Optimal. Leaf size=108 \[ \frac {121}{14 \sqrt {1-2 x} (2+3 x)^3}-\frac {467 \sqrt {1-2 x}}{126 (2+3 x)^3}-\frac {905 \sqrt {1-2 x}}{882 (2+3 x)^2}-\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}-\frac {905 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
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Rubi [A]
time = 0.02, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 44, 65,
212} \begin {gather*} -\frac {905 \sqrt {1-2 x}}{2058 (3 x+2)}-\frac {905 \sqrt {1-2 x}}{882 (3 x+2)^2}-\frac {467 \sqrt {1-2 x}}{126 (3 x+2)^3}+\frac {121}{14 \sqrt {1-2 x} (3 x+2)^3}-\frac {905 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^4} \, dx &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^3}-\frac {1}{14} \int \frac {-973+175 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^3}-\frac {467 \sqrt {1-2 x}}{126 (2+3 x)^3}+\frac {905}{63} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^3}-\frac {467 \sqrt {1-2 x}}{126 (2+3 x)^3}-\frac {905 \sqrt {1-2 x}}{882 (2+3 x)^2}+\frac {905}{294} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^3}-\frac {467 \sqrt {1-2 x}}{126 (2+3 x)^3}-\frac {905 \sqrt {1-2 x}}{882 (2+3 x)^2}-\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {905 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2058}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^3}-\frac {467 \sqrt {1-2 x}}{126 (2+3 x)^3}-\frac {905 \sqrt {1-2 x}}{882 (2+3 x)^2}-\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}-\frac {905 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2058}\\ &=\frac {121}{14 \sqrt {1-2 x} (2+3 x)^3}-\frac {467 \sqrt {1-2 x}}{126 (2+3 x)^3}-\frac {905 \sqrt {1-2 x}}{882 (2+3 x)^2}-\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}-\frac {905 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 65, normalized size = 0.60 \begin {gather*} \frac {\frac {21 \left (2316+13747 x+26245 x^2+16290 x^3\right )}{2 \sqrt {1-2 x} (2+3 x)^3}-905 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 66, normalized size = 0.61
method | result | size |
risch | \(\frac {16290 x^{3}+26245 x^{2}+13747 x +2316}{2058 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {905 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(51\) |
derivativedivides | \(\frac {484}{2401 \sqrt {1-2 x}}+\frac {\frac {5937 \left (1-2 x \right )^{\frac {5}{2}}}{2401}-\frac {11476 \left (1-2 x \right )^{\frac {3}{2}}}{1029}+\frac {1849 \sqrt {1-2 x}}{147}}{\left (-4-6 x \right )^{3}}-\frac {905 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(66\) |
default | \(\frac {484}{2401 \sqrt {1-2 x}}+\frac {\frac {5937 \left (1-2 x \right )^{\frac {5}{2}}}{2401}-\frac {11476 \left (1-2 x \right )^{\frac {3}{2}}}{1029}+\frac {1849 \sqrt {1-2 x}}{147}}{\left (-4-6 x \right )^{3}}-\frac {905 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(66\) |
trager | \(-\frac {\left (16290 x^{3}+26245 x^{2}+13747 x +2316\right ) \sqrt {1-2 x}}{2058 \left (2+3 x \right )^{3} \left (-1+2 x \right )}+\frac {905 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{43218}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 101, normalized size = 0.94 \begin {gather*} \frac {905}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8145 \, {\left (2 \, x - 1\right )}^{3} + 50680 \, {\left (2 \, x - 1\right )}^{2} + 208838 \, x - 33271}{1029 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 99, normalized size = 0.92 \begin {gather*} \frac {905 \, \sqrt {21} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (16290 \, x^{3} + 26245 \, x^{2} + 13747 \, x + 2316\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, 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 = 2.26, size = 93, normalized size = 0.86 \begin {gather*} \frac {905}{43218} \, \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 {484}{2401 \, \sqrt {-2 \, x + 1}} - \frac {17811 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 80332 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 90601 \, \sqrt {-2 \, x + 1}}{57624 \, {\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 = 0.07, size = 82, normalized size = 0.76 \begin {gather*} \frac {\frac {4262\,x}{567}+\frac {7240\,{\left (2\,x-1\right )}^2}{3969}+\frac {905\,{\left (2\,x-1\right )}^3}{3087}-\frac {97}{81}}{\frac {343\,\sqrt {1-2\,x}}{27}-\frac {49\,{\left (1-2\,x\right )}^{3/2}}{3}+7\,{\left (1-2\,x\right )}^{5/2}-{\left (1-2\,x\right )}^{7/2}}-\frac {905\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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