Integrand size = 31, antiderivative size = 340 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {4 b \sqrt {e} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{(e f-d g)^{3/2}}+\frac {2 b \sqrt {e} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{(e f-d g)^{3/2}}+\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) \sqrt {f+g x}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^{3/2}}-\frac {4 b \sqrt {e} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{3/2}}-\frac {2 b \sqrt {e} n \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{(e f-d g)^{3/2}} \]
4*b*n*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*e^(1/2)/(-d*g+e*f)^( 3/2)+2*b*n*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))^2*e^(1/2)/(-d*g +e*f)^(3/2)-2*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/2))*(a+b*ln(c*(e *x+d)^n))*e^(1/2)/(-d*g+e*f)^(3/2)-4*b*n*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^(1/2)/(- d*g+e*f)^(3/2)-2*b*n*polylog(2,1-2/(1-e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)^(1/ 2)))*e^(1/2)/(-d*g+e*f)^(3/2)+2*(a+b*ln(c*(e*x+d)^n))/(-d*g+e*f)/(g*x+f)^( 1/2)
Time = 0.38 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.55 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx=\frac {8 b \sqrt {e} n \sqrt {f+g x} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )+4 \sqrt {e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right )+2 \sqrt {e} \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-2 \sqrt {e} \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-b \sqrt {e} n \sqrt {f+g x} \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+b \sqrt {e} n \sqrt {f+g x} \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )}{2 (e f-d g)^{3/2} \sqrt {f+g x}} \]
(8*b*Sqrt[e]*n*Sqrt[f + g*x]*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d* g]] + 4*Sqrt[e*f - d*g]*(a + b*Log[c*(d + e*x)^n]) + 2*Sqrt[e]*Sqrt[f + g* x]*(a + b*Log[c*(d + e*x)^n])*Log[Sqrt[e*f - d*g] - Sqrt[e]*Sqrt[f + g*x]] - 2*Sqrt[e]*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n])*Log[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]] - b*Sqrt[e]*n*Sqrt[f + g*x]*(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])]) + b*Sqrt[e]*n*Sqrt[f + g*x ]*(Log[Sqrt[e*f - d*g] + Sqrt[e]*Sqrt[f + g*x]]*(Log[Sqrt[e*f - d*g] + Sqr t[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]))/(2 *(e*f - d*g)^(3/2)*Sqrt[f + g*x])
Time = 1.86 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.44, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2858, 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.
| \(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )^{3/2}}d(d+e x)}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {2 b e n \int \frac {1}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{g}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}-\frac {g \left (\frac {4 b e^2 n \int \frac {1}{d+\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{g}-\frac {e f}{g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{g^2}-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {e \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}d(d+e x)}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle \frac {\frac {e \left (-b n \int -\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} (d+e x)}d(d+e x)-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \left (\frac {2 b \sqrt {e} n \int \frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d+e x}d(d+e x)}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {\frac {e \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{d g-e \left (\frac {d g}{e}-\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 2092 |
| \(\displaystyle \frac {\frac {e \left (\frac {4 b e^{3/2} n \int \frac {\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{-e f+d g+e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {\frac {e \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\int \frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {\frac {e \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}-\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{1-\frac {e \left (f-\frac {d g}{e}+\frac {g (d+e x)}{e}\right )}{e f-d g}}d\sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {\frac {e \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}}d\frac {1}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}}{\sqrt {e}}+\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {\frac {e \left (\frac {4 b e^{3/2} n \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )^2}{2 e}-\frac {\frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}}\right )}{\sqrt {e}}+\frac {\sqrt {e f-d g} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g (d+e x)}{e}}}{\sqrt {e f-d g}}}\right )}{2 \sqrt {e}}}{\sqrt {e} \sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}\right )}{e f-d g}-\frac {g \left (-\frac {2 e \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}-\frac {4 b e^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {\frac {g (d+e x)}{e}-\frac {d g}{e}+f}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}\right )}{e f-d g}}{e}\) |
(-((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]])/(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]))/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])/Sqrt[e*f - d*g])])/(2*Sqrt[e]))/(Sqrt[e]*Sq rt[e*f - d*g])))/Sqrt[e*f - d*g]))/(e*f - d*g))/e
3.3.2.3.1 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] :> 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[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]]
\[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\left (e x +d \right ) \left (g x +f \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (e x + d\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
integral((sqrt(g*x + f)*b*log((e*x + d)^n*c) + sqrt(g*x + f)*a)/(e*g^2*x^3 + d*f^2 + (2*e*f*g + d*g^2)*x^2 + (e*f^2 + 2*d*f*g)*x), x)
Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (e x + d\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (f+g\,x\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]