Optimal. Leaf size=54 \[ \frac {3}{10} (1-2 x)^{3/2}-\frac {111}{50} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {88, 63, 206} \[ \frac {3}{10} (1-2 x)^{3/2}-\frac {111}{50} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx &=\int \left (\frac {111}{50 \sqrt {1-2 x}}-\frac {9}{10} \sqrt {1-2 x}+\frac {1}{25 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=-\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}+\frac {1}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}-\frac {1}{25} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {111}{50} \sqrt {1-2 x}+\frac {3}{10} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 46, normalized size = 0.85 \[ -\frac {3}{25} \sqrt {1-2 x} (5 x+16)-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 45, normalized size = 0.83 \[ -\frac {3}{25} \, {\left (5 \, x + 16\right )} \sqrt {-2 \, x + 1} + \frac {1}{1375} \, \sqrt {55} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 58, normalized size = 1.07 \[ \frac {3}{10} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {111}{50} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.70 \[ -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1375}+\frac {3 \left (-2 x +1\right )^{\frac {3}{2}}}{10}-\frac {111 \sqrt {-2 x +1}}{50} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 55, normalized size = 1.02 \[ \frac {3}{10} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {111}{50} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 37, normalized size = 0.69 \[ \frac {3\,{\left (1-2\,x\right )}^{3/2}}{10}-\frac {111\,\sqrt {1-2\,x}}{50}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1375} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.73, size = 90, normalized size = 1.67 \[ \frac {3 \left (1 - 2 x\right )^{\frac {3}{2}}}{10} - \frac {111 \sqrt {1 - 2 x}}{50} + \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{55} & \text {for}\: \frac {1}{1 - 2 x} > \frac {5}{11} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55}}{5 \sqrt {1 - 2 x}} \right )}}{55} & \text {for}\: \frac {1}{1 - 2 x} < \frac {5}{11} \end {cases}\right )}{25} \]
Verification of antiderivative is not currently implemented for this CAS.
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