Problem number 216

3.1 Problem number 216

$\int \frac{1}{(1 - x)^{7 / 2} x^{5}} d x$

Optimal antiderivative $\frac{3003}{320 {(1 - x)}^{\frac{5}{2}}} + \frac{1001}{64 {(1 - x)}^{\frac{3}{2}}} - \frac{1}{4 {(1 - x)}^{\frac{5}{2}} x^{4}} - \frac{13}{24 {(1 - x)}^{\frac{5}{2}} x^{3}} - \frac{143}{96 {(1 - x)}^{\frac{5}{2}} x^{2}} - \frac{429}{64 {(1 - x)}^{\frac{5}{2}} x} - \frac{3003 arctanh (\sqrt{1 - x})}{64} + \frac{3003}{64 \sqrt{1 - x}}$

command

integrate(1/(1-x)**(7/2)/x**5,x)

Sympy 1.10.1 under Python 3.10.4 output

$Timed out$

Sympy 1.8 under Python 3.8.8 output

${\begin{cases} - \frac{45045 i x^{7} asin (\frac{1}{\sqrt{x}})}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} + \frac{45045 i x^{6} \sqrt{x - 1}}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} + \frac{135135 i x^{6} asin (\frac{1}{\sqrt{x}})}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} - \frac{105105 i x^{5} \sqrt{x - 1}}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} - \frac{135135 i x^{5} asin (\frac{1}{\sqrt{x}})}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} + \frac{69069 i x^{4} \sqrt{x - 1}}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} + \frac{45045 i x^{4} asin (\frac{1}{\sqrt{x}})}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} - \frac{6435 i x^{3} \sqrt{x - 1}}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} - \frac{1430 i x^{2} \sqrt{x - 1}}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} - \frac{520 i x \sqrt{x - 1}}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} - \frac{240 i \sqrt{x - 1}}{- 960 x^{7} + 2880 x^{6} - 2880 x^{5} + 960 x^{4}} & for | x | > 1 \\ - \frac{45045 x^{7} \log (x)}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac{90090 x^{7} \log (\sqrt{1 - x} + 1)}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{45045 i π x^{7}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac{90090 x^{6} \sqrt{1 - x}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac{135135 x^{6} \log (x)}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{270270 x^{6} \log (\sqrt{1 - x} + 1)}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac{135135 i π x^{6}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{210210 x^{5} \sqrt{1 - x}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{135135 x^{5} \log (x)}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac{270270 x^{5} \log (\sqrt{1 - x} + 1)}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{135135 i π x^{5}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac{138138 x^{4} \sqrt{1 - x}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac{45045 x^{4} \log (x)}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{90090 x^{4} \log (\sqrt{1 - x} + 1)}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac{45045 i π x^{4}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{12870 x^{3} \sqrt{1 - x}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{2860 x^{2} \sqrt{1 - x}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{1040 x \sqrt{1 - x}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac{480 \sqrt{1 - x}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} & otherwise \end{cases}$

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