*e + (b^3*d*
log(c) + a*b^2*d)*x)*log(x^(1/3*n)))*n*log(d*x^(2/3) + e) + 12*((b^3*log(c) + a*b^2)*x^(1/3)*e + (b^3*d*log(c)
+ a*b^2*d)*x)*log(x^(1/3*n))^2 + (b^3*log(c)^3 + 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + a^3)*x^(1/3)*e + (b^3*d*
log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x - 6*((b^3*log(c)^2 + 2*a*b^2*log(c) + a^2*b)*x^(1/
3)*e + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x)*log(x^(1/3*n)))/(d*x^3 + x^(7/3)*e), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*log(c*((d*x + x^(1/3)*e)/x)^n)^3 + 3*a*b^2*log(c*((d*x + x^(1/3)*e)/x)^n)^2 + 3*a^2*b*log(c*((d*
x + x^(1/3)*e)/x)^n) + a^3)/x^2, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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