Optimal. Leaf size=73 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{21 (3 x+2)}-\frac{10}{189} \sqrt{1-2 x} (95 x+214)-\frac{208 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}} \]
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Rubi [A] time = 0.0175322, antiderivative size = 73, normalized size of antiderivative = 1., 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 = {98, 147, 63, 206} \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{21 (3 x+2)}-\frac{10}{189} \sqrt{1-2 x} (95 x+214)-\frac{208 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^2} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac{1}{21} \int \frac{(-92-190 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{\sqrt{1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac{10}{189} \sqrt{1-2 x} (214+95 x)+\frac{104}{189} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{\sqrt{1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac{10}{189} \sqrt{1-2 x} (214+95 x)-\frac{104}{189} \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} (3+5 x)^2}{21 (2+3 x)}-\frac{10}{189} \sqrt{1-2 x} (214+95 x)-\frac{208 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{189 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0303162, size = 58, normalized size = 0.79 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (2625 x^2+8050 x+4199\right )}{3 x+2}-208 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 54, normalized size = 0.7 \begin{align*}{\frac{125}{54} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{725}{54}\sqrt{1-2\,x}}-{\frac{2}{567}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{208\,\sqrt{21}}{3969}{\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 = 2.70011, size = 96, normalized size = 1.32 \begin{align*} \frac{125}{54} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{104}{3969} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{725}{54} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{189 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65918, size = 190, normalized size = 2.6 \begin{align*} \frac{104 \, \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 (2625 \, x^{2} + 8050 \, x + 4199\right )} \sqrt{-2 \, x + 1}}{3969 \,{\left (3 \, 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 = 1.24405, size = 100, normalized size = 1.37 \begin{align*} \frac{125}{54} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{104}{3969} \, \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{725}{54} \, \sqrt{-2 \, x + 1} + \frac{\sqrt{-2 \, x + 1}}{189 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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