Integrand size = 26, antiderivative size = 583 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx=-\frac {16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac {128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt {f+g x}}+\frac {368 b^2 e^{7/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac {8 b^2 e^{7/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac {8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac {8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt {f+g x}}-\frac {8 b e^{7/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac {16 b^2 e^{7/2} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}-\frac {8 b^2 e^{7/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{7 g (e f-d g)^{7/2}} \]
-16/105*b^2*e^2*n^2/g/(-d*g+e*f)^2/(g*x+f)^(3/2)+368/105*b^2*e^(7/2)*n^2*a rctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))/g/(-d*g+e*f)^(7/2)+8/7*b^2* e^(7/2)*n^2*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))^2/g/(-d*g+e*f) ^(7/2)+8/35*b*e*n*(a+b*ln(c*(e*x+d)^n))/g/(-d*g+e*f)/(g*x+f)^(5/2)+8/21*b* e^2*n*(a+b*ln(c*(e*x+d)^n))/g/(-d*g+e*f)^2/(g*x+f)^(3/2)-8/7*b*e^(7/2)*n*a rctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*(a+b*ln(c*(e*x+d)^n))/g/(-d *g+e*f)^(7/2)-2/7*(a+b*ln(c*(e*x+d)^n))^2/g/(g*x+f)^(7/2)-16/7*b^2*e^(7/2) *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)))/g/(-d*g+e*f)^(7/2)-8/7*b^2*e^(7/2)*n^2*polylog (2,1-2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))/g/(-d*g+e*f)^(7/2)-128/ 105*b^2*e^3*n^2/g/(-d*g+e*f)^3/(g*x+f)^(1/2)+8/7*b*e^3*n*(a+b*ln(c*(e*x+d) ^n))/g/(-d*g+e*f)^3/(g*x+f)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.21 (sec) , antiderivative size = 728, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx=\frac {2 \left (-15 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+\frac {b e n (f+g x) \left (120 b e^{5/2} n (f+g x)^{5/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )-8 b e (e f-d g)^{3/2} n (f+g x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {e (f+g x)}{e f-d g}\right )-40 b e^2 \sqrt {e f-d g} n (f+g x)^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {e (f+g x)}{e f-d g}\right )+12 (e f-d g)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+20 e (e f-d g)^{3/2} (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )+60 e^2 \sqrt {e f-d g} (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+30 e^{5/2} (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-30 e^{5/2} (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-15 b e^{5/2} n (f+g x)^{5/2} \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+15 b e^{5/2} n (f+g x)^{5/2} \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )\right )}{(e f-d g)^{7/2}}\right )}{105 g (f+g x)^{7/2}} \]
(2*(-15*(a + b*Log[c*(d + e*x)^n])^2 + (b*e*n*(f + g*x)*(120*b*e^(5/2)*n*( f + g*x)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]] - 8*b*e*(e *f - d*g)^(3/2)*n*(f + g*x)*Hypergeometric2F1[-3/2, 1, -1/2, (e*(f + g*x)) /(e*f - d*g)] - 40*b*e^2*Sqrt[e*f - d*g]*n*(f + g*x)^2*Hypergeometric2F1[- 1/2, 1, 1/2, (e*(f + g*x))/(e*f - d*g)] + 12*(e*f - d*g)^(5/2)*(a + b*Log[ c*(d + e*x)^n]) + 20*e*(e*f - d*g)^(3/2)*(f + g*x)*(a + b*Log[c*(d + e*x)^ n]) + 60*e^2*Sqrt[e*f - d*g]*(f + g*x)^2*(a + b*Log[c*(d + e*x)^n]) + 30*e ^(5/2)*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n])*Log[Sqrt[e*f - d*g] - Sq rt[e]*Sqrt[f + g*x]] - 30*e^(5/2)*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n ])*Log[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]] - 15*b*e^(5/2)*n*(f + g*x) ^(5/2)*(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])] ) + 15*b*e^(5/2)*n*(f + g*x)^(5/2)*(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])))/(e*f - d*g)^(7/2)))/(105*g*(f + g*x)^(7/2) )
Time = 4.25 (sec) , antiderivative size = 940, normalized size of antiderivative = 1.61, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.962, Rules used = {2845, 2858, 2789, 2756, 61, 61, 73, 221, 2789, 2756, 61, 73, 221, 2789, 2756, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {4 b e n \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{7/2}}dx}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {4 b n \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{7/2}}d(d+e x)}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {4 b n \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e f-d g}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{7/2}}d(d+e x)}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {4 b n \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \int \frac {1}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {4 b n \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \int \frac {1}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {4 b n \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {e \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}+\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 b n \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e^2 \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g (e f-d g)}+\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 b n \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}d(d+e x)}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \int \frac {1}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}+\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e^2 \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g (e f-d g)}+\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {4 b e^2 n \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g^2}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \left (-b n \int -\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} (d+e x)}d(d+e x)-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \left (\frac {2 b \sqrt {e} n \int \frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d+e x}d(d+e x)}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {e \left (\frac {2 e}{(e f-d g) \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (\frac {g (d+e x)}{e}-\frac {d g}{e}+f\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d g-e \left (\frac {d g}{e}-\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {4 b n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) e^{3/2}}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right ) e}{g \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {\left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right ) e}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{-e f+d g+e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {4 b n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right ) e^{3/2}}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right ) e}{g \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {\left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}\right ) e}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \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 )}{e f-d g}-\frac {g \left (-\frac {4 b n \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^{3/2}}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right ) e}{g \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \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)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {\left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \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)^{3/2}}\right ) e}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \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 )}{e f-d g}-\frac {g \left (-\frac {4 b n \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^{3/2}}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right ) e}{g \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \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)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {\left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \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)^{3/2}}\right ) e}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \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 )}{e f-d g}-\frac {g \left (-\frac {4 b n \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^{3/2}}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right ) e}{g \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \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)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {\left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \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)^{3/2}}\right ) e}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {4 b n \left (\frac {e \left (\frac {e \left (\frac {e \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 )}{e f-d g}-\frac {g \left (-\frac {4 b n \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^{3/2}}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right ) e}{g \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \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)^{3/2}}\right )}{3 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{e f-d g}\right )}{e f-d g}-\frac {g \left (\frac {2 b e n \left (\frac {\left (\frac {2 e}{(e f-d g) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}-\frac {2 e^{3/2} \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)^{3/2}}\right ) e}{e f-d g}+\frac {2 e}{3 (e f-d g) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}\right )}{5 g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{5/2}}\right )}{e f-d g}\right )}{7 g}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}\) |
(-2*(a + b*Log[c*(d + e*x)^n])^2)/(7*g*(f + g*x)^(7/2)) + (4*b*n*(-((g*((2 *b*e*n*((2*e)/(3*(e*f - d*g)*(f - (d*g)/e + (g*(d + e*x))/e)^(3/2)) + (e*( (2*e)/((e*f - d*g)*Sqrt[f - (d*g)/e + (g*(d + e*x))/e]) - (2*e^(3/2)*ArcTa nh[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]])/(e*f - d*g)^(3/2)))/(e*f - d*g)))/(5*g) - (2*e*(a + b*Log[c*(d + e*x)^n]))/(5*g*( f - (d*g)/e + (g*(d + e*x))/e)^(5/2))))/(e*f - d*g)) + (e*(-((g*((2*b*e*n* ((2*e)/((e*f - d*g)*Sqrt[f - (d*g)/e + (g*(d + e*x))/e]) - (2*e^(3/2)*ArcT anh[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]])/(e*f - d*g)^(3/2)))/(3*g) - (2*e*(a + b*Log[c*(d + e*x)^n]))/(3*g*(f - (d*g)/e + (g*(d + e*x))/e)^(3/2))))/(e*f - d*g)) + (e*(-((g*((-4*b*e^(3/2)*n*ArcTan h[(Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]])/(g*Sqrt[ e*f - d*g]) - (2*e*(a + b*Log[c*(d + e*x)^n]))/(g*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])))/(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]))/Sqr t[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]*S qrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*S qrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g])])/Sqrt[e] + (Sqrt[e*f - d*g]*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e] )/Sqrt[e*f - d*g])])/(2*Sqrt[e]))/(Sqrt[e]*Sqrt[e*f - d*g])))/Sqrt[e*f ...
3.2.51.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && 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]]
\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{\left (g x +f \right )^{\frac {9}{2}}}d x\]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{\frac {9}{2}}} \,d x } \]
integral((sqrt(g*x + f)*b^2*log((e*x + d)^n*c)^2 + 2*sqrt(g*x + f)*a*b*log ((e*x + d)^n*c) + sqrt(g*x + f)*a^2)/(g^5*x^5 + 5*f*g^4*x^4 + 10*f^2*g^3*x ^3 + 10*f^3*g^2*x^2 + 5*f^4*g*x + f^5), x)
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^{9/2}} \,d x \]