Optimal. Leaf size=74 \[ -\frac {(1-2 x)^{3/2}}{275 (5 x+3)}-\frac {3}{25} (1-2 x)^{3/2}+\frac {26}{275} \sqrt {1-2 x}-\frac {26 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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Rubi [A] time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 80, 50, 63, 206} \begin {gather*} -\frac {(1-2 x)^{3/2}}{275 (5 x+3)}-\frac {3}{25} (1-2 x)^{3/2}+\frac {26}{275} \sqrt {1-2 x}-\frac {26 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{3/2}}{275 (3+5 x)}+\frac {1}{275} \int \frac {\sqrt {1-2 x} (362+495 x)}{3+5 x} \, dx\\ &=-\frac {3}{25} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{275 (3+5 x)}+\frac {13}{55} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {26}{275} \sqrt {1-2 x}-\frac {3}{25} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{275 (3+5 x)}+\frac {13}{25} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {26}{275} \sqrt {1-2 x}-\frac {3}{25} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{275 (3+5 x)}-\frac {13}{25} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {26}{275} \sqrt {1-2 x}-\frac {3}{25} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{275 (3+5 x)}-\frac {26 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 58, normalized size = 0.78 \begin {gather*} \frac {\sqrt {1-2 x} \left (30 x^2+15 x-2\right )}{25 (5 x+3)}-\frac {26 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 70, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {1-2 x} \left (15 (1-2 x)^2-45 (1-2 x)+26\right )}{25 (5 (1-2 x)-11)}-\frac {26 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 64, normalized size = 0.86 \begin {gather*} \frac {13 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (30 \, x^{2} + 15 \, x - 2\right )} \sqrt {-2 \, x + 1}}{1375 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 74, normalized size = 1.00 \begin {gather*} -\frac {3}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {13}{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 {12}{125} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.73 \begin {gather*} -\frac {26 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1375}-\frac {3 \left (-2 x +1\right )^{\frac {3}{2}}}{25}+\frac {12 \sqrt {-2 x +1}}{125}+\frac {2 \sqrt {-2 x +1}}{625 \left (-2 x -\frac {6}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 71, normalized size = 0.96 \begin {gather*} -\frac {3}{25} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {13}{1375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12}{125} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{125 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 55, normalized size = 0.74 \begin {gather*} \frac {12\,\sqrt {1-2\,x}}{125}-\frac {2\,\sqrt {1-2\,x}}{625\,\left (2\,x+\frac {6}{5}\right )}-\frac {3\,{\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 )\,26{}\mathrm {i}}{1375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 119.60, size = 190, normalized size = 2.57 \begin {gather*} - \frac {3 \left (1 - 2 x\right )^{\frac {3}{2}}}{25} + \frac {12 \sqrt {1 - 2 x}}{125} - \frac {44 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{125} + \frac {128 \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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