Optimal. Leaf size=128 \[ \frac{347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac{(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac{8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac{8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac{8051 \sqrt{1-2 x}}{31752 (3 x+2)}+\frac{8051 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}} \]
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Rubi [A] time = 0.0385309, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 47, 63, 206} \[ \frac{347 (1-2 x)^{7/2}}{8820 (3 x+2)^4}-\frac{(1-2 x)^{7/2}}{315 (3 x+2)^5}-\frac{8051 (1-2 x)^{5/2}}{26460 (3 x+2)^3}+\frac{8051 (1-2 x)^{3/2}}{31752 (3 x+2)^2}-\frac{8051 \sqrt{1-2 x}}{31752 (3 x+2)}+\frac{8051 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 47
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^6} \, dx &=-\frac{(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac{1}{315} \int \frac{(1-2 x)^{5/2} (1403+2625 x)}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac{347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}+\frac{8051 \int \frac{(1-2 x)^{5/2}}{(2+3 x)^4} \, dx}{2940}\\ &=-\frac{(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac{347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac{8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}-\frac{8051 \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3} \, dx}{5292}\\ &=-\frac{(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac{347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac{8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac{8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}+\frac{8051 \int \frac{\sqrt{1-2 x}}{(2+3 x)^2} \, dx}{10584}\\ &=-\frac{(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac{347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac{8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac{8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac{8051 \sqrt{1-2 x}}{31752 (2+3 x)}-\frac{8051 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{31752}\\ &=-\frac{(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac{347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac{8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac{8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac{8051 \sqrt{1-2 x}}{31752 (2+3 x)}+\frac{8051 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{31752}\\ &=-\frac{(1-2 x)^{7/2}}{315 (2+3 x)^5}+\frac{347 (1-2 x)^{7/2}}{8820 (2+3 x)^4}-\frac{8051 (1-2 x)^{5/2}}{26460 (2+3 x)^3}+\frac{8051 (1-2 x)^{3/2}}{31752 (2+3 x)^2}-\frac{8051 \sqrt{1-2 x}}{31752 (2+3 x)}+\frac{8051 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15876 \sqrt{21}}\\ \end{align*}
Mathematica [A] time = 0.0684588, size = 84, normalized size = 0.66 \[ \frac{21 \left (14646690 x^5+17489565 x^4+4147953 x^3-2438512 x^2-1912794 x-503276\right )+80510 \sqrt{21-42 x} (3 x+2)^5 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3333960 \sqrt{1-2 x} (3 x+2)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.6 \begin{align*} -3888\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{54247\, \left ( 1-2\,x \right ) ^{9/2}}{2286144}}+{\frac{12269\, \left ( 1-2\,x \right ) ^{7/2}}{69984}}-{\frac{16102\, \left ( 1-2\,x \right ) ^{5/2}}{32805}}+{\frac{394499\, \left ( 1-2\,x \right ) ^{3/2}}{629856}}-{\frac{394499\,\sqrt{1-2\,x}}{1259712}} \right ) }+{\frac{8051\,\sqrt{21}}{333396}{\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 = 2.05024, size = 173, normalized size = 1.35 \begin{align*} -\frac{8051}{666792} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{7323345 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 54106290 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 151487616 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 193304510 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 96652255 \, \sqrt{-2 \, x + 1}}{79380 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30098, size = 366, normalized size = 2.86 \begin{align*} \frac{40255 \, \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (7323345 \, x^{4} + 12406455 \, x^{3} + 8277204 \, x^{2} + 2919346 \, x + 503276\right )} \sqrt{-2 \, x + 1}}{3333960 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 2.45654, size = 157, normalized size = 1.23 \begin{align*} -\frac{8051}{666792} \, \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{7323345 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 54106290 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 151487616 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 193304510 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 96652255 \, \sqrt{-2 \, x + 1}}{2540160 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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