Optimal. Leaf size=738 \[ \frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \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 )}{(-d)^{3/2}}-\frac {12 b^3 e^{3/2} n^3 \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 )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {12 b^3 e^{3/2} n^3 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \text {Li}_2\left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {2 b e^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 )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.81, 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 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx &=3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+(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}{d+\frac {e}{x^2}} \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+(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}{d}-\frac {e \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2}{d \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {(6 b e n) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (6 b e^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 )}{d}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {\left (6 b e^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 )}{d}+\frac {\left (24 b^2 e^2 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^2} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {\left (6 b e^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 )}{d}+\frac {\left (24 b^2 e^2 n^2\right ) \text {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {\left (6 b e^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 )}{d}+\frac {\left (12 b^2 e^{3/2} n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^2 e^{3/2} n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x^2}\right )^n\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^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 )}{d}+\frac {\left (24 b^3 e^{5/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (24 b^3 e^{5/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^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 )}{d}+\frac {\left (24 b^3 e^{5/2} n^3\right ) \text {Subst}\left (\int \left (\frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{e x}-\frac {d x \log \left (\sqrt {e}-\sqrt {-d} x\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (24 b^3 e^{5/2} n^3\right ) \text {Subst}\left (\int \left (\frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{e x}-\frac {d x \log \left (\sqrt {e}+\sqrt {-d} x\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^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 )}{d}+\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {x \log \left (\sqrt {e}-\sqrt {-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {x \log \left (\sqrt {e}+\sqrt {-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {-d}}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^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 )}{d}+\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {-d} \log \left (\sqrt {e}-\sqrt {-d} x\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \log \left (\sqrt {e}-\sqrt {-d} x\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {-d} \log \left (\sqrt {e}+\sqrt {-d} x\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \log \left (\sqrt {e}+\sqrt {-d} x\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {-d}}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {\sqrt {-d} x}{\sqrt {e}}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (24 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {\sqrt {-d} x}{\sqrt {e}}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^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 )}{d}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \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 )}{(-d)^{3/2}}-\frac {12 b^3 e^{3/2} n^3 \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 )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^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 )}{d}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {\sqrt {e}-\sqrt {-d} x}{2 \sqrt {e}}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {\sqrt {e}+\sqrt {-d} x}{2 \sqrt {e}}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\sqrt [3]{x}\right )}{d}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \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 )}{(-d)^{3/2}}-\frac {12 b^3 e^{3/2} n^3 \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 )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^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 )}{d}+\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {e}}\right )}{x} \, dx,x,\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {\left (12 b^3 e^{3/2} n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {e}}\right )}{x} \, dx,x,\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}\\ &=\frac {6 b e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}-\frac {12 b^2 e^{3/2} n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {6 b^3 e^{3/2} n^3 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \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 )}{(-d)^{3/2}}-\frac {12 b^3 e^{3/2} n^3 \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 )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}+\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {12 b^3 e^{3/2} n^3 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )}{(-d)^{3/2}}+\frac {12 b^3 e^{3/2} n^3 \text {Li}_2\left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{(-d)^{3/2}}-\frac {24 b^3 e^{3/2} n^3 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{(-d)^{3/2}}-\frac {\left (6 b e^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 )}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.52, size = 824, normalized size = 1.12 \begin {gather*} \frac {3 b^2 n^2 \left (-3 e^2 \left (e+d x^{2/3}\right ) \, _4F_3\left (1,1,1,\frac {5}{2};2,2,2;1+\frac {e}{d x^{2/3}}\right )-d x^{2/3} \log \left (d+\frac {e}{x^{2/3}}\right ) \left (4 e \left (e-\frac {e}{\sqrt {-\frac {e}{d x^{2/3}}}}\right )+4 e^2 \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {e}{d x^{2/3}}}\right )\right )+\left (-e^2+d^2 \sqrt {-\frac {e}{d x^{2/3}}} x^{4/3}\right ) \log \left (d+\frac {e}{x^{2/3}}\right )\right )\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 )}{d^3 \sqrt {-\frac {e}{d x^{2/3}}} x}+\frac {6 b e n \sqrt [3]{x} \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}{d}-\frac {6 b e^{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}{d^{3/2}}+3 b n x \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 \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 )^3+\frac {b^3 n^3 \sqrt [3]{x} \left (\sqrt {d} \log ^2\left (d+\frac {e}{x^{2/3}}\right ) \left (6 e+d x^{2/3} \log \left (d+\frac {e}{x^{2/3}}\right )\right )-6 e \sqrt {\frac {e}{e+d 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+\frac {e}{x^{2/3}}}\right )+\log \left (d+\frac {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+\frac {e}{x^{2/3}}}\right )+\sqrt {d+\frac {e}{x^{2/3}}} \sin ^{-1}\left (\frac {\sqrt {d}}{\sqrt {d+\frac {e}{x^{2/3}}}}\right ) \log \left (d+\frac {e}{x^{2/3}}\right )\right )\right )+6 \sqrt {d} e \left (\frac {4 \sqrt {\frac {e}{x^{2/3}}} \tanh ^{-1}\left (\frac {\sqrt {\frac {e}{x^{2/3}}}}{\sqrt {-d}}\right ) \left (\log \left (d+\frac {e}{x^{2/3}}\right )-\log \left (1+\frac {e}{d x^{2/3}}\right )\right )}{\sqrt {-d}}-\sqrt {-\frac {e}{d x^{2/3}}} \left (2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {e}{d x^{2/3}}}\right )\right )-4 \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {e}{d x^{2/3}}}\right )\right ) \log \left (1+\frac {e}{d x^{2/3}}\right )+\log ^2\left (1+\frac {e}{d x^{2/3}}\right )-4 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {e}{d x^{2/3}}}\right )\right )\right )\right )}{d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________