Optimal. Leaf size=69 \[ \frac {9}{50} (1-2 x)^{5/2}-\frac {37}{50} (1-2 x)^{3/2}+\frac {2}{125} \sqrt {1-2 x}-\frac {2}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \begin {gather*} \frac {9}{50} (1-2 x)^{5/2}-\frac {37}{50} (1-2 x)^{3/2}+\frac {2}{125} \sqrt {1-2 x}-\frac {2}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^2}{3+5 x} \, dx &=\int \left (\frac {111}{50} \sqrt {1-2 x}-\frac {9}{10} (1-2 x)^{3/2}+\frac {\sqrt {1-2 x}}{25 (3+5 x)}\right ) \, dx\\ &=-\frac {37}{50} (1-2 x)^{3/2}+\frac {9}{50} (1-2 x)^{5/2}+\frac {1}{25} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {2}{125} \sqrt {1-2 x}-\frac {37}{50} (1-2 x)^{3/2}+\frac {9}{50} (1-2 x)^{5/2}+\frac {11}{125} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {2}{125} \sqrt {1-2 x}-\frac {37}{50} (1-2 x)^{3/2}+\frac {9}{50} (1-2 x)^{5/2}-\frac {11}{125} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {2}{125} \sqrt {1-2 x}-\frac {37}{50} (1-2 x)^{3/2}+\frac {9}{50} (1-2 x)^{5/2}-\frac {2}{125} \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 = 51, normalized size = 0.74 \begin {gather*} \frac {1}{625} \left (5 \sqrt {1-2 x} \left (90 x^2+95 x-68\right )-2 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 61, normalized size = 0.88 \begin {gather*} \frac {1}{250} \left (45 (1-2 x)^2-185 (1-2 x)+4\right ) \sqrt {1-2 x}-\frac {2}{125} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 56, normalized size = 0.81 \begin {gather*} \frac {1}{625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{125} \, {\left (90 \, x^{2} + 95 \, x - 68\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 74, normalized size = 1.07 \begin {gather*} \frac {9}{50} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {37}{50} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{625} \, \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 {2}{125} \, \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 = 47, normalized size = 0.68 \begin {gather*} -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{625}-\frac {37 \left (-2 x +1\right )^{\frac {3}{2}}}{50}+\frac {9 \left (-2 x +1\right )^{\frac {5}{2}}}{50}+\frac {2 \sqrt {-2 x +1}}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 64, normalized size = 0.93 \begin {gather*} \frac {9}{50} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {37}{50} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{125} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 48, normalized size = 0.70 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{125}-\frac {37\,{\left (1-2\,x\right )}^{3/2}}{50}+\frac {9\,{\left (1-2\,x\right )}^{5/2}}{50}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.21, size = 102, normalized size = 1.48 \begin {gather*} \frac {9 \left (1 - 2 x\right )^{\frac {5}{2}}}{50} - \frac {37 \left (1 - 2 x\right )^{\frac {3}{2}}}{50} + \frac {2 \sqrt {1 - 2 x}}{125} + \frac {22 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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