Optimal. Leaf size=113 \[ \frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3}+\frac {49 \sqrt {1-2 x}}{9 (2+3 x)^2}+\frac {1138 \sqrt {1-2 x}}{21 (2+3 x)}+\frac {78506 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-110 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 156, 162,
65, 212} \begin {gather*} \frac {1138 \sqrt {1-2 x}}{21 (3 x+2)}+\frac {49 \sqrt {1-2 x}}{9 (3 x+2)^2}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3}+\frac {78506 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-110 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {120-163 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3}+\frac {49 \sqrt {1-2 x}}{9 (2+3 x)^2}+\frac {1}{126} \int \frac {9072-10290 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3}+\frac {49 \sqrt {1-2 x}}{9 (2+3 x)^2}+\frac {1138 \sqrt {1-2 x}}{21 (2+3 x)}+\frac {1}{882} \int \frac {390222-238980 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3}+\frac {49 \sqrt {1-2 x}}{9 (2+3 x)^2}+\frac {1138 \sqrt {1-2 x}}{21 (2+3 x)}-\frac {39253}{21} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+3025 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3}+\frac {49 \sqrt {1-2 x}}{9 (2+3 x)^2}+\frac {1138 \sqrt {1-2 x}}{21 (2+3 x)}+\frac {39253}{21} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-3025 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3}+\frac {49 \sqrt {1-2 x}}{9 (2+3 x)^2}+\frac {1138 \sqrt {1-2 x}}{21 (2+3 x)}+\frac {78506 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-110 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 83, normalized size = 0.73 \begin {gather*} \frac {\sqrt {1-2 x} \left (4797+13999 x+10242 x^2\right )}{21 (2+3 x)^3}+\frac {78506 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-110 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 75, normalized size = 0.66
method | result | size |
risch | \(-\frac {20484 x^{3}+17756 x^{2}-4405 x -4797}{21 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-110 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {78506 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(69\) |
derivativedivides | \(-110 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1138 \left (1-2 x \right )^{\frac {5}{2}}}{63}-\frac {6926 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {8204 \sqrt {1-2 x}}{81}\right )}{\left (-4-6 x \right )^{3}}+\frac {78506 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(75\) |
default | \(-110 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1138 \left (1-2 x \right )^{\frac {5}{2}}}{63}-\frac {6926 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {8204 \sqrt {1-2 x}}{81}\right )}{\left (-4-6 x \right )^{3}}+\frac {78506 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(75\) |
trager | \(\frac {\left (10242 x^{2}+13999 x +4797\right ) \sqrt {1-2 x}}{21 \left (2+3 x \right )^{3}}-55 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )+\frac {39253 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{441}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 128, normalized size = 1.13 \begin {gather*} 55 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {39253}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (5121 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 24241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 28714 \, \sqrt {-2 \, x + 1}\right )}}{21 \, {\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 = 0.61, size = 130, normalized size = 1.15 \begin {gather*} \frac {24255 \, \sqrt {55} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 39253 \, \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 (10242 \, x^{2} + 13999 \, x + 4797\right )} \sqrt {-2 \, x + 1}}{441 \, {\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 = 0.78, size = 123, normalized size = 1.09 \begin {gather*} 55 \, \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 {39253}{441} \, \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 {5121 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 24241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 28714 \, \sqrt {-2 \, x + 1}}{42 \, {\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 = 1.19, size = 89, normalized size = 0.79 \begin {gather*} \frac {78506\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}-110\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {16408\,\sqrt {1-2\,x}}{81}-\frac {13852\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {2276\,{\left (1-2\,x\right )}^{5/2}}{63}}{\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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