Optimal. Leaf size=100 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{84 (3 x+2)^4}+\frac{\sqrt{1-2 x} (4955 x+3168)}{10584 (3 x+2)^3}-\frac{42995 \sqrt{1-2 x}}{74088 (3 x+2)}-\frac{42995 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{37044 \sqrt{21}} \]
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Rubi [A] time = 0.0261442, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 145, 51, 63, 206} \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{84 (3 x+2)^4}+\frac{\sqrt{1-2 x} (4955 x+3168)}{10584 (3 x+2)^3}-\frac{42995 \sqrt{1-2 x}}{74088 (3 x+2)}-\frac{42995 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{37044 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 145
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^5} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^2}{84 (2+3 x)^4}-\frac{1}{84} \int \frac{(-389-685 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=\frac{\sqrt{1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}+\frac{42995 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{10584}\\ &=-\frac{42995 \sqrt{1-2 x}}{74088 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}+\frac{42995 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{74088}\\ &=-\frac{42995 \sqrt{1-2 x}}{74088 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}-\frac{42995 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{74088}\\ &=-\frac{42995 \sqrt{1-2 x}}{74088 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^2}{84 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3168+4955 x)}{10584 (2+3 x)^3}-\frac{42995 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{37044 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0224584, size = 52, normalized size = 0.52 \[ \frac{\sqrt{1-2 x} \left (\frac{1029 \left (31500 x^2+41823 x+13885\right )}{(3 x+2)^4}-1031880 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{2333772} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 66, normalized size = 0.7 \begin{align*} -324\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ( -{\frac{42995\, \left ( 1-2\,x \right ) ^{7/2}}{444528}}+{\frac{374945\, \left ( 1-2\,x \right ) ^{5/2}}{571536}}-{\frac{363407\, \left ( 1-2\,x \right ) ^{3/2}}{244944}}+{\frac{274027\,\sqrt{1-2\,x}}{244944}} \right ) }-{\frac{42995\,\sqrt{21}}{777924}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70982, size = 149, normalized size = 1.49 \begin{align*} \frac{42995}{1555848} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1160865 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 7873845 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 17806943 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13427323 \, \sqrt{-2 \, x + 1}}{37044 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6366, size = 311, normalized size = 3.11 \begin{align*} \frac{42995 \, \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (1160865 \, x^{3} + 2195625 \, x^{2} + 1385462 \, x + 291670\right )} \sqrt{-2 \, x + 1}}{1555848 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = 2.20456, size = 135, normalized size = 1.35 \begin{align*} \frac{42995}{1555848} \, \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{1160865 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 7873845 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 17806943 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 13427323 \, \sqrt{-2 \, x + 1}}{592704 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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