Optimal. Leaf size=154 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{5 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}-\frac{653 \sqrt{1-2 x} (5 x+3)^2}{2520 (3 x+2)^4}-\frac{\sqrt{1-2 x} (664915 x+413424)}{317520 (3 x+2)^3}-\frac{15313 \sqrt{1-2 x}}{444528 (3 x+2)}-\frac{15313 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]
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Rubi [A] time = 0.0498541, antiderivative size = 154, 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 = {97, 149, 145, 51, 63, 206} \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{5 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}-\frac{653 \sqrt{1-2 x} (5 x+3)^2}{2520 (3 x+2)^4}-\frac{\sqrt{1-2 x} (664915 x+413424)}{317520 (3 x+2)^3}-\frac{15313 \sqrt{1-2 x}}{444528 (3 x+2)}-\frac{15313 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 145
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{1}{18} \int \frac{(6-45 x) \sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac{1}{270} \int \frac{(3+5 x)^2 (-1179+1170 x)}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=-\frac{653 \sqrt{1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac{\int \frac{(3+5 x) (-83907+75645 x)}{\sqrt{1-2 x} (2+3 x)^4} \, dx}{22680}\\ &=-\frac{653 \sqrt{1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac{\sqrt{1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}+\frac{15313 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{63504}\\ &=-\frac{15313 \sqrt{1-2 x}}{444528 (2+3 x)}-\frac{653 \sqrt{1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac{\sqrt{1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}+\frac{15313 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{444528}\\ &=-\frac{15313 \sqrt{1-2 x}}{444528 (2+3 x)}-\frac{653 \sqrt{1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac{\sqrt{1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}-\frac{15313 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{444528}\\ &=-\frac{15313 \sqrt{1-2 x}}{444528 (2+3 x)}-\frac{653 \sqrt{1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac{(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac{2 \sqrt{1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac{\sqrt{1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}-\frac{15313 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.034196, size = 52, normalized size = 0.34 \[ \frac{(1-2 x)^{5/2} \left (\frac{16807 \left (26250 x^2+34911 x+11609\right )}{(3 x+2)^6}-490016 \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{31765230} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 84, normalized size = 0.6 \begin{align*} -11664\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{6}} \left ( -{\frac{15313\, \left ( 1-2\,x \right ) ^{11/2}}{10668672}}-{\frac{3037\, \left ( 1-2\,x \right ) ^{9/2}}{41150592}}+{\frac{256271\, \left ( 1-2\,x \right ) ^{7/2}}{4898880}}-{\frac{923549\, \left ( 1-2\,x \right ) ^{5/2}}{4898880}}+{\frac{1822247\, \left ( 1-2\,x \right ) ^{3/2}}{7558272}}-{\frac{750337\,\sqrt{1-2\,x}}{7558272}} \right ) }-{\frac{15313\,\sqrt{21}}{4667544}{\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 = 1.56945, size = 197, normalized size = 1.28 \begin{align*} \frac{15313}{9335088} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{18605295 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 956655 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 678093066 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 2443710654 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 3125153605 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1286827955 \, \sqrt{-2 \, x + 1}}{1111320 \,{\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 = 1.34077, size = 429, normalized size = 2.79 \begin{align*} \frac{76565 \, \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 (18605295 \, x^{5} - 46991565 \, x^{4} - 122053374 \, x^{3} - 75153042 \, x^{2} - 10947400 \, x + 1660816\right )} \sqrt{-2 \, x + 1}}{46675440 \,{\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 = 1.3711, size = 178, normalized size = 1.16 \begin{align*} \frac{15313}{9335088} \, \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{18605295 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - 956655 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 678093066 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 2443710654 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 3125153605 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1286827955 \, \sqrt{-2 \, x + 1}}{71124480 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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