Optimal. Leaf size=145 \[ -\frac {39520}{11319 (1-2 x)^{3/2}}-\frac {446660}{290521 \sqrt {1-2 x}}+\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac {127710 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401}-\frac {6250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {105, 156, 157,
162, 65, 212} \begin {gather*} -\frac {446660}{290521 \sqrt {1-2 x}}+\frac {582}{49 (1-2 x)^{3/2} (3 x+2)}-\frac {39520}{11319 (1-2 x)^{3/2}}+\frac {57}{49 (1-2 x)^{3/2} (3 x+2)^2}+\frac {1}{7 (1-2 x)^{3/2} (3 x+2)^3}+\frac {127710 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401}-\frac {6250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 105
Rule 156
Rule 157
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx &=\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {1}{21} \int \frac {24-135 x}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {1}{294} \int \frac {168-11970 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac {\int \frac {-109410-611100 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx}{2058}\\ &=-\frac {39520}{11319 (1-2 x)^{3/2}}+\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {582}{49 (1-2 x)^{3/2} (2+3 x)}-\frac {\int \frac {-2301705+18673200 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{237699}\\ &=-\frac {39520}{11319 (1-2 x)^{3/2}}-\frac {446660}{290521 \sqrt {1-2 x}}+\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac {2 \int \frac {\frac {346068765}{2}-105523425 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{18302823}\\ &=-\frac {39520}{11319 (1-2 x)^{3/2}}-\frac {446660}{290521 \sqrt {1-2 x}}+\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {582}{49 (1-2 x)^{3/2} (2+3 x)}-\frac {191565 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2401}+\frac {15625}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {39520}{11319 (1-2 x)^{3/2}}-\frac {446660}{290521 \sqrt {1-2 x}}+\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac {191565 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2401}-\frac {15625}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {39520}{11319 (1-2 x)^{3/2}}-\frac {446660}{290521 \sqrt {1-2 x}}+\frac {1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac {57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac {582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac {127710 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401}-\frac {6250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 99, normalized size = 0.68 \begin {gather*} \frac {8496203-9083055 x-47036214 x^2+26376300 x^3+72358920 x^4}{871563 (1-2 x)^{3/2} (2+3 x)^3}+\frac {127710 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401}-\frac {6250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 93, normalized size = 0.64
method | result | size |
derivativedivides | \(-\frac {6250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {32}{79233 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5344}{2033647 \sqrt {1-2 x}}-\frac {1458 \left (\frac {1438 \left (1-2 x \right )^{\frac {5}{2}}}{3}-\frac {61250 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {72520 \sqrt {1-2 x}}{27}\right )}{16807 \left (-4-6 x \right )^{3}}+\frac {127710 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{16807}\) | \(93\) |
default | \(-\frac {6250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {32}{79233 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5344}{2033647 \sqrt {1-2 x}}-\frac {1458 \left (\frac {1438 \left (1-2 x \right )^{\frac {5}{2}}}{3}-\frac {61250 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {72520 \sqrt {1-2 x}}{27}\right )}{16807 \left (-4-6 x \right )^{3}}+\frac {127710 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{16807}\) | \(93\) |
trager | \(\frac {\left (72358920 x^{4}+26376300 x^{3}-47036214 x^{2}-9083055 x +8496203\right ) \sqrt {1-2 x}}{871563 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2}}+\frac {63855 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{16807}-\frac {3125 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{1331}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 146, normalized size = 1.01 \begin {gather*} \frac {3125}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {63855}{16807} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (9044865 \, {\left (2 \, x - 1\right )}^{4} + 42773535 \, {\left (2 \, x - 1\right )}^{3} + 50533308 \, {\left (2 \, x - 1\right )}^{2} - 315168 \, x + 187768\right )}}{871563 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 343 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.80, size = 182, normalized size = 1.26 \begin {gather*} \frac {157565625 \, \sqrt {11} \sqrt {5} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 254973015 \, \sqrt {7} \sqrt {3} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (72358920 \, x^{4} + 26376300 \, x^{3} - 47036214 \, x^{2} - 9083055 \, x + 8496203\right )} \sqrt {-2 \, x + 1}}{67110351 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 15.31, size = 5506, normalized size = 37.97 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.05, size = 134, normalized size = 0.92 \begin {gather*} \frac {3125}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {63855}{16807} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {4 \, {\left (9044865 \, {\left (2 \, x - 1\right )}^{4} + 42773535 \, {\left (2 \, x - 1\right )}^{3} + 50533308 \, {\left (2 \, x - 1\right )}^{2} - 315168 \, x + 187768\right )}}{871563 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 109, normalized size = 0.75 \begin {gather*} \frac {\frac {50928\,{\left (2\,x-1\right )}^2}{5929}-\frac {8576\,x}{160083}+\frac {905260\,{\left (2\,x-1\right )}^3}{124509}+\frac {446660\,{\left (2\,x-1\right )}^4}{290521}+\frac {15328}{480249}}{\frac {343\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {49\,{\left (1-2\,x\right )}^{5/2}}{3}+7\,{\left (1-2\,x\right )}^{7/2}-{\left (1-2\,x\right )}^{9/2}}+\frac {127710\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{16807}-\frac {6250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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