Optimal. Leaf size=89 \[ \frac{(1-2 x)^{7/2}}{21 (3 x+2)}+\frac{16}{63} (1-2 x)^{5/2}+\frac{80}{81} (1-2 x)^{3/2}+\frac{560}{81} \sqrt{1-2 x}-\frac{560}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0230484, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 50, 63, 206} \[ \frac{(1-2 x)^{7/2}}{21 (3 x+2)}+\frac{16}{63} (1-2 x)^{5/2}+\frac{80}{81} (1-2 x)^{3/2}+\frac{560}{81} \sqrt{1-2 x}-\frac{560}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^2} \, dx &=\frac{(1-2 x)^{7/2}}{21 (2+3 x)}+\frac{40}{21} \int \frac{(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=\frac{16}{63} (1-2 x)^{5/2}+\frac{(1-2 x)^{7/2}}{21 (2+3 x)}+\frac{40}{9} \int \frac{(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac{80}{81} (1-2 x)^{3/2}+\frac{16}{63} (1-2 x)^{5/2}+\frac{(1-2 x)^{7/2}}{21 (2+3 x)}+\frac{280}{27} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{560}{81} \sqrt{1-2 x}+\frac{80}{81} (1-2 x)^{3/2}+\frac{16}{63} (1-2 x)^{5/2}+\frac{(1-2 x)^{7/2}}{21 (2+3 x)}+\frac{1960}{81} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{560}{81} \sqrt{1-2 x}+\frac{80}{81} (1-2 x)^{3/2}+\frac{16}{63} (1-2 x)^{5/2}+\frac{(1-2 x)^{7/2}}{21 (2+3 x)}-\frac{1960}{81} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{560}{81} \sqrt{1-2 x}+\frac{80}{81} (1-2 x)^{3/2}+\frac{16}{63} (1-2 x)^{5/2}+\frac{(1-2 x)^{7/2}}{21 (2+3 x)}-\frac{560}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0333211, size = 63, normalized size = 0.71 \[ \frac{1}{243} \left (\frac{3 \sqrt{1-2 x} \left (216 x^3-516 x^2+1474 x+1325\right )}{3 x+2}-560 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 63, normalized size = 0.7 \begin{align*}{\frac{2}{9} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{74}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{182}{27}\sqrt{1-2\,x}}-{\frac{98}{243}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{560\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.10995, size = 108, normalized size = 1.21 \begin{align*} \frac{2}{9} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{74}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{280}{243} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{182}{27} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47486, size = 219, normalized size = 2.46 \begin{align*} \frac{280 \, \sqrt{7} \sqrt{3}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \,{\left (216 \, x^{3} - 516 \, x^{2} + 1474 \, x + 1325\right )} \sqrt{-2 \, x + 1}}{243 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4392, size = 122, normalized size = 1.37 \begin{align*} \frac{2}{9} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{74}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{280}{243} \, \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{182}{27} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{81 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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