Optimal. Leaf size=88 \[ -\frac {\sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}+\frac {21}{125} \sqrt {1-2 x} (3 x+2)^2-\frac {294}{625} \sqrt {1-2 x}-\frac {196 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 153, 12, 80, 63, 206} \[ -\frac {\sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}+\frac {21}{125} \sqrt {1-2 x} (3 x+2)^2-\frac {294}{625} \sqrt {1-2 x}-\frac {196 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 80
Rule 97
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^2} \, dx &=-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}+\frac {1}{5} \int \frac {(7-21 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}-\frac {1}{125} \int -\frac {98 (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}+\frac {98}{125} \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {294}{625} \sqrt {1-2 x}+\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}+\frac {98}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {294}{625} \sqrt {1-2 x}+\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}-\frac {98}{625} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {294}{625} \sqrt {1-2 x}+\frac {21}{125} \sqrt {1-2 x} (2+3 x)^2-\frac {\sqrt {1-2 x} (2+3 x)^3}{5 (3+5 x)}-\frac {196 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.72 \[ \frac {\sqrt {1-2 x} \left (1350 x^3+2385 x^2-90 x-622\right )}{625 (5 x+3)}-\frac {196 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{625 \sqrt {55}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 69, normalized size = 0.78 \[ \frac {98 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (1350 \, x^{3} + 2385 \, x^{2} - 90 \, x - 622\right )} \sqrt {-2 \, x + 1}}{34375 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 90, normalized size = 1.02 \[ \frac {27}{250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {117}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {98}{34375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {18}{625} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.72 \[ -\frac {196 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{34375}+\frac {27 \left (-2 x +1\right )^{\frac {5}{2}}}{250}-\frac {117 \left (-2 x +1\right )^{\frac {3}{2}}}{250}+\frac {18 \sqrt {-2 x +1}}{625}+\frac {2 \sqrt {-2 x +1}}{3125 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 80, normalized size = 0.91 \[ \frac {27}{250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {117}{250} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {98}{34375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {18}{625} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{625 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 64, normalized size = 0.73 \[ \frac {18\,\sqrt {1-2\,x}}{625}-\frac {2\,\sqrt {1-2\,x}}{3125\,\left (2\,x+\frac {6}{5}\right )}-\frac {117\,{\left (1-2\,x\right )}^{3/2}}{250}+\frac {27\,{\left (1-2\,x\right )}^{5/2}}{250}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,196{}\mathrm {i}}{34375} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 133.29, size = 202, normalized size = 2.30 \[ \frac {27 \left (1 - 2 x\right )^{\frac {5}{2}}}{250} - \frac {117 \left (1 - 2 x\right )^{\frac {3}{2}}}{250} + \frac {18 \sqrt {1 - 2 x}}{625} - \frac {44 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{625} + \frac {194 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{625} \]
Verification of antiderivative is not currently implemented for this CAS.
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