Optimal. Leaf size=89 \[ -\frac {(1-2 x)^{7/2}}{55 (5 x+3)}+\frac {56 (1-2 x)^{5/2}}{1375}+\frac {56}{375} (1-2 x)^{3/2}+\frac {616}{625} \sqrt {1-2 x}-\frac {616}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 50, 63, 206} \[ -\frac {(1-2 x)^{7/2}}{55 (5 x+3)}+\frac {56 (1-2 x)^{5/2}}{1375}+\frac {56}{375} (1-2 x)^{3/2}+\frac {616}{625} \sqrt {1-2 x}-\frac {616}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}+\frac {28}{55} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}+\frac {28}{25} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}+\frac {308}{125} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {616}{625} \sqrt {1-2 x}+\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}+\frac {3388}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {616}{625} \sqrt {1-2 x}+\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}-\frac {3388}{625} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {616}{625} \sqrt {1-2 x}+\frac {56}{375} (1-2 x)^{3/2}+\frac {56 (1-2 x)^{5/2}}{1375}-\frac {(1-2 x)^{7/2}}{55 (3+5 x)}-\frac {616}{625} \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 = 0.05, size = 63, normalized size = 0.71 \[ \frac {\frac {5 \sqrt {1-2 x} \left (1800 x^3-3820 x^2+8630 x+6579\right )}{5 x+3}-1848 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{9375} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 75, normalized size = 0.84 \[ \frac {924 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (1800 \, x^{3} - 3820 \, x^{2} + 8630 \, x + 6579\right )} \sqrt {-2 \, x + 1}}{9375 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 90, normalized size = 1.01 \[ \frac {6}{125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {62}{375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {308}{3125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {638}{625} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.71 \[ -\frac {616 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{3125}+\frac {6 \left (-2 x +1\right )^{\frac {5}{2}}}{125}+\frac {62 \left (-2 x +1\right )^{\frac {3}{2}}}{375}+\frac {638 \sqrt {-2 x +1}}{625}+\frac {242 \sqrt {-2 x +1}}{3125 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 80, normalized size = 0.90 \[ \frac {6}{125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {62}{375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {308}{3125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {638}{625} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 64, normalized size = 0.72 \[ \frac {638\,\sqrt {1-2\,x}}{625}-\frac {242\,\sqrt {1-2\,x}}{3125\,\left (2\,x+\frac {6}{5}\right )}+\frac {62\,{\left (1-2\,x\right )}^{3/2}}{375}+\frac {6\,{\left (1-2\,x\right )}^{5/2}}{125}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,616{}\mathrm {i}}{3125} \]
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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