Optimal. Leaf size=88 \[ -\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac {2381 \sqrt {1-2 x}}{2646 (2+3 x)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1323 \sqrt {21}} \]
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Rubi [A]
time = 0.02, antiderivative size = 88, 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, 43, 65,
212} \begin {gather*} \frac {23 (1-2 x)^{3/2}}{294 (3 x+2)^2}-\frac {(1-2 x)^{3/2}}{189 (3 x+2)^3}-\frac {2381 \sqrt {1-2 x}}{2646 (3 x+2)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1323 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {1}{189} \int \frac {\sqrt {1-2 x} (843+1575 x)}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}+\frac {2381}{882} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac {2381 \sqrt {1-2 x}}{2646 (2+3 x)}-\frac {2381 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2646}\\ &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac {2381 \sqrt {1-2 x}}{2646 (2+3 x)}+\frac {2381 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2646}\\ &=-\frac {(1-2 x)^{3/2}}{189 (2+3 x)^3}+\frac {23 (1-2 x)^{3/2}}{294 (2+3 x)^2}-\frac {2381 \sqrt {1-2 x}}{2646 (2+3 x)}+\frac {2381 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1323 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 60, normalized size = 0.68 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (9124+28751 x+22671 x^2\right )}{2 (2+3 x)^3}+2381 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27783} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 57, normalized size = 0.65
method | result | size |
risch | \(\frac {45342 x^{3}+34831 x^{2}-10503 x -9124}{2646 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {2381 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{27783}\) | \(51\) |
derivativedivides | \(-\frac {108 \left (-\frac {2519 \left (1-2 x \right )^{\frac {5}{2}}}{15876}+\frac {3673 \left (1-2 x \right )^{\frac {3}{2}}}{5103}-\frac {2381 \sqrt {1-2 x}}{2916}\right )}{\left (-4-6 x \right )^{3}}+\frac {2381 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{27783}\) | \(57\) |
default | \(-\frac {108 \left (-\frac {2519 \left (1-2 x \right )^{\frac {5}{2}}}{15876}+\frac {3673 \left (1-2 x \right )^{\frac {3}{2}}}{5103}-\frac {2381 \sqrt {1-2 x}}{2916}\right )}{\left (-4-6 x \right )^{3}}+\frac {2381 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{27783}\) | \(57\) |
trager | \(-\frac {\left (22671 x^{2}+28751 x +9124\right ) \sqrt {1-2 x}}{2646 \left (2+3 x \right )^{3}}-\frac {2381 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{55566}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 92, normalized size = 1.05 \begin {gather*} -\frac {2381}{55566} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {22671 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 102844 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 116669 \, \sqrt {-2 \, x + 1}}{1323 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.34, size = 85, normalized size = 0.97 \begin {gather*} \frac {2381 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (22671 \, x^{2} + 28751 \, x + 9124\right )} \sqrt {-2 \, x + 1}}{55566 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 84, normalized size = 0.95 \begin {gather*} -\frac {2381}{55566} \, \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 {22671 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 102844 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 116669 \, \sqrt {-2 \, x + 1}}{10584 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 72, normalized size = 0.82 \begin {gather*} \frac {2381\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{27783}-\frac {\frac {2381\,\sqrt {1-2\,x}}{729}-\frac {14692\,{\left (1-2\,x\right )}^{3/2}}{5103}+\frac {2519\,{\left (1-2\,x\right )}^{5/2}}{3969}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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