Optimal. Leaf size=81 \[ \frac {409 \sqrt {1-2 x}}{3025}-\frac {(1-2 x)^{3/2}}{550 (3+5 x)^2}-\frac {133 (1-2 x)^{3/2}}{6050 (3+5 x)}-\frac {409 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \]
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Rubi [A]
time = 0.02, antiderivative size = 81, 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, 79, 52, 65,
212} \begin {gather*} -\frac {133 (1-2 x)^{3/2}}{6050 (5 x+3)}-\frac {(1-2 x)^{3/2}}{550 (5 x+3)^2}+\frac {409 \sqrt {1-2 x}}{3025}-\frac {409 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{3/2}}{550 (3+5 x)^2}+\frac {1}{550} \int \frac {\sqrt {1-2 x} (727+990 x)}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{550 (3+5 x)^2}-\frac {133 (1-2 x)^{3/2}}{6050 (3+5 x)}+\frac {409 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{1210}\\ &=\frac {409 \sqrt {1-2 x}}{3025}-\frac {(1-2 x)^{3/2}}{550 (3+5 x)^2}-\frac {133 (1-2 x)^{3/2}}{6050 (3+5 x)}+\frac {409}{550} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {409 \sqrt {1-2 x}}{3025}-\frac {(1-2 x)^{3/2}}{550 (3+5 x)^2}-\frac {133 (1-2 x)^{3/2}}{6050 (3+5 x)}-\frac {409}{550} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {409 \sqrt {1-2 x}}{3025}-\frac {(1-2 x)^{3/2}}{550 (3+5 x)^2}-\frac {133 (1-2 x)^{3/2}}{6050 (3+5 x)}-\frac {409 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 58, normalized size = 0.72 \begin {gather*} \frac {\sqrt {1-2 x} \left (632+2245 x+1980 x^2\right )}{550 (3+5 x)^2}-\frac {409 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{275 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 57, normalized size = 0.70
method | result | size |
risch | \(-\frac {3960 x^{3}+2510 x^{2}-981 x -632}{550 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {409 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\) | \(51\) |
derivativedivides | \(\frac {18 \sqrt {1-2 x}}{125}+\frac {\frac {131 \left (1-2 x \right )^{\frac {3}{2}}}{275}-\frac {133 \sqrt {1-2 x}}{125}}{\left (-6-10 x \right )^{2}}-\frac {409 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\) | \(57\) |
default | \(\frac {18 \sqrt {1-2 x}}{125}+\frac {\frac {131 \left (1-2 x \right )^{\frac {3}{2}}}{275}-\frac {133 \sqrt {1-2 x}}{125}}{\left (-6-10 x \right )^{2}}-\frac {409 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15125}\) | \(57\) |
trager | \(\frac {\left (1980 x^{2}+2245 x +632\right ) \sqrt {1-2 x}}{550 \left (3+5 x \right )^{2}}+\frac {409 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{30250}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 83, normalized size = 1.02 \begin {gather*} \frac {409}{30250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {18}{125} \, \sqrt {-2 \, x + 1} + \frac {655 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1463 \, \sqrt {-2 \, x + 1}}{1375 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.68, size = 74, normalized size = 0.91 \begin {gather*} \frac {409 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (1980 \, x^{2} + 2245 \, x + 632\right )} \sqrt {-2 \, x + 1}}{30250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 233.27, size = 360, normalized size = 4.44 \begin {gather*} \frac {18 \sqrt {1 - 2 x}}{125} - \frac {256 \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 {88 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{125} + \frac {174 \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 = 1.29, size = 77, normalized size = 0.95 \begin {gather*} \frac {409}{30250} \, \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 {18}{125} \, \sqrt {-2 \, x + 1} + \frac {655 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1463 \, \sqrt {-2 \, x + 1}}{5500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 63, normalized size = 0.78 \begin {gather*} \frac {18\,\sqrt {1-2\,x}}{125}-\frac {409\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{15125}-\frac {\frac {133\,\sqrt {1-2\,x}}{3125}-\frac {131\,{\left (1-2\,x\right )}^{3/2}}{6875}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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