Optimal. Leaf size=148 \[ \frac{83 (1-2 x)^{7/2}}{2646 (3 x+2)^5}-\frac{(1-2 x)^{7/2}}{378 (3 x+2)^6}-\frac{263 (1-2 x)^{5/2}}{1176 (3 x+2)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (3 x+2)^3}+\frac{1315 \sqrt{1-2 x}}{148176 (3 x+2)}-\frac{1315 \sqrt{1-2 x}}{21168 (3 x+2)^2}+\frac{1315 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{74088 \sqrt{21}} \]
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Rubi [A] time = 0.0473317, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {89, 78, 47, 51, 63, 206} \[ \frac{83 (1-2 x)^{7/2}}{2646 (3 x+2)^5}-\frac{(1-2 x)^{7/2}}{378 (3 x+2)^6}-\frac{263 (1-2 x)^{5/2}}{1176 (3 x+2)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (3 x+2)^3}+\frac{1315 \sqrt{1-2 x}}{148176 (3 x+2)}-\frac{1315 \sqrt{1-2 x}}{21168 (3 x+2)^2}+\frac{1315 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{74088 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 47
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx &=-\frac{(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac{1}{378} \int \frac{(1-2 x)^{5/2} (1685+3150 x)}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac{83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}+\frac{263}{98} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac{83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac{263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}-\frac{1315 \int \frac{(1-2 x)^{3/2}}{(2+3 x)^4} \, dx}{1176}\\ &=-\frac{(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac{83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac{263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}+\frac{1315 \int \frac{\sqrt{1-2 x}}{(2+3 x)^3} \, dx}{3528}\\ &=-\frac{(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac{83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac{263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac{1315 \sqrt{1-2 x}}{21168 (2+3 x)^2}-\frac{1315 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{21168}\\ &=-\frac{(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac{83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac{263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac{1315 \sqrt{1-2 x}}{21168 (2+3 x)^2}+\frac{1315 \sqrt{1-2 x}}{148176 (2+3 x)}-\frac{1315 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{148176}\\ &=-\frac{(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac{83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac{263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac{1315 \sqrt{1-2 x}}{21168 (2+3 x)^2}+\frac{1315 \sqrt{1-2 x}}{148176 (2+3 x)}+\frac{1315 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{148176}\\ &=-\frac{(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac{83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac{263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac{1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac{1315 \sqrt{1-2 x}}{21168 (2+3 x)^2}+\frac{1315 \sqrt{1-2 x}}{148176 (2+3 x)}+\frac{1315 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{74088 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0297792, size = 47, normalized size = 0.32 \[ \frac{(1-2 x)^{7/2} \left (\frac{352947 (83 x+53)}{(3 x+2)^6}-227232 \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{311299254} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 84, normalized size = 0.6 \begin{align*} 23328\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{6}} \left ( -{\frac{1315\, \left ( 1-2\,x \right ) ^{11/2}}{7112448}}-{\frac{112405\, \left ( 1-2\,x \right ) ^{9/2}}{82301184}}+{\frac{8345\, \left ( 1-2\,x \right ) ^{7/2}}{653184}}-{\frac{2893\, \left ( 1-2\,x \right ) ^{5/2}}{93312}}+{\frac{156485\, \left ( 1-2\,x \right ) ^{3/2}}{5038848}}-{\frac{64435\,\sqrt{1-2\,x}}{5038848}} \right ) }+{\frac{1315\,\sqrt{21}}{1555848}{\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 = 3.87044, size = 197, normalized size = 1.33 \begin{align*} -\frac{1315}{3111696} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{319545 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 2360505 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 22080870 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 53584146 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 53674355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 22101205 \, \sqrt{-2 \, x + 1}}{74088 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29073, size = 412, normalized size = 2.78 \begin{align*} \frac{1315 \, \sqrt{21}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (319545 \, x^{5} - 1979115 \, x^{4} - 2360850 \, x^{3} - 587502 \, x^{2} - 106808 \, x - 81568\right )} \sqrt{-2 \, x + 1}}{3111696 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.61753, size = 178, normalized size = 1.2 \begin{align*} -\frac{1315}{3111696} \, \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{319545 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - 2360505 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 22080870 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 53584146 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 53674355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 22101205 \, \sqrt{-2 \, x + 1}}{4741632 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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