Optimal. Leaf size=56 \[ -\frac {5}{9} (1-2 x)^{3/2}-\frac {2}{9} \sqrt {1-2 x}+\frac {2}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {80, 50, 63, 206} \begin {gather*} -\frac {5}{9} (1-2 x)^{3/2}-\frac {2}{9} \sqrt {1-2 x}+\frac {2}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)}{2+3 x} \, dx &=-\frac {5}{9} (1-2 x)^{3/2}-\frac {1}{3} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {2}{9} \sqrt {1-2 x}-\frac {5}{9} (1-2 x)^{3/2}-\frac {7}{9} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {2}{9} \sqrt {1-2 x}-\frac {5}{9} (1-2 x)^{3/2}+\frac {7}{9} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {2}{9} \sqrt {1-2 x}-\frac {5}{9} (1-2 x)^{3/2}+\frac {2}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 0.82 \begin {gather*} \frac {1}{27} \left (3 \sqrt {1-2 x} (10 x-7)+2 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 52, normalized size = 0.93 \begin {gather*} \frac {2}{9} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1}{9} (5 (1-2 x)+2) \sqrt {1-2 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.50, size = 52, normalized size = 0.93 \begin {gather*} \frac {1}{27} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac {1}{9} \, {\left (10 \, x - 7\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 58, normalized size = 1.04 \begin {gather*} -\frac {5}{9} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1}{27} \, \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 {2}{9} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.68 \begin {gather*} \frac {2 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{27}-\frac {5 \left (-2 x +1\right )^{\frac {3}{2}}}{9}-\frac {2 \sqrt {-2 x +1}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 55, normalized size = 0.98 \begin {gather*} -\frac {5}{9} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1}{27} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2}{9} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 37, normalized size = 0.66 \begin {gather*} \frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{27}-\frac {2\,\sqrt {1-2\,x}}{9}-\frac {5\,{\left (1-2\,x\right )}^{3/2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.63, size = 92, normalized size = 1.64 \begin {gather*} - \frac {5 \left (1 - 2 x\right )^{\frac {3}{2}}}{9} - \frac {2 \sqrt {1 - 2 x}}{9} - \frac {14 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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