Optimal. Leaf size=74 \[ \frac {26}{275} \sqrt {1-2 x}-\frac {3}{25} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{275 (3+5 x)}-\frac {26 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{25 \sqrt {55}} \]
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Rubi [A]
time = 0.01, 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 = {91, 81, 52, 65,
212} \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 52
Rule 65
Rule 81
Rule 91
Rule 212
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} \text {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 = 0.11, size = 58, normalized size = 0.78 \begin {gather*} \frac {\sqrt {1-2 x} \left (-2+15 x+30 x^2\right )}{25 (3+5 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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Maple [A]
time = 0.10, size = 54, normalized size = 0.73
method | result | size |
risch | \(-\frac {60 x^{3}-19 x +2}{25 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {26 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(46\) |
derivativedivides | \(-\frac {3 \left (1-2 x \right )^{\frac {3}{2}}}{25}+\frac {12 \sqrt {1-2 x}}{125}+\frac {2 \sqrt {1-2 x}}{625 \left (-\frac {6}{5}-2 x \right )}-\frac {26 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(54\) |
default | \(-\frac {3 \left (1-2 x \right )^{\frac {3}{2}}}{25}+\frac {12 \sqrt {1-2 x}}{125}+\frac {2 \sqrt {1-2 x}}{625 \left (-\frac {6}{5}-2 x \right )}-\frac {26 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1375}\) | \(54\) |
trager | \(\frac {\left (30 x^{2}+15 x -2\right ) \sqrt {1-2 x}}{75+125 x}-\frac {13 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{1375}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, 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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Fricas [A]
time = 0.79, 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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Sympy [A]
time = 53.47, size = 204, normalized size = 2.76 \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}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{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}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.81, 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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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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