Optimal. Leaf size=56 \[ \frac{2}{121 \sqrt{1-2 x}}+\frac{7}{33 (1-2 x)^{3/2}}-\frac{2}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0137696, antiderivative size = 56, 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} \[ \frac{2}{121 \sqrt{1-2 x}}+\frac{7}{33 (1-2 x)^{3/2}}-\frac{2}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\frac{7}{33 (1-2 x)^{3/2}}+\frac{1}{11} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac{7}{33 (1-2 x)^{3/2}}+\frac{2}{121 \sqrt{1-2 x}}+\frac{5}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7}{33 (1-2 x)^{3/2}}+\frac{2}{121 \sqrt{1-2 x}}-\frac{5}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7}{33 (1-2 x)^{3/2}}+\frac{2}{121 \sqrt{1-2 x}}-\frac{2}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [C] time = 0.0130728, size = 38, normalized size = 0.68 \[ \frac{(6-12 x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+77}{363 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 38, normalized size = 0.7 \begin{align*}{\frac{7}{33} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{121}{\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 = 1.78161, size = 69, normalized size = 1.23 \begin{align*} \frac{1}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{12 \, x - 83}{363 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.39125, size = 211, normalized size = 3.77 \begin{align*} \frac{3 \, \sqrt{11} \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \,{\left (12 \, x - 83\right )} \sqrt{-2 \, x + 1}}{3993 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.3294, size = 90, normalized size = 1.61 \begin{align*} \frac{10 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right )}{121} + \frac{2}{121 \sqrt{1 - 2 x}} + \frac{7}{33 \left (1 - 2 x\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.51851, size = 82, normalized size = 1.46 \begin{align*} \frac{1}{1331} \, \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{12 \, x - 83}{363 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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