Optimal. Leaf size=76 \[ -\frac {(1-2 x)^{5/2}}{55 (5 x+3)}+\frac {4}{55} (1-2 x)^{3/2}+\frac {12}{25} \sqrt {1-2 x}-\frac {12}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 50, 63, 206} \[ -\frac {(1-2 x)^{5/2}}{55 (5 x+3)}+\frac {4}{55} (1-2 x)^{3/2}+\frac {12}{25} \sqrt {1-2 x}-\frac {12}{25} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}+\frac {6}{11} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}+\frac {6}{5} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {12}{25} \sqrt {1-2 x}+\frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}+\frac {66}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {12}{25} \sqrt {1-2 x}+\frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}-\frac {66}{25} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {12}{25} \sqrt {1-2 x}+\frac {4}{55} (1-2 x)^{3/2}-\frac {(1-2 x)^{5/2}}{55 (3+5 x)}-\frac {12}{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 = 0.04, size = 58, normalized size = 0.76 \[ \frac {1}{125} \left (\frac {5 \sqrt {1-2 x} \left (-20 x^2+60 x+41\right )}{5 x+3}-12 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 70, normalized size = 0.92 \[ \frac {6 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (20 \, x^{2} - 60 \, x - 41\right )} \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 74, normalized size = 0.97 \[ \frac {2}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6}{125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {62}{125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.71 \[ -\frac {12 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{125}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{25}+\frac {62 \sqrt {-2 x +1}}{125}+\frac {22 \sqrt {-2 x +1}}{625 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 71, normalized size = 0.93 \[ \frac {2}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6}{125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {62}{125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 55, normalized size = 0.72 \[ \frac {62\,\sqrt {1-2\,x}}{125}-\frac {22\,\sqrt {1-2\,x}}{625\,\left (2\,x+\frac {6}{5}\right )}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{25}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,12{}\mathrm {i}}{125} \]
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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