Optimal. Leaf size=141 \[ -\frac{(1-2 x)^{5/2} (3 x+2)^4}{5 (5 x+3)}+\frac{39}{275} (1-2 x)^{5/2} (3 x+2)^3-\frac{32 (1-2 x)^{5/2} (3 x+2)^2}{4125}+\frac{254 (1-2 x)^{3/2}}{46875}-\frac{(1-2 x)^{5/2} (1110975 x+1347116)}{3609375}+\frac{2794 \sqrt{1-2 x}}{78125}-\frac{2794 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]
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Rubi [A] time = 0.0519603, antiderivative size = 141, 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 = {97, 153, 147, 50, 63, 206} \[ -\frac{(1-2 x)^{5/2} (3 x+2)^4}{5 (5 x+3)}+\frac{39}{275} (1-2 x)^{5/2} (3 x+2)^3-\frac{32 (1-2 x)^{5/2} (3 x+2)^2}{4125}+\frac{254 (1-2 x)^{3/2}}{46875}-\frac{(1-2 x)^{5/2} (1110975 x+1347116)}{3609375}+\frac{2794 \sqrt{1-2 x}}{78125}-\frac{2794 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^2} \, dx &=-\frac{(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}+\frac{1}{5} \int \frac{(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{3+5 x} \, dx\\ &=\frac{39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac{1}{275} \int \frac{(-337-96 x) (1-2 x)^{3/2} (2+3 x)^2}{3+5 x} \, dx\\ &=-\frac{32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac{39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}+\frac{\int \frac{(1-2 x)^{3/2} (2+3 x) (29178+44439 x)}{3+5 x} \, dx}{12375}\\ &=-\frac{32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac{39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac{127 \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx}{3125}\\ &=\frac{254 (1-2 x)^{3/2}}{46875}-\frac{32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac{39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac{1397 \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx}{15625}\\ &=\frac{2794 \sqrt{1-2 x}}{78125}+\frac{254 (1-2 x)^{3/2}}{46875}-\frac{32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac{39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}+\frac{15367 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{78125}\\ &=\frac{2794 \sqrt{1-2 x}}{78125}+\frac{254 (1-2 x)^{3/2}}{46875}-\frac{32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac{39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}-\frac{15367 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{78125}\\ &=\frac{2794 \sqrt{1-2 x}}{78125}+\frac{254 (1-2 x)^{3/2}}{46875}-\frac{32 (1-2 x)^{5/2} (2+3 x)^2}{4125}+\frac{39}{275} (1-2 x)^{5/2} (2+3 x)^3-\frac{(1-2 x)^{5/2} (2+3 x)^4}{5 (3+5 x)}-\frac{(1-2 x)^{5/2} (1347116+1110975 x)}{3609375}-\frac{2794 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125}\\ \end{align*}
Mathematica [A] time = 0.0798603, size = 78, normalized size = 0.55 \[ \frac{\frac{5 \sqrt{1-2 x} \left (212625000 x^6+237037500 x^5-173598750 x^4-214071975 x^3+85482115 x^2+50081215 x-15982128\right )}{5 x+3}-645414 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{90234375} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 90, normalized size = 0.6 \begin{align*} -{\frac{81}{1100} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}+{\frac{111}{250} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{12393}{17500} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{24}{15625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{52}{9375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2816}{78125}\sqrt{1-2\,x}}+{\frac{242}{390625}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{2794\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.99443, size = 144, normalized size = 1.02 \begin{align*} -\frac{81}{1100} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{111}{250} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{12393}{17500} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{24}{15625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{52}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1397}{390625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2816}{78125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{78125 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39761, size = 323, normalized size = 2.29 \begin{align*} \frac{322707 \, \sqrt{11} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \,{\left (212625000 \, x^{6} + 237037500 \, x^{5} - 173598750 \, x^{4} - 214071975 \, x^{3} + 85482115 \, x^{2} + 50081215 \, x - 15982128\right )} \sqrt{-2 \, x + 1}}{90234375 \,{\left (5 \, x + 3\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 = 1.99159, size = 186, normalized size = 1.32 \begin{align*} \frac{81}{1100} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{111}{250} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{12393}{17500} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{24}{15625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{52}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1397}{390625} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2816}{78125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{78125 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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