Optimal. Leaf size=68 \[ -\frac{1091 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)}+\frac{134 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]
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Rubi [A] time = 0.0165094, antiderivative size = 68, 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 = {89, 78, 63, 206} \[ -\frac{1091 \sqrt{1-2 x}}{294 (3 x+2)}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)}+\frac{134 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)}-\frac{1}{14} \int \frac{-247+175 x}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)}-\frac{1091 \sqrt{1-2 x}}{294 (2+3 x)}-\frac{67}{147} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)}-\frac{1091 \sqrt{1-2 x}}{294 (2+3 x)}+\frac{67}{147} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)}-\frac{1091 \sqrt{1-2 x}}{294 (2+3 x)}+\frac{134 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0313163, size = 62, normalized size = 0.91 \[ \frac{21 (1091 x+725)+134 \sqrt{21-42 x} (3 x+2) \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3087 \sqrt{1-2 x} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 45, normalized size = 0.7 \begin{align*}{\frac{2}{441}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}+{\frac{134\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{121}{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 = 2.92635, size = 88, normalized size = 1.29 \begin{align*} -\frac{67}{3087} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1091 \, x + 725\right )}}{147 \,{\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 = 1.58924, size = 189, normalized size = 2.78 \begin{align*} \frac{67 \, \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 (1091 \, x + 725\right )} \sqrt{-2 \, x + 1}}{3087 \,{\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 = 1.64795, size = 92, normalized size = 1.35 \begin{align*} -\frac{67}{3087} \, \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 (1091 \, x + 725\right )}}{147 \,{\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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