Optimal. Leaf size=92 \[ -\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}-\frac {58}{75} \sqrt {1-2 x}-\frac {98}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {836}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 92, 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, 154, 156, 63, 206} \[ -\frac {11 (1-2 x)^{3/2}}{5 (5 x+3)}-\frac {58}{75} \sqrt {1-2 x}-\frac {98}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {836}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 154
Rule 156
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx &=-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {1}{5} \int \frac {\sqrt {1-2 x} (101+29 x)}{(2+3 x) (3+5 x)} \, dx\\ &=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {2}{75} \int \frac {\frac {1863}{2}-\frac {1493 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}+\frac {343}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {4598}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {343}{3} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {4598}{25} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {58}{75} \sqrt {1-2 x}-\frac {11 (1-2 x)^{3/2}}{5 (3+5 x)}-\frac {98}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {836}{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 = 0.14, size = 83, normalized size = 0.90 \[ \frac {1}{375} \left (\frac {5 \sqrt {1-2 x} (40 x-339)}{5 x+3}+2508 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {98}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 107, normalized size = 1.16 \[ \frac {3762 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 6125 \, \sqrt {7} \sqrt {3} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 15 \, {\left (40 \, x - 339\right )} \sqrt {-2 \, x + 1}}{1125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 104, normalized size = 1.13 \[ -\frac {418}{125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {49}{9} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {8}{75} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.68 \[ -\frac {98 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{9}+\frac {836 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{125}+\frac {8 \sqrt {-2 x +1}}{75}+\frac {242 \sqrt {-2 x +1}}{125 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 98, normalized size = 1.07 \[ -\frac {418}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {49}{9} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8}{75} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 66, normalized size = 0.72 \[ \frac {8\,\sqrt {1-2\,x}}{75}-\frac {242\,\sqrt {1-2\,x}}{125\,\left (2\,x+\frac {6}{5}\right )}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{9}-\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,836{}\mathrm {i}}{125} \]
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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