Integrand size = 23, antiderivative size = 255 \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {16}{3} b d n \sqrt {d+e x}-\frac {4}{9} b n (d+e x)^{3/2}+\frac {16}{3} b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )-2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-2 b d^{3/2} n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \]
-4/9*b*n*(e*x+d)^(3/2)+16/3*b*d^(3/2)*n*arctanh((e*x+d)^(1/2)/d^(1/2))+2*b *d^(3/2)*n*arctanh((e*x+d)^(1/2)/d^(1/2))^2+2/3*(e*x+d)^(3/2)*(a+b*ln(c*x^ n))-2*d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*(a+b*ln(c*x^n))-4*b*d^(3/2)*n *arctanh((e*x+d)^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)-(e*x+d)^(1/2)))-2*b* d^(3/2)*n*polylog(2,1-2*d^(1/2)/(d^(1/2)-(e*x+d)^(1/2)))-16/3*b*d*n*(e*x+d )^(1/2)+2*d*(a+b*ln(c*x^n))*(e*x+d)^(1/2)
Time = 0.19 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=2 a d \sqrt {d+e x}-4 b d n \sqrt {d+e x}-\frac {4}{9} b n \sqrt {d+e x} (4 d+e x)+\frac {16}{3} b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b d \sqrt {d+e x} \log \left (c x^n\right )+\frac {2}{3} (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )+d^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-d^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )+\frac {1}{2} b d^{3/2} n \log \left (\sqrt {d}+\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )-\frac {1}{2} b d^{3/2} n \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )-b d^{3/2} n \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+b d^{3/2} n \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right ) \]
2*a*d*Sqrt[d + e*x] - 4*b*d*n*Sqrt[d + e*x] - (4*b*n*Sqrt[d + e*x]*(4*d + e*x))/9 + (16*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/3 + 2*b*d*Sqrt[d + e*x]*Log[c*x^n] + (2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/3 + d^(3/2)*(a + b*Log[c*x^n])*Log[Sqrt[d] - Sqrt[d + e*x]] - d^(3/2)*(a + b*Log[c*x^n]) *Log[Sqrt[d] + Sqrt[d + e*x]] + (b*d^(3/2)*n*Log[Sqrt[d] + Sqrt[d + e*x]]* (Log[Sqrt[d] + Sqrt[d + e*x]] + 2*Log[1/2 - Sqrt[d + e*x]/(2*Sqrt[d])]))/2 - (b*d^(3/2)*n*Log[Sqrt[d] - Sqrt[d + e*x]]*(Log[Sqrt[d] - Sqrt[d + e*x]] + 2*Log[(1 + Sqrt[d + e*x]/Sqrt[d])/2]))/2 - b*d^(3/2)*n*PolyLog[2, 1/2 - Sqrt[d + e*x]/(2*Sqrt[d])] + b*d^(3/2)*n*PolyLog[2, (1 + Sqrt[d + e*x]/Sq rt[d])/2]
Time = 1.45 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.21, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {2788, 2756, 60, 60, 73, 221, 2788, 2756, 60, 73, 221, 2790, 27, 7267, 25, 6546, 27, 6470, 27, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle e \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )dx+d \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}dx\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \int \frac {(d+e x)^{3/2}}{x}dx}{3 e}\right )+d \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \int \frac {\sqrt {d+e x}}{x}dx+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )+d \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (d \int \frac {1}{x \sqrt {d+e x}}dx+2 \sqrt {d+e x}\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )+d \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}dx\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (\frac {2 d \int \frac {1}{\frac {d+e x}{e}-\frac {d}{e}}d\sqrt {d+e x}}{e}+2 \sqrt {d+e x}\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )+d \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}dx\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle d \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}dx+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle d \left (e \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}}dx+d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle d \left (e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \int \frac {\sqrt {d+e x}}{x}dx}{e}\right )+d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle d \left (e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (d \int \frac {1}{x \sqrt {d+e x}}dx+2 \sqrt {d+e x}\right )}{e}\right )+d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle d \left (e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (\frac {2 d \int \frac {1}{\frac {d+e x}{e}-\frac {d}{e}}d\sqrt {d+e x}}{e}+2 \sqrt {d+e x}\right )}{e}\right )+d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle d \left (d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}}dx+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 2790 |
| \(\displaystyle d \left (d \left (-b n \int -\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d} x}dx-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (d \left (\frac {2 b n \int \frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x}dx}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle d \left (d \left (\frac {4 b n \int \frac {\sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (d \left (-\frac {4 b n \int -\frac {\sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle d \left (d \left (\frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\frac {\int \frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}-\sqrt {d+e x}}d\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (d \left (\frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\int \frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}-\sqrt {d+e x}}d\sqrt {d+e x}\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle d \left (d \left (\frac {4 b n \left (\frac {\int -\frac {d \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{e x}d\sqrt {d+e x}}{\sqrt {d}}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (d \left (\frac {4 b n \left (\sqrt {d} \int -\frac {\log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{e x}d\sqrt {d+e x}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle d \left (d \left (\frac {4 b n \left (-\sqrt {d} \int \frac {\log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}}d\frac {1}{\sqrt {d}-\sqrt {d+e x}}+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle d \left (d \left (\frac {4 b n \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2-\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )\right )}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}\right )+e \left (\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b n \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e}\right )\right )+e \left (\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {2 b n \left (d \left (2 \sqrt {d+e x}-2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+\frac {2}{3} (d+e x)^{3/2}\right )}{3 e}\right )\) |
e*((-2*b*n*((2*(d + e*x)^(3/2))/3 + d*(2*Sqrt[d + e*x] - 2*Sqrt[d]*ArcTanh [Sqrt[d + e*x]/Sqrt[d]])))/(3*e) + (2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/ (3*e)) + d*(e*((-2*b*n*(2*Sqrt[d + e*x] - 2*Sqrt[d]*ArcTanh[Sqrt[d + e*x]/ Sqrt[d]]))/e + (2*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e) + d*((-2*ArcTanh[Sq rt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/Sqrt[d] + (4*b*n*(ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2/2 - ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt [d] - Sqrt[d + e*x])] - PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x ])]/2))/Sqrt[d]))
3.2.41.3.1 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[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[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_.) + 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]]
\[\int \frac {\left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x}d x\]
\[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
1/3*(3*d^(3/2)*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d))) + 2*(e*x + d)^(3/2) + 6*sqrt(e*x + d)*d)*a + b*integrate((e*x*log(c) + d*log (c) + (e*x + d)*log(x^n))*sqrt(e*x + d)/x, x)
\[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^{3/2}}{x} \,d x \]