Optimal. Leaf size=92 \[ \frac {7 (1-2 x)^{3/2}}{3 (3 x+2)}+\frac {26}{15} \sqrt {1-2 x}+\frac {140}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {242}{5} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 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} \[ \frac {7 (1-2 x)^{3/2}}{3 (3 x+2)}+\frac {26}{15} \sqrt {1-2 x}+\frac {140}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {242}{5} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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)^2 (3+5 x)} \, dx &=\frac {7 (1-2 x)^{3/2}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\sqrt {1-2 x} (96+39 x)}{(2+3 x) (3+5 x)} \, dx\\ &=\frac {26}{15} \sqrt {1-2 x}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x)}+\frac {2}{45} \int \frac {954-\frac {813 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac {26}{15} \sqrt {1-2 x}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x)}-\frac {490}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {1331}{5} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {26}{15} \sqrt {1-2 x}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x)}+\frac {490}{3} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {1331}{5} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {26}{15} \sqrt {1-2 x}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x)}+\frac {140}{3} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {242}{5} \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 = 0.09, size = 78, normalized size = 0.85 \[ \frac {1}{225} \left (\frac {15 \sqrt {1-2 x} (8 x+87)}{3 x+2}+3500 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-2178 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 107, normalized size = 1.16 \[ \frac {1089 \, \sqrt {11} \sqrt {5} {\left (3 \, x + 2\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 1750 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 15 \, {\left (8 \, x + 87\right )} \sqrt {-2 \, x + 1}}{225 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 104, normalized size = 1.13 \[ \frac {121}{25} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {70}{9} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {8}{45} \, \sqrt {-2 \, x + 1} + \frac {49 \, \sqrt {-2 \, x + 1}}{9 \, {\left (3 \, x + 2\right )}} \]
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 \[ \frac {140 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{9}-\frac {242 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{25}+\frac {8 \sqrt {-2 x +1}}{45}-\frac {98 \sqrt {-2 x +1}}{27 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 98, normalized size = 1.07 \[ \frac {121}{25} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {70}{9} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8}{45} \, \sqrt {-2 \, x + 1} + \frac {49 \, \sqrt {-2 \, x + 1}}{9 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 66, normalized size = 0.72 \[ \frac {98\,\sqrt {1-2\,x}}{27\,\left (2\,x+\frac {4}{3}\right )}+\frac {8\,\sqrt {1-2\,x}}{45}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,140{}\mathrm {i}}{9}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{25} \]
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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