Optimal. Leaf size=118 \[ -\frac {1}{4 (1-x)^{5/2} x^4}-\frac {13}{24 (1-x)^{5/2} x^3}-\frac {143}{96 (1-x)^{5/2} x^2}+\frac {3003}{64 \sqrt {1-x}}-\frac {429}{64 (1-x)^{5/2} x}+\frac {1001}{64 (1-x)^{3/2}}+\frac {3003}{320 (1-x)^{5/2}}-\frac {3003}{64} \tanh ^{-1}\left (\sqrt {1-x}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 127, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 206} \[ -\frac {1001 \sqrt {1-x}}{32 x^2}-\frac {1001 \sqrt {1-x}}{40 x^3}-\frac {429 \sqrt {1-x}}{20 x^4}+\frac {286}{15 \sqrt {1-x} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {2}{5 (1-x)^{5/2} x^4}-\frac {3003 \sqrt {1-x}}{64 x}-\frac {3003}{64} \tanh ^{-1}\left (\sqrt {1-x}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{7/2} x^5} \, dx &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {13}{5} \int \frac {1}{(1-x)^{5/2} x^5} \, dx\\ &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {143}{15} \int \frac {1}{(1-x)^{3/2} x^5} \, dx\\ &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {286}{15 \sqrt {1-x} x^4}+\frac {429}{5} \int \frac {1}{\sqrt {1-x} x^5} \, dx\\ &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {286}{15 \sqrt {1-x} x^4}-\frac {429 \sqrt {1-x}}{20 x^4}+\frac {3003}{40} \int \frac {1}{\sqrt {1-x} x^4} \, dx\\ &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {286}{15 \sqrt {1-x} x^4}-\frac {429 \sqrt {1-x}}{20 x^4}-\frac {1001 \sqrt {1-x}}{40 x^3}+\frac {1001}{16} \int \frac {1}{\sqrt {1-x} x^3} \, dx\\ &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {286}{15 \sqrt {1-x} x^4}-\frac {429 \sqrt {1-x}}{20 x^4}-\frac {1001 \sqrt {1-x}}{40 x^3}-\frac {1001 \sqrt {1-x}}{32 x^2}+\frac {3003}{64} \int \frac {1}{\sqrt {1-x} x^2} \, dx\\ &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {286}{15 \sqrt {1-x} x^4}-\frac {429 \sqrt {1-x}}{20 x^4}-\frac {1001 \sqrt {1-x}}{40 x^3}-\frac {1001 \sqrt {1-x}}{32 x^2}-\frac {3003 \sqrt {1-x}}{64 x}+\frac {3003}{128} \int \frac {1}{\sqrt {1-x} x} \, dx\\ &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {286}{15 \sqrt {1-x} x^4}-\frac {429 \sqrt {1-x}}{20 x^4}-\frac {1001 \sqrt {1-x}}{40 x^3}-\frac {1001 \sqrt {1-x}}{32 x^2}-\frac {3003 \sqrt {1-x}}{64 x}-\frac {3003}{64} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x}\right )\\ &=\frac {2}{5 (1-x)^{5/2} x^4}+\frac {26}{15 (1-x)^{3/2} x^4}+\frac {286}{15 \sqrt {1-x} x^4}-\frac {429 \sqrt {1-x}}{20 x^4}-\frac {1001 \sqrt {1-x}}{40 x^3}-\frac {1001 \sqrt {1-x}}{32 x^2}-\frac {3003 \sqrt {1-x}}{64 x}-\frac {3003}{64} \tanh ^{-1}\left (\sqrt {1-x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.22 \[ \frac {2 \, _2F_1\left (-\frac {5}{2},5;-\frac {3}{2};1-x\right )}{5 (1-x)^{5/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 66, normalized size = 0.56 \[ \frac {\sqrt {1-x} \left (-45045 x^6+105105 x^5-69069 x^4+6435 x^3+1430 x^2+520 x+240\right )}{960 (x-1)^3 x^4}-\frac {3003}{64} \tanh ^{-1}\left (\sqrt {1-x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 125, normalized size = 1.06 \[ -\frac {45045 \, {\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )} \log \left (\sqrt {-x + 1} + 1\right ) - 45045 \, {\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )} \log \left (\sqrt {-x + 1} - 1\right ) + 2 \, {\left (45045 \, x^{6} - 105105 \, x^{5} + 69069 \, x^{4} - 6435 \, x^{3} - 1430 \, x^{2} - 520 \, x - 240\right )} \sqrt {-x + 1}}{1920 \, {\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 104, normalized size = 0.88 \[ \frac {2 \, {\left (225 \, {\left (x - 1\right )}^{2} - 25 \, x + 28\right )}}{15 \, {\left (x - 1\right )}^{2} \sqrt {-x + 1}} - \frac {3249 \, {\left (x - 1\right )}^{3} \sqrt {-x + 1} + 10633 \, {\left (x - 1\right )}^{2} \sqrt {-x + 1} - 11767 \, {\left (-x + 1\right )}^{\frac {3}{2}} + 4431 \, \sqrt {-x + 1}}{192 \, x^{4}} - \frac {3003}{128} \, \log \left (\sqrt {-x + 1} + 1\right ) + \frac {3003}{128} \, \log \left ({\left | \sqrt {-x + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 59, normalized size = 0.50
method | result | size |
risch | \(\frac {45045 x^{6}-105105 x^{5}+69069 x^{4}-6435 x^{3}-1430 x^{2}-520 x -240}{960 x^{4} \sqrt {1-x}\, \left (-1+x \right )^{2}}-\frac {3003 \arctanh \left (\sqrt {1-x}\right )}{64}\) | \(59\) |
trager | \(-\frac {\left (45045 x^{6}-105105 x^{5}+69069 x^{4}-6435 x^{3}-1430 x^{2}-520 x -240\right ) \sqrt {1-x}}{960 \left (-1+x \right )^{3} x^{4}}+\frac {3003 \ln \left (\frac {-2+x +2 \sqrt {1-x}}{x}\right )}{128}\) | \(68\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \left (-329177 x^{4}+110880 x^{3}+30240 x^{2}+8960 x +1920\right )}{7680 x^{4}}-\frac {\sqrt {\pi }\, \left (-180180 x^{6}+420420 x^{5}-276276 x^{4}+25740 x^{3}+5720 x^{2}+2080 x +960\right )}{3840 x^{4} \left (1-x \right )^{\frac {5}{2}}}-\frac {3003 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-x}}{2}\right )}{64}+\frac {3003 \left (\frac {329177}{180180}-2 \ln \relax (2)+\ln \relax (x )+i \pi \right ) \sqrt {\pi }}{128}-\frac {\sqrt {\pi }}{4 x^{4}}-\frac {7 \sqrt {\pi }}{6 x^{3}}-\frac {63 \sqrt {\pi }}{16 x^{2}}-\frac {231 \sqrt {\pi }}{16 x}}{\sqrt {\pi }}\) | \(146\) |
derivativedivides | \(-\frac {1}{64 \left (\sqrt {1-x}-1\right )^{4}}+\frac {17}{96 \left (\sqrt {1-x}-1\right )^{3}}-\frac {159}{128 \left (\sqrt {1-x}-1\right )^{2}}+\frac {1083}{128 \left (\sqrt {1-x}-1\right )}+\frac {3003 \ln \left (\sqrt {1-x}-1\right )}{128}+\frac {2}{5 \left (1-x \right )^{\frac {5}{2}}}+\frac {10}{3 \left (1-x \right )^{\frac {3}{2}}}+\frac {30}{\sqrt {1-x}}+\frac {1}{64 \left (\sqrt {1-x}+1\right )^{4}}+\frac {17}{96 \left (\sqrt {1-x}+1\right )^{3}}+\frac {159}{128 \left (\sqrt {1-x}+1\right )^{2}}+\frac {1083}{128 \left (\sqrt {1-x}+1\right )}-\frac {3003 \ln \left (\sqrt {1-x}+1\right )}{128}\) | \(157\) |
default | \(-\frac {1}{64 \left (\sqrt {1-x}-1\right )^{4}}+\frac {17}{96 \left (\sqrt {1-x}-1\right )^{3}}-\frac {159}{128 \left (\sqrt {1-x}-1\right )^{2}}+\frac {1083}{128 \left (\sqrt {1-x}-1\right )}+\frac {3003 \ln \left (\sqrt {1-x}-1\right )}{128}+\frac {2}{5 \left (1-x \right )^{\frac {5}{2}}}+\frac {10}{3 \left (1-x \right )^{\frac {3}{2}}}+\frac {30}{\sqrt {1-x}}+\frac {1}{64 \left (\sqrt {1-x}+1\right )^{4}}+\frac {17}{96 \left (\sqrt {1-x}+1\right )^{3}}+\frac {159}{128 \left (\sqrt {1-x}+1\right )^{2}}+\frac {1083}{128 \left (\sqrt {1-x}+1\right )}-\frac {3003 \ln \left (\sqrt {1-x}+1\right )}{128}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 111, normalized size = 0.94 \[ \frac {45045 \, {\left (x - 1\right )}^{6} + 165165 \, {\left (x - 1\right )}^{5} + 219219 \, {\left (x - 1\right )}^{4} + 119691 \, {\left (x - 1\right )}^{3} + 18304 \, {\left (x - 1\right )}^{2} - 1664 \, x + 2048}{960 \, {\left ({\left (-x + 1\right )}^{\frac {13}{2}} - 4 \, {\left (-x + 1\right )}^{\frac {11}{2}} + 6 \, {\left (-x + 1\right )}^{\frac {9}{2}} - 4 \, {\left (-x + 1\right )}^{\frac {7}{2}} + {\left (-x + 1\right )}^{\frac {5}{2}}\right )}} - \frac {3003}{128} \, \log \left (\sqrt {-x + 1} + 1\right ) + \frac {3003}{128} \, \log \left (\sqrt {-x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 96, normalized size = 0.81 \[ \frac {\frac {286\,{\left (x-1\right )}^2}{15}-\frac {26\,x}{15}+\frac {39897\,{\left (x-1\right )}^3}{320}+\frac {73073\,{\left (x-1\right )}^4}{320}+\frac {11011\,{\left (x-1\right )}^5}{64}+\frac {3003\,{\left (x-1\right )}^6}{64}+\frac {32}{15}}{{\left (1-x\right )}^{5/2}-4\,{\left (1-x\right )}^{7/2}+6\,{\left (1-x\right )}^{9/2}-4\,{\left (1-x\right )}^{11/2}+{\left (1-x\right )}^{13/2}}-\frac {3003\,\mathrm {atanh}\left (\sqrt {1-x}\right )}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 20.37, size = 971, normalized size = 8.23 \[ \begin {cases} - \frac {45045 i x^{7} \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )}}{- 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} \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )}}{- 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} \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )}}{- 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} \operatorname {asin}{\left (\frac {1}{\sqrt {x}} \right )}}{- 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}} & \text {for}\: \left |{x}\right | > 1 \\- \frac {45045 x^{7} \log {\relax (x )}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac {90090 x^{7} \log {\left (\sqrt {1 - x} + 1 \right )}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac {45045 i \pi 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 {\relax (x )}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac {270270 x^{6} \log {\left (\sqrt {1 - x} + 1 \right )}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac {135135 i \pi 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 {\relax (x )}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac {270270 x^{5} \log {\left (\sqrt {1 - x} + 1 \right )}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac {135135 i \pi 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 {\relax (x )}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} - \frac {90090 x^{4} \log {\left (\sqrt {1 - x} + 1 \right )}}{- 1920 x^{7} + 5760 x^{6} - 5760 x^{5} + 1920 x^{4}} + \frac {45045 i \pi 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}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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