Optimal. Leaf size=69 \[ \frac {2}{27} \sqrt {1-2 x}-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}-\frac {2}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 212}
\begin {gather*} \frac {5}{6} (1-2 x)^{5/2}-\frac {155}{54} (1-2 x)^{3/2}+\frac {2}{27} \sqrt {1-2 x}-\frac {2}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^2}{2+3 x} \, dx &=\int \left (\frac {155}{18} \sqrt {1-2 x}-\frac {25}{6} (1-2 x)^{3/2}+\frac {\sqrt {1-2 x}}{9 (2+3 x)}\right ) \, dx\\ &=-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}+\frac {1}{9} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {2}{27} \sqrt {1-2 x}-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}+\frac {7}{27} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {2}{27} \sqrt {1-2 x}-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}-\frac {7}{27} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {2}{27} \sqrt {1-2 x}-\frac {155}{54} (1-2 x)^{3/2}+\frac {5}{6} (1-2 x)^{5/2}-\frac {2}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 51, normalized size = 0.74 \begin {gather*} \frac {1}{81} \left (3 \sqrt {1-2 x} \left (-53+65 x+90 x^2\right )-2 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 47, normalized size = 0.68
method | result | size |
risch | \(-\frac {\left (90 x^{2}+65 x -53\right ) \left (-1+2 x \right )}{27 \sqrt {1-2 x}}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}\) | \(44\) |
derivativedivides | \(-\frac {155 \left (1-2 x \right )^{\frac {3}{2}}}{54}+\frac {5 \left (1-2 x \right )^{\frac {5}{2}}}{6}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}+\frac {2 \sqrt {1-2 x}}{27}\) | \(47\) |
default | \(-\frac {155 \left (1-2 x \right )^{\frac {3}{2}}}{54}+\frac {5 \left (1-2 x \right )^{\frac {5}{2}}}{6}-\frac {2 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{81}+\frac {2 \sqrt {1-2 x}}{27}\) | \(47\) |
trager | \(\left (\frac {10}{3} x^{2}+\frac {65}{27} x -\frac {53}{27}\right ) \sqrt {1-2 x}-\frac {\RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{81}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 64, normalized size = 0.93 \begin {gather*} \frac {5}{6} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {155}{54} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{81} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{27} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.37, size = 56, normalized size = 0.81 \begin {gather*} \frac {1}{81} \, \sqrt {7} \sqrt {3} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + \frac {1}{27} \, {\left (90 \, x^{2} + 65 \, x - 53\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.12, size = 95, normalized size = 1.38 \begin {gather*} \frac {5 \left (1 - 2 x\right )^{\frac {5}{2}}}{6} - \frac {155 \left (1 - 2 x\right )^{\frac {3}{2}}}{54} + \frac {2 \sqrt {1 - 2 x}}{27} + \frac {14 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 74, normalized size = 1.07 \begin {gather*} \frac {5}{6} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {155}{54} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{81} \, \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 {2}{27} \, \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 48, normalized size = 0.70 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{27}-\frac {155\,{\left (1-2\,x\right )}^{3/2}}{54}+\frac {5\,{\left (1-2\,x\right )}^{5/2}}{6}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,2{}\mathrm {i}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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