Optimal. Leaf size=54 \[ \frac{25}{18} (1-2 x)^{3/2}-\frac{155}{18} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]
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Rubi [A] time = 0.019101, 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 = {88, 63, 206} \[ \frac{25}{18} (1-2 x)^{3/2}-\frac{155}{18} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)} \, dx &=\int \left (\frac{155}{18 \sqrt{1-2 x}}-\frac{25}{6} \sqrt{1-2 x}+\frac{1}{9 \sqrt{1-2 x} (2+3 x)}\right ) \, dx\\ &=-\frac{155}{18} \sqrt{1-2 x}+\frac{25}{18} (1-2 x)^{3/2}+\frac{1}{9} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{155}{18} \sqrt{1-2 x}+\frac{25}{18} (1-2 x)^{3/2}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{155}{18} \sqrt{1-2 x}+\frac{25}{18} (1-2 x)^{3/2}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0278732, size = 46, normalized size = 0.85 \[ -\frac{5}{9} \sqrt{1-2 x} (5 x+13)-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 38, normalized size = 0.7 \begin{align*}{\frac{25}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2\,\sqrt{21}}{189}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{155}{18}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62914, size = 74, normalized size = 1.37 \begin{align*} \frac{25}{18} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{189} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{155}{18} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77962, size = 135, normalized size = 2.5 \begin{align*} -\frac{5}{9} \,{\left (5 \, x + 13\right )} \sqrt{-2 \, x + 1} + \frac{1}{189} \, \sqrt{21} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.3985, size = 90, normalized size = 1.67 \begin{align*} \frac{25 \left (1 - 2 x\right )^{\frac{3}{2}}}{18} - \frac{155 \sqrt{1 - 2 x}}{18} + \frac{2 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21}}{3 \sqrt{1 - 2 x}} \right )}}{21} & \text{for}\: \frac{1}{1 - 2 x} > \frac{3}{7} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21}}{3 \sqrt{1 - 2 x}} \right )}}{21} & \text{for}\: \frac{1}{1 - 2 x} < \frac{3}{7} \end{cases}\right )}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.02442, size = 78, normalized size = 1.44 \begin{align*} \frac{25}{18} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{189} \, \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{155}{18} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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