Optimal. Leaf size=61 \[ \frac{64}{147 \sqrt{1-2 x}}+\frac{1}{21 \sqrt{1-2 x} (3 x+2)}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]
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Rubi [A] time = 0.0130735, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ \frac{64}{147 \sqrt{1-2 x}}+\frac{1}{21 \sqrt{1-2 x} (3 x+2)}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac{1}{21 \sqrt{1-2 x} (2+3 x)}+\frac{32}{21} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac{64}{147 \sqrt{1-2 x}}+\frac{1}{21 \sqrt{1-2 x} (2+3 x)}+\frac{32}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{64}{147 \sqrt{1-2 x}}+\frac{1}{21 \sqrt{1-2 x} (2+3 x)}-\frac{32}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{64}{147 \sqrt{1-2 x}}+\frac{1}{21 \sqrt{1-2 x} (2+3 x)}-\frac{64 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0135888, size = 46, normalized size = 0.75 \[ \frac{64 (3 x+2) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+7}{147 \sqrt{1-2 x} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 45, normalized size = 0.7 \begin{align*} -{\frac{2}{147}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{64\,\sqrt{21}}{1029}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{22}{49}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 4.58474, size = 88, normalized size = 1.44 \begin{align*} \frac{32}{1029} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (64 \, x + 45\right )}}{49 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68333, size = 185, normalized size = 3.03 \begin{align*} \frac{32 \, \sqrt{21}{\left (6 \, x^{2} + x - 2\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (64 \, x + 45\right )} \sqrt{-2 \, x + 1}}{1029 \,{\left (6 \, x^{2} + x - 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.31826, size = 92, normalized size = 1.51 \begin{align*} \frac{32}{1029} \, \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 \,{\left (64 \, x + 45\right )}}{49 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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