Optimal. Leaf size=97 \[ -\frac {11 (1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac {803 \sqrt {1-2 x}}{50 (5 x+3)}+98 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2523}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 149, 156, 63, 206} \[ -\frac {11 (1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac {803 \sqrt {1-2 x}}{50 (5 x+3)}+98 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2523}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 149
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^3} \, dx &=-\frac {11 (1-2 x)^{3/2}}{10 (3+5 x)^2}-\frac {1}{10} \int \frac {(136-41 x) \sqrt {1-2 x}}{(2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {11 (1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac {803 \sqrt {1-2 x}}{50 (3+5 x)}-\frac {1}{50} \int \frac {-4056+2491 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {11 (1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac {803 \sqrt {1-2 x}}{50 (3+5 x)}-343 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {27753}{50} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {11 (1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac {803 \sqrt {1-2 x}}{50 (3+5 x)}+343 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {27753}{50} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {11 (1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac {803 \sqrt {1-2 x}}{50 (3+5 x)}+98 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2523}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 81, normalized size = 0.84 \[ 98 \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1}{250} \left (\frac {55 \sqrt {1-2 x} (375 x+214)}{(5 x+3)^2}-5046 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 122, normalized size = 1.26 \[ \frac {7569 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 12250 \, \sqrt {7} \sqrt {3} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 165 \, {\left (375 \, x + 214\right )} \sqrt {-2 \, x + 1}}{750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 107, normalized size = 1.10 \[ \frac {2523}{250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {49}{3} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {11 \, {\left (375 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 803 \, \sqrt {-2 \, x + 1}\right )}}{100 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.68 \[ \frac {98 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{3}-\frac {2523 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{125}+\frac {-165 \left (-2 x +1\right )^{\frac {3}{2}}+\frac {8833 \sqrt {-2 x +1}}{25}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 110, normalized size = 1.13 \[ \frac {2523}{250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {49}{3} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {11 \, {\left (375 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 803 \, \sqrt {-2 \, x + 1}\right )}}{25 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 71, normalized size = 0.73 \[ \frac {98\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3}-\frac {2523\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{125}+\frac {\frac {8833\,\sqrt {1-2\,x}}{625}-\frac {33\,{\left (1-2\,x\right )}^{3/2}}{5}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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