Optimal. Leaf size=68 \[ \frac {137 \sqrt {1-2 x}}{882 (3 x+2)}-\frac {\sqrt {1-2 x}}{126 (3 x+2)^2}-\frac {257 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \]
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Rubi [A] time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {89, 78, 63, 206} \[ \frac {137 \sqrt {1-2 x}}{882 (3 x+2)}-\frac {\sqrt {1-2 x}}{126 (3 x+2)^2}-\frac {257 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx &=-\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {563+1050 x}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}+\frac {257}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}-\frac {257}{98} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}-\frac {257 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.78 \[ \frac {\frac {7 \sqrt {1-2 x} (137 x+89)}{(3 x+2)^2}-514 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2058} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 69, normalized size = 1.01 \[ \frac {257 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (137 \, x + 89\right )} \sqrt {-2 \, x + 1}}{2058 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 68, normalized size = 1.00 \[ \frac {257}{2058} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 315 \, \sqrt {-2 \, x + 1}}{588 \, {\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.71 \[ -\frac {257 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1029}+\frac {-\frac {137 \left (-2 x +1\right )^{\frac {3}{2}}}{147}+\frac {15 \sqrt {-2 x +1}}{7}}{\left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 74, normalized size = 1.09 \[ \frac {257}{2058} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 315 \, \sqrt {-2 \, x + 1}}{147 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 53, normalized size = 0.78 \[ \frac {\frac {5\,\sqrt {1-2\,x}}{21}-\frac {137\,{\left (1-2\,x\right )}^{3/2}}{1323}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {257\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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