Integrand size = 26, antiderivative size = 590 \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {368 b^2 (e f-d g)^2 n^2 \sqrt {f+g x}}{75 e^2 g}+\frac {128 b^2 (e f-d g) n^2 (f+g x)^{3/2}}{225 e g}+\frac {16 b^2 n^2 (f+g x)^{5/2}}{125 g}-\frac {368 b^2 (e f-d g)^{5/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{75 e^{5/2} g}-\frac {8 b^2 (e f-d g)^{5/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{5 e^{5/2} g}-\frac {8 b (e f-d g)^2 n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e^2 g}-\frac {8 b (e f-d g) n (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{15 e g}-\frac {8 b n (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{25 g}+\frac {8 b (e f-d g)^{5/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 e^{5/2} g}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}+\frac {16 b^2 (e f-d g)^{5/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{5 e^{5/2} g}+\frac {8 b^2 (e f-d g)^{5/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{5 e^{5/2} g} \]
128/225*b^2*(-d*g+e*f)*n^2*(g*x+f)^(3/2)/e/g+16/125*b^2*n^2*(g*x+f)^(5/2)/ g-368/75*b^2*(-d*g+e*f)^(5/2)*n^2*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f) ^(1/2))/e^(5/2)/g-8/5*b^2*(-d*g+e*f)^(5/2)*n^2*arctanh(e^(1/2)*(g*x+f)^(1/ 2)/(-d*g+e*f)^(1/2))^2/e^(5/2)/g-8/15*b*(-d*g+e*f)*n*(g*x+f)^(3/2)*(a+b*ln (c*(e*x+d)^n))/e/g-8/25*b*n*(g*x+f)^(5/2)*(a+b*ln(c*(e*x+d)^n))/g+8/5*b*(- d*g+e*f)^(5/2)*n*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*(a+b*ln(c *(e*x+d)^n))/e^(5/2)/g+2/5*(g*x+f)^(5/2)*(a+b*ln(c*(e*x+d)^n))^2/g+16/5*b^ 2*(-d*g+e*f)^(5/2)*n^2*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*ln( 2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))/e^(5/2)/g+8/5*b^2*(-d*g+e*f) ^(5/2)*n^2*polylog(2,1-2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))/e^(5/ 2)/g+368/75*b^2*(-d*g+e*f)^2*n^2*(g*x+f)^(1/2)/e^2/g-8/5*b*(-d*g+e*f)^2*n* (a+b*ln(c*(e*x+d)^n))*(g*x+f)^(1/2)/e^2/g
Time = 1.21 (sec) , antiderivative size = 854, normalized size of antiderivative = 1.45 \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {2 \left ((f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac {b n \left (900 a \sqrt {e} (e f-d g)^2 \sqrt {f+g x}-1800 b \sqrt {e} (e f-d g)^2 n \sqrt {f+g x}+1800 b (e f-d g)^{5/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )-200 b (e f-d g) n \left (\sqrt {e} \sqrt {f+g x} (4 e f-3 d g+e g x)-3 (e f-d g)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )-24 b n \left (3 e^{5/2} (f+g x)^{5/2}+5 (e f-d g) \left (\sqrt {e} \sqrt {f+g x} (4 e f-3 d g+e g x)-3 (e f-d g)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+900 b \sqrt {e} (e f-d g)^2 \sqrt {f+g x} \log \left (c (d+e x)^n\right )+300 e^{3/2} (e f-d g) (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+180 e^{5/2} (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+450 (e f-d g)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-450 (e f-d g)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-225 b (e f-d g)^{5/2} n \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+225 b (e f-d g)^{5/2} n \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )\right )}{225 e^{5/2}}\right )}{5 g} \]
(2*((f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n])^2 - (b*n*(900*a*Sqrt[e]*(e* f - d*g)^2*Sqrt[f + g*x] - 1800*b*Sqrt[e]*(e*f - d*g)^2*n*Sqrt[f + g*x] + 1800*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g] ] - 200*b*(e*f - d*g)*n*(Sqrt[e]*Sqrt[f + g*x]*(4*e*f - 3*d*g + e*g*x) - 3 *(e*f - d*g)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]) - 24* b*n*(3*e^(5/2)*(f + g*x)^(5/2) + 5*(e*f - d*g)*(Sqrt[e]*Sqrt[f + g*x]*(4*e *f - 3*d*g + e*g*x) - 3*(e*f - d*g)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/ Sqrt[e*f - d*g]])) + 900*b*Sqrt[e]*(e*f - d*g)^2*Sqrt[f + g*x]*Log[c*(d + e*x)^n] + 300*e^(3/2)*(e*f - d*g)*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n ]) + 180*e^(5/2)*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n]) + 450*(e*f - d *g)^(5/2)*(a + b*Log[c*(d + e*x)^n])*Log[Sqrt[e*f - d*g] - Sqrt[e]*Sqrt[f + g*x]] - 450*(e*f - d*g)^(5/2)*(a + b*Log[c*(d + e*x)^n])*Log[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]] - 225*b*(e*f - d*g)^(5/2)*n*(Log[Sqrt[e*f - d*g] - Sqrt[e]*Sqrt[f + g*x]]*(Log[Sqrt[e*f - d*g] - Sqrt[e]*Sqrt[f + g*x] ] + 2*Log[(1 + (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])/2]) + 2*PolyLog[2, 1/2 - (Sqrt[e]*Sqrt[f + g*x])/(2*Sqrt[e*f - d*g])]) + 225*b*(e*f - d*g)^( 5/2)*n*(Log[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]]*(Log[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]] + 2*Log[1/2 - (Sqrt[e]*Sqrt[f + g*x])/(2*Sqrt[e*f - d*g])]) + 2*PolyLog[2, (1 + (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])/2] )))/(225*e^(5/2))))/(5*g)
Time = 4.35 (sec) , antiderivative size = 1007, normalized size of antiderivative = 1.71, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {2845, 2858, 2788, 2756, 60, 60, 60, 73, 221, 2788, 2756, 60, 60, 73, 221, 2788, 2756, 60, 73, 221, 2790, 27, 7267, 2092, 6546, 6470, 2849, 2752}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b e n \int \frac {(f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}dx}{5 g}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)}{5 g}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \int \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )d(d+e x)}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )}{5 g}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}{d+e x}d(d+e x)}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )}{5 g}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}{d+e x}d(d+e x)+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )}{5 g}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{d+e x}d(d+e x)+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )}{5 g}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)+2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )}{5 g}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (\frac {2 e \left (f-\frac {d g}{e}\right ) \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g}+2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )}{5 g}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \int \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )d(d+e x)}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \int \frac {\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}{d+e x}d(d+e x)}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{d+e x}d(d+e x)+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)+2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {2 e \left (f-\frac {d g}{e}\right ) \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g}+2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{d+e x}d(d+e x)}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)+2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (\frac {2 e \left (f-\frac {d g}{e}\right ) \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g}+2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )}{3 g}\right )}{e}\right )+\frac {g \left (\frac {2 e \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \left (\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )+\frac {2}{3} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}\right )+\frac {2}{5} \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}\right )}{5 g}\right )}{e}\right )}{5 g}\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {2}{5} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}+\left (f-\frac {d g}{e}\right ) \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}-b n \int -\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} (d+e x)}d(d+e x)\right )\right )\right )\right )}{5 g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {2}{5} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}+\left (f-\frac {d g}{e}\right ) \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {2 b \sqrt {e} n \int \frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d+e x}d(d+e x)}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )\right )\right )\right )}{5 g}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {2}{5} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}+\left (f-\frac {d g}{e}\right ) \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d g-e \left (\frac {d g}{e}-\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )\right )\right )\right )}{5 g}\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {2}{5} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}+\left (f-\frac {d g}{e}\right ) \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{-e f+d g+e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )\right )\right )\right )}{5 g}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {2}{5} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}+\left (f-\frac {d g}{e}\right ) \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\int \frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )\right )\right )\right )}{5 g}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {2}{5} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}+\left (f-\frac {d g}{e}\right ) \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}-\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{1-\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{e f-d g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )\right )\right )\right )}{5 g}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {2}{5} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}+\left (f-\frac {d g}{e}\right ) \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}+\frac {\sqrt {e f-d g} \int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}}d\frac {1}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}}{\sqrt {e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )\right )\right )\right )}{5 g}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{5 g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {2}{5} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}+\left (f-\frac {d g}{e}\right ) \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )\right )}{5 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {2 b e n \left (\frac {2}{3} \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}+\left (f-\frac {d g}{e}\right ) \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )\right )}{3 g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {g \left (\frac {2 e \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}+\frac {\sqrt {e f-d g} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{2 \sqrt {e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )\right )\right )\right )}{5 g}\) |
(2*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n])^2)/(5*g) - (4*b*n*((g*((-2*b *e*n*((2*(f - (d*g)/e + (g*(d + e*x))/e)^(5/2))/5 + (f - (d*g)/e)*((2*(f - (d*g)/e + (g*(d + e*x))/e)^(3/2))/3 + (f - (d*g)/e)*(2*Sqrt[f - (d*g)/e + (g*(d + e*x))/e] - (2*Sqrt[e]*(f - (d*g)/e)*ArcTanh[(Sqrt[e]*Sqrt[f - (d* g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]])/Sqrt[e*f - d*g]))))/(5*g) + (2* e*(f - (d*g)/e + (g*(d + e*x))/e)^(5/2)*(a + b*Log[c*(d + e*x)^n]))/(5*g)) )/e + (f - (d*g)/e)*((g*((-2*b*e*n*((2*(f - (d*g)/e + (g*(d + e*x))/e)^(3/ 2))/3 + (f - (d*g)/e)*(2*Sqrt[f - (d*g)/e + (g*(d + e*x))/e] - (2*Sqrt[e]* (f - (d*g)/e)*ArcTanh[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e *f - d*g]])/Sqrt[e*f - d*g])))/(3*g) + (2*e*(f - (d*g)/e + (g*(d + e*x))/e )^(3/2)*(a + b*Log[c*(d + e*x)^n]))/(3*g)))/e + (f - (d*g)/e)*((g*((-2*b*e *n*(2*Sqrt[f - (d*g)/e + (g*(d + e*x))/e] - (2*Sqrt[e]*(f - (d*g)/e)*ArcTa nh[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]])/Sqrt[e* f - d*g]))/g + (2*e*Sqrt[f - (d*g)/e + (g*(d + e*x))/e]*(a + b*Log[c*(d + e*x)^n]))/g))/e + (f - (d*g)/e)*((-2*Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt[f - (d* g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/Sqrt [e*f - d*g] + (4*b*e^(3/2)*n*(ArcTanh[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]]^2/(2*e) - ((Sqrt[e*f - d*g]*ArcTanh[(Sqrt[e]*Sq rt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sq rt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g])])/Sqrt[e] + (Sqrt[e...
3.2.45.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && BinomialQ[z, x] && BinomialQ[u , x] && !(BinomialMatchQ[z, x] && BinomialMatchQ[u, x])
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) /(x_), x_Symbol] :> Simp[d Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) , x], x] + Simp[e Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.)) /(x_), x_Symbol] :> With[{u = IntHide[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*L og[c*x^n]), x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, n , r}, x] && IntegerQ[q - 1/2]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 , 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
\[\int \left (g x +f \right )^{\frac {3}{2}} {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}d x\]
\[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (g x + f\right )}^{\frac {3}{2}} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \,d x } \]
integral((b^2*g*x + b^2*f)*sqrt(g*x + f)*log((e*x + d)^n*c)^2 + 2*(a*b*g*x + a*b*f)*sqrt(g*x + f)*log((e*x + d)^n*c) + (a^2*g*x + a^2*f)*sqrt(g*x + f), x)
\[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right )^{\frac {3}{2}}\, dx \]
Exception generated. \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(d*g-e*f)>0)', see `assume?` f or more de
\[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int { {\left (g x + f\right )}^{\frac {3}{2}} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\int {\left (f+g\,x\right )}^{3/2}\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2 \,d x \]