Optimal. Leaf size=81 \[ -\frac{130}{1029 \sqrt{1-2 x}}-\frac{365}{294 \sqrt{1-2 x} (3 x+2)}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)}+\frac{130 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{343 \sqrt{21}} \]
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Rubi [A] time = 0.0217992, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 51, 63, 206} \[ -\frac{130}{1029 \sqrt{1-2 x}}-\frac{365}{294 \sqrt{1-2 x} (3 x+2)}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)}+\frac{130 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{343 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac{1}{42} \int \frac{-15+525 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac{365}{294 \sqrt{1-2 x} (2+3 x)}-\frac{65}{147} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=-\frac{130}{1029 \sqrt{1-2 x}}+\frac{121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac{365}{294 \sqrt{1-2 x} (2+3 x)}-\frac{65}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{130}{1029 \sqrt{1-2 x}}+\frac{121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac{365}{294 \sqrt{1-2 x} (2+3 x)}+\frac{65}{343} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{130}{1029 \sqrt{1-2 x}}+\frac{121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac{365}{294 \sqrt{1-2 x} (2+3 x)}+\frac{130 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{343 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0167756, size = 55, normalized size = 0.68 \[ -\frac{-130 \left (6 x^2+x-2\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-7 (365 x+241)}{1029 (1-2 x)^{3/2} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 54, normalized size = 0.7 \begin{align*}{\frac{2}{1029}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}+{\frac{130\,\sqrt{21}}{7203}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{121}{147} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{44}{343}{\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 = 3.62382, size = 100, normalized size = 1.23 \begin{align*} -\frac{65}{7203} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (195 \,{\left (2 \, x - 1\right )}^{2} + 3465 \, x + 1232\right )}}{1029 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 7 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65208, size = 232, normalized size = 2.86 \begin{align*} \frac{65 \, \sqrt{21}{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \,{\left (780 \, x^{2} + 2685 \, x + 1427\right )} \sqrt{-2 \, x + 1}}{7203 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, 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 = 2.17235, size = 104, normalized size = 1.28 \begin{align*} -\frac{65}{7203} \, \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{11 \,{\left (24 \, x + 65\right )}}{1029 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{343 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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