Optimal. Leaf size=80 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{5 \sqrt{1-2 x} (857 x+541)}{2058 (3 x+2)^2}+\frac{2245 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]
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Rubi [A] time = 0.0193927, antiderivative size = 80, 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 = {98, 145, 63, 206} \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{5 \sqrt{1-2 x} (857 x+541)}{2058 (3 x+2)^2}+\frac{2245 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 145
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{1}{7} \int \frac{(3+5 x) (25+5 x)}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{5 \sqrt{1-2 x} (541+857 x)}{2058 (2+3 x)^2}-\frac{2245 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2058}\\ &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{5 \sqrt{1-2 x} (541+857 x)}{2058 (2+3 x)^2}+\frac{2245 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2058}\\ &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{5 \sqrt{1-2 x} (541+857 x)}{2058 (2+3 x)^2}+\frac{2245 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0182952, size = 59, normalized size = 0.74 \[ \frac{7 \left (36750 x^2+48795 x+16199\right )-4490 (3 x+2)^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )}{6174 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 57, normalized size = 0.7 \begin{align*} -{\frac{18}{343\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{203}{54} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{469}{54}\sqrt{1-2\,x}} \right ) }+{\frac{2245\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{1331}{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 = 1.56263, size = 112, normalized size = 1.4 \begin{align*} -\frac{2245}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (18070 \,{\left (2 \, x - 1\right )}^{2} + 168175 \, x + 13741\right )}}{1029 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60925, size = 246, normalized size = 3.08 \begin{align*} \frac{2245 \, \sqrt{21}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (72280 \, x^{2} + 95895 \, x + 31811\right )} \sqrt{-2 \, x + 1}}{43218 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 = 2.50053, size = 104, normalized size = 1.3 \begin{align*} -\frac{2245}{43218} \, \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{1331}{343 \, \sqrt{-2 \, x + 1}} + \frac{29 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 67 \, \sqrt{-2 \, x + 1}}{588 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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