Optimal. Leaf size=69 \[ \frac{5}{6} (1-2 x)^{5/2}-\frac{155}{54} (1-2 x)^{3/2}+\frac{2}{27} \sqrt{1-2 x}-\frac{2}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0244974, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ \frac{5}{6} (1-2 x)^{5/2}-\frac{155}{54} (1-2 x)^{3/2}+\frac{2}{27} \sqrt{1-2 x}-\frac{2}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 88
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^2}{2+3 x} \, dx &=\int \left (\frac{155}{18} \sqrt{1-2 x}-\frac{25}{6} (1-2 x)^{3/2}+\frac{\sqrt{1-2 x}}{9 (2+3 x)}\right ) \, dx\\ &=-\frac{155}{54} (1-2 x)^{3/2}+\frac{5}{6} (1-2 x)^{5/2}+\frac{1}{9} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{2}{27} \sqrt{1-2 x}-\frac{155}{54} (1-2 x)^{3/2}+\frac{5}{6} (1-2 x)^{5/2}+\frac{7}{27} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{2}{27} \sqrt{1-2 x}-\frac{155}{54} (1-2 x)^{3/2}+\frac{5}{6} (1-2 x)^{5/2}-\frac{7}{27} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{2}{27} \sqrt{1-2 x}-\frac{155}{54} (1-2 x)^{3/2}+\frac{5}{6} (1-2 x)^{5/2}-\frac{2}{27} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0293299, size = 51, normalized size = 0.74 \[ \frac{1}{81} \left (3 \sqrt{1-2 x} \left (90 x^2+65 x-53\right )-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 47, normalized size = 0.7 \begin{align*} -{\frac{155}{54} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{5}{6} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2\,\sqrt{21}}{81}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{2}{27}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60818, size = 86, normalized size = 1.25 \begin{align*} \frac{5}{6} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{155}{54} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{81} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{27} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57456, size = 166, normalized size = 2.41 \begin{align*} \frac{1}{81} \, \sqrt{7} \sqrt{3} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + \frac{1}{27} \,{\left (90 \, x^{2} + 65 \, x - 53\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.55769, size = 102, normalized size = 1.48 \begin{align*} \frac{5 \left (1 - 2 x\right )^{\frac{5}{2}}}{6} - \frac{155 \left (1 - 2 x\right )^{\frac{3}{2}}}{54} + \frac{2 \sqrt{1 - 2 x}}{27} + \frac{14 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.69974, size = 100, normalized size = 1.45 \begin{align*} \frac{5}{6} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{155}{54} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{81} \, \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{2}{27} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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