Optimal. Leaf size=114 \[ \frac{211 (1-2 x)^{7/2}}{2646 (3 x+2)^2}-\frac{(1-2 x)^{7/2}}{189 (3 x+2)^3}-\frac{887 (1-2 x)^{5/2}}{882 (3 x+2)}-\frac{4435 (1-2 x)^{3/2}}{3969}-\frac{4435}{567} \sqrt{1-2 x}+\frac{4435 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]
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Rubi [A] time = 0.0345993, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {89, 78, 47, 50, 63, 206} \[ \frac{211 (1-2 x)^{7/2}}{2646 (3 x+2)^2}-\frac{(1-2 x)^{7/2}}{189 (3 x+2)^3}-\frac{887 (1-2 x)^{5/2}}{882 (3 x+2)}-\frac{4435 (1-2 x)^{3/2}}{3969}-\frac{4435}{567} \sqrt{1-2 x}+\frac{4435 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 47
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac{1}{189} \int \frac{(1-2 x)^{5/2} (839+1575 x)}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac{211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}+\frac{887}{294} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac{211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac{887 (1-2 x)^{5/2}}{882 (2+3 x)}-\frac{4435}{882} \int \frac{(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac{4435 (1-2 x)^{3/2}}{3969}-\frac{(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac{211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac{887 (1-2 x)^{5/2}}{882 (2+3 x)}-\frac{4435}{378} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=-\frac{4435}{567} \sqrt{1-2 x}-\frac{4435 (1-2 x)^{3/2}}{3969}-\frac{(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac{211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac{887 (1-2 x)^{5/2}}{882 (2+3 x)}-\frac{4435}{162} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{4435}{567} \sqrt{1-2 x}-\frac{4435 (1-2 x)^{3/2}}{3969}-\frac{(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac{211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac{887 (1-2 x)^{5/2}}{882 (2+3 x)}+\frac{4435}{162} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{4435}{567} \sqrt{1-2 x}-\frac{4435 (1-2 x)^{3/2}}{3969}-\frac{(1-2 x)^{7/2}}{189 (2+3 x)^3}+\frac{211 (1-2 x)^{7/2}}{2646 (2+3 x)^2}-\frac{887 (1-2 x)^{5/2}}{882 (2+3 x)}+\frac{4435 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0216508, size = 54, normalized size = 0.47 \[ \frac{(1-2 x)^{7/2} \left (343 (211 x+136)-10644 (3 x+2)^3 \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{302526 (3 x+2)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.7 \begin{align*} -{\frac{100}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1480}{243}\sqrt{1-2\,x}}-{\frac{4}{9\, \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{3091}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{31675}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{144305}{108}\sqrt{1-2\,x}} \right ) }+{\frac{4435\,\sqrt{21}}{1701}{\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 = 3.82404, size = 149, normalized size = 1.31 \begin{align*} -\frac{100}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{4435}{3402} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1480}{243} \, \sqrt{-2 \, x + 1} - \frac{27819 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 126700 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 144305 \, \sqrt{-2 \, x + 1}}{243 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40804, size = 278, normalized size = 2.44 \begin{align*} \frac{4435 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (3600 \, x^{4} - 21240 \, x^{3} - 61353 \, x^{2} - 48697 \, x - 12212\right )} \sqrt{-2 \, x + 1}}{3402 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 = 2.45585, size = 138, normalized size = 1.21 \begin{align*} -\frac{100}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{4435}{3402} \, \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{1480}{243} \, \sqrt{-2 \, x + 1} - \frac{27819 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 126700 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 144305 \, \sqrt{-2 \, x + 1}}{1944 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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