Optimal. Leaf size=61 \[ -\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (3 x+2)}+\frac {46 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
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Rubi [A] time = 0.01, antiderivative size = 61, 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, 80, 63, 206} \[ -\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (3 x+2)}+\frac {46 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx &=-\frac {\sqrt {1-2 x}}{63 (2+3 x)}+\frac {1}{63} \int \frac {281+525 x}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (2+3 x)}-\frac {23}{21} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (2+3 x)}+\frac {23}{21} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {25}{9} \sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{63 (2+3 x)}+\frac {46 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 51, normalized size = 0.84 \[ \frac {46 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-\frac {\sqrt {1-2 x} (175 x+117)}{63 x+42} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 60, normalized size = 0.98 \[ \frac {23 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (175 \, x + 117\right )} \sqrt {-2 \, x + 1}}{441 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.84, size = 65, normalized size = 1.07 \[ -\frac {23}{441} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {25}{9} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{63 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.74 \[ \frac {46 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{441}-\frac {25 \sqrt {-2 x +1}}{9}+\frac {2 \sqrt {-2 x +1}}{189 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 62, normalized size = 1.02 \[ -\frac {23}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {25}{9} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{63 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 44, normalized size = 0.72 \[ \frac {46\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}-\frac {2\,\sqrt {1-2\,x}}{189\,\left (2\,x+\frac {4}{3}\right )}-\frac {25\,\sqrt {1-2\,x}}{9} \]
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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