Optimal. Leaf size=61 \[ -\frac{9}{25} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{275 (5 x+3)}-\frac{134 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]
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Rubi [A] time = 0.0140236, antiderivative size = 61, 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, 80, 63, 206} \[ -\frac{9}{25} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{275 (5 x+3)}-\frac{134 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 80
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^2} \, dx &=-\frac{\sqrt{1-2 x}}{275 (3+5 x)}+\frac{1}{275} \int \frac{364+495 x}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{9}{25} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{275 (3+5 x)}+\frac{67}{275} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{9}{25} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{275 (3+5 x)}-\frac{67}{275} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{9}{25} \sqrt{1-2 x}-\frac{\sqrt{1-2 x}}{275 (3+5 x)}-\frac{134 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0276832, size = 53, normalized size = 0.87 \[ -\frac{\sqrt{1-2 x} (495 x+298)}{275 (5 x+3)}-\frac{134 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275 \sqrt{55}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 45, normalized size = 0.7 \begin{align*} -{\frac{9}{25}\sqrt{1-2\,x}}+{\frac{2}{1375}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{134\,\sqrt{55}}{15125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49958, size = 84, normalized size = 1.38 \begin{align*} \frac{67}{15125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9}{25} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{275 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61941, size = 173, normalized size = 2.84 \begin{align*} \frac{67 \, \sqrt{55}{\left (5 \, x + 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (495 \, x + 298\right )} \sqrt{-2 \, x + 1}}{15125 \,{\left (5 \, x + 3\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 = 1.75371, size = 88, normalized size = 1.44 \begin{align*} \frac{67}{15125} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{9}{25} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{275 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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