Optimal. Leaf size=82 \[ \frac {9}{70} (1-2 x)^{7/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {2}{375} (1-2 x)^{3/2}+\frac {22}{625} \sqrt {1-2 x}-\frac {22}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ \frac {9}{70} (1-2 x)^{7/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {2}{375} (1-2 x)^{3/2}+\frac {22}{625} \sqrt {1-2 x}-\frac {22}{625} \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^2}{3+5 x} \, dx &=\int \left (\frac {111}{50} (1-2 x)^{3/2}-\frac {9}{10} (1-2 x)^{5/2}+\frac {(1-2 x)^{3/2}}{25 (3+5 x)}\right ) \, dx\\ &=-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}+\frac {1}{25} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}+\frac {11}{125} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {22}{625} \sqrt {1-2 x}+\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}+\frac {121}{625} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {22}{625} \sqrt {1-2 x}+\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}-\frac {121}{625} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {22}{625} \sqrt {1-2 x}+\frac {2}{375} (1-2 x)^{3/2}-\frac {111}{250} (1-2 x)^{5/2}+\frac {9}{70} (1-2 x)^{7/2}-\frac {22}{625} \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 = 0.06, size = 56, normalized size = 0.68 \[ \frac {-5 \sqrt {1-2 x} \left (13500 x^3+3060 x^2-13045 x+3608\right )-462 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{65625} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 61, normalized size = 0.74 \[ \frac {11}{3125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{13125} \, {\left (13500 \, x^{3} + 3060 \, x^{2} - 13045 \, x + 3608\right )} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 90, normalized size = 1.10 \[ -\frac {9}{70} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {111}{250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{3125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22}{625} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.68 \[ -\frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{3125}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{375}-\frac {111 \left (-2 x +1\right )^{\frac {5}{2}}}{250}+\frac {9 \left (-2 x +1\right )^{\frac {7}{2}}}{70}+\frac {22 \sqrt {-2 x +1}}{625} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 73, normalized size = 0.89 \[ \frac {9}{70} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {111}{250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{3125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{625} \, \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 57, normalized size = 0.70 \[ \frac {22\,\sqrt {1-2\,x}}{625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{375}-\frac {111\,{\left (1-2\,x\right )}^{5/2}}{250}+\frac {9\,{\left (1-2\,x\right )}^{7/2}}{70}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{3125} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.93, size = 114, normalized size = 1.39 \[ \frac {9 \left (1 - 2 x\right )^{\frac {7}{2}}}{70} - \frac {111 \left (1 - 2 x\right )^{\frac {5}{2}}}{250} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{375} + \frac {22 \sqrt {1 - 2 x}}{625} + \frac {242 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{625} \]
Verification of antiderivative is not currently implemented for this CAS.
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