Integrand size = 26, antiderivative size = 418 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}} \, dx=\frac {16 b^2 n^2 \sqrt {f+g x}}{g}-\frac {16 b^2 \sqrt {e f-d g} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e} g}-\frac {8 b^2 \sqrt {e f-d g} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{\sqrt {e} g}-\frac {8 b n \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {8 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} g}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac {16 b^2 \sqrt {e f-d g} n^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} g}+\frac {8 b^2 \sqrt {e f-d g} n^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} g} \]
-16*b^2*n^2*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*(-d*g+e*f)^(1/ 2)/g/e^(1/2)-8*b^2*n^2*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))^2*( -d*g+e*f)^(1/2)/g/e^(1/2)+8*b*n*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^( 1/2))*(a+b*ln(c*(e*x+d)^n))*(-d*g+e*f)^(1/2)/g/e^(1/2)+16*b^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)))*(-d*g+e*f)^(1/2)/g/e^(1/2)+8*b^2*n^2*polylog(2,1-2/(1-e^(1/ 2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2)))*(-d*g+e*f)^(1/2)/g/e^(1/2)+16*b^2*n^2* (g*x+f)^(1/2)/g-8*b*n*(a+b*ln(c*(e*x+d)^n))*(g*x+f)^(1/2)/g+2*(a+b*ln(c*(e *x+d)^n))^2*(g*x+f)^(1/2)/g
Time = 0.69 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \left (\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2-\frac {b n \left (4 a \sqrt {e} \sqrt {f+g x}-8 b \sqrt {e} n \sqrt {f+g x}+8 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )+4 b \sqrt {e} \sqrt {f+g x} \log \left (c (d+e x)^n\right )+2 \sqrt {e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-2 \sqrt {e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-b \sqrt {e f-d g} n \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+b \sqrt {e f-d g} n \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )\right )}{\sqrt {e}}\right )}{g} \]
(2*(Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n])^2 - (b*n*(4*a*Sqrt[e]*Sqrt[f + g*x] - 8*b*Sqrt[e]*n*Sqrt[f + g*x] + 8*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt [e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]] + 4*b*Sqrt[e]*Sqrt[f + g*x]*Log[c*(d + e*x)^n] + 2*Sqrt[e*f - d*g]*(a + b*Log[c*(d + e*x)^n])*Log[Sqrt[e*f - d*g ] - Sqrt[e]*Sqrt[f + g*x]] - 2*Sqrt[e*f - d*g]*(a + b*Log[c*(d + e*x)^n])* Log[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]] - b*Sqrt[e*f - d*g]*n*(Log[Sq rt[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*P olyLog[2, 1/2 - (Sqrt[e]*Sqrt[f + g*x])/(2*Sqrt[e*f - d*g])]) + b*Sqrt[e*f - d*g]*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])))/Sqrt[e]))/g
Time = 2.09 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.32, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {2845, 2858, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}} \, dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b e n \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}dx}{g}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)}{g}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)\right )}{g}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{d+e x}d(d+e x)}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)\right )}{g}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (\left (f-\frac {d g}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)+2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)\right )}{g}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (\frac {2 e \left (f-\frac {d g}{e}\right ) \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g}+2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}\right )}{g}\right )}{e}+\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)\right )}{g}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (-b n \int -\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} (d+e x)}d(d+e x)-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {2 b \sqrt {e} n \int \frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d+e x}d(d+e x)}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d g-e \left (\frac {d g}{e}-\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{-e f+d g+e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\int \frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}-\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{1-\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{e f-d g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}}d\frac {1}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}}{\sqrt {e}}+\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {4 b n \left (\left (f-\frac {d g}{e}\right ) \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}+\frac {\sqrt {e f-d g} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{2 \sqrt {e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )+\frac {g \left (\frac {2 e \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {2 b e n \left (2 \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}-\frac {2 \sqrt {e} \left (f-\frac {d g}{e}\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{g}\right )}{e}\right )}{g}\) |
(2*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n])^2)/g - (4*b*n*((g*((-2*b*e*n*( 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]))/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]*Sqrt[f - (d*g)/e + (g*(d + e*x))/e])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[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])/Sqr t[e*f - d*g])])/(2*Sqrt[e]))/(Sqrt[e]*Sqrt[e*f - d*g])))/Sqrt[e*f - d*g])) )/g
3.2.47.3.1 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]]
\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}{\sqrt {g x +f}}d x\]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{\sqrt {g x + f}} \,d x } \]
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*x + f), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}{\sqrt {f + g x}}\, dx \]
Exception generated. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{\sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {f+g x}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{\sqrt {f+g\,x}} \,d x \]