Optimal. Leaf size=54 \[ -\frac{217}{242 \sqrt{1-2 x}}+\frac{49}{66 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]
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Rubi [A] time = 0.0237088, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {87, 63, 206} \[ -\frac{217}{242 \sqrt{1-2 x}}+\frac{49}{66 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 87
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\int \left (\frac{49}{22 (1-2 x)^{5/2}}-\frac{217}{242 (1-2 x)^{3/2}}+\frac{1}{121 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac{49}{66 (1-2 x)^{3/2}}-\frac{217}{242 \sqrt{1-2 x}}+\frac{1}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{49}{66 (1-2 x)^{3/2}}-\frac{217}{242 \sqrt{1-2 x}}-\frac{1}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{49}{66 (1-2 x)^{3/2}}-\frac{217}{242 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{121 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0192421, size = 40, normalized size = 0.74 \[ \frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )+33 (45 x-4)}{825 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 38, normalized size = 0.7 \begin{align*}{\frac{49}{66} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{6655}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{217}{242}{\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 = 1.51611, size = 69, normalized size = 1.28 \begin{align*} \frac{1}{6655} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{7 \,{\left (93 \, x - 8\right )}}{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 [A] time = 1.25351, size = 190, normalized size = 3.52 \begin{align*} \frac{3 \, \sqrt{55}{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 385 \,{\left (93 \, x - 8\right )} \sqrt{-2 \, x + 1}}{19965 \,{\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 = 23.2063, size = 90, normalized size = 1.67 \begin{align*} \frac{2 \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{217}{242 \sqrt{1 - 2 x}} + \frac{49}{66 \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 = 1.34021, size = 82, normalized size = 1.52 \begin{align*} \frac{1}{6655} \, \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{7 \,{\left (93 \, x - 8\right )}}{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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