Optimal. Leaf size=74 \[ -\frac {(1-2 x)^{3/2}}{63 (3 x+2)}-\frac {25}{27} (1-2 x)^{3/2}-\frac {142}{189} \sqrt {1-2 x}+\frac {142 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \]
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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} \[ -\frac {(1-2 x)^{3/2}}{63 (3 x+2)}-\frac {25}{27} (1-2 x)^{3/2}-\frac {142}{189} \sqrt {1-2 x}+\frac {142 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \]
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} (3+5 x)^2}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{3/2}}{63 (2+3 x)}+\frac {1}{63} \int \frac {\sqrt {1-2 x} (279+525 x)}{2+3 x} \, dx\\ &=-\frac {25}{27} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{63 (2+3 x)}-\frac {71}{63} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=-\frac {142}{189} \sqrt {1-2 x}-\frac {25}{27} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{63 (2+3 x)}-\frac {71}{27} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {142}{189} \sqrt {1-2 x}-\frac {25}{27} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{63 (2+3 x)}+\frac {71}{27} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {142}{189} \sqrt {1-2 x}-\frac {25}{27} (1-2 x)^{3/2}-\frac {(1-2 x)^{3/2}}{63 (2+3 x)}+\frac {142 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 55, normalized size = 0.74 \[ \frac {\sqrt {1-2 x} \left (150 x^2-35 x-91\right )}{81 x+54}+\frac {142 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 65, normalized size = 0.88 \[ \frac {71 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (150 \, x^{2} - 35 \, x - 91\right )} \sqrt {-2 \, x + 1}}{567 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 74, normalized size = 1.00 \[ -\frac {25}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {71}{567} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {20}{27} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \]
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 \[ \frac {142 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{567}-\frac {25 \left (-2 x +1\right )^{\frac {3}{2}}}{27}-\frac {20 \sqrt {-2 x +1}}{27}+\frac {2 \sqrt {-2 x +1}}{81 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 71, normalized size = 0.96 \[ -\frac {25}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {71}{567} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20}{27} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 55, normalized size = 0.74 \[ -\frac {2\,\sqrt {1-2\,x}}{81\,\left (2\,x+\frac {4}{3}\right )}-\frac {20\,\sqrt {1-2\,x}}{27}-\frac {25\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,142{}\mathrm {i}}{567} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 121.34, size = 192, normalized size = 2.59 \[ - \frac {25 \left (1 - 2 x\right )^{\frac {3}{2}}}{27} - \frac {20 \sqrt {1 - 2 x}}{27} - \frac {28 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {2}{3} \end {cases}\right )}{27} - \frac {16 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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