Optimal. Leaf size=133 \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{5 (5 x+3)}+\frac{11}{75} \sqrt{1-2 x} (3 x+2)^4+\frac{64 \sqrt{1-2 x} (3 x+2)^3}{2625}-\frac{172 \sqrt{1-2 x} (3 x+2)^2}{3125}-\frac{4 \sqrt{1-2 x} (3625 x+10998)}{15625}-\frac{328 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}} \]
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Rubi [A] time = 0.0479381, antiderivative size = 133, 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 = {97, 153, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^5}{5 (5 x+3)}+\frac{11}{75} \sqrt{1-2 x} (3 x+2)^4+\frac{64 \sqrt{1-2 x} (3 x+2)^3}{2625}-\frac{172 \sqrt{1-2 x} (3 x+2)^2}{3125}-\frac{4 \sqrt{1-2 x} (3625 x+10998)}{15625}-\frac{328 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^5}{5 (3+5 x)}+\frac{1}{5} \int \frac{(13-33 x) (2+3 x)^4}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{11}{75} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac{1}{225} \int \frac{(2+3 x)^3 (-180+192 x)}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{64 \sqrt{1-2 x} (2+3 x)^3}{2625}+\frac{11}{75} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{5 (3+5 x)}+\frac{\int \frac{(2+3 x)^2 (8568+10836 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{7875}\\ &=-\frac{172 \sqrt{1-2 x} (2+3 x)^2}{3125}+\frac{64 \sqrt{1-2 x} (2+3 x)^3}{2625}+\frac{11}{75} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac{\int \frac{(-558432-913500 x) (2+3 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{196875}\\ &=-\frac{172 \sqrt{1-2 x} (2+3 x)^2}{3125}+\frac{64 \sqrt{1-2 x} (2+3 x)^3}{2625}+\frac{11}{75} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac{4 \sqrt{1-2 x} (10998+3625 x)}{15625}+\frac{164 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{15625}\\ &=-\frac{172 \sqrt{1-2 x} (2+3 x)^2}{3125}+\frac{64 \sqrt{1-2 x} (2+3 x)^3}{2625}+\frac{11}{75} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac{4 \sqrt{1-2 x} (10998+3625 x)}{15625}-\frac{164 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{15625}\\ &=-\frac{172 \sqrt{1-2 x} (2+3 x)^2}{3125}+\frac{64 \sqrt{1-2 x} (2+3 x)^3}{2625}+\frac{11}{75} \sqrt{1-2 x} (2+3 x)^4-\frac{\sqrt{1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac{4 \sqrt{1-2 x} (10998+3625 x)}{15625}-\frac{328 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0692084, size = 73, normalized size = 0.55 \[ \frac{\frac{55 \sqrt{1-2 x} \left (1181250 x^5+3864375 x^4+4760100 x^3+2225760 x^2-1133340 x-862072\right )}{5 x+3}-2296 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6015625} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 81, normalized size = 0.6 \begin{align*}{\frac{27}{200} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{8829}{7000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{107109}{25000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{144681}{25000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{6}{3125}\sqrt{1-2\,x}}+{\frac{2}{78125}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{328\,\sqrt{55}}{859375}{\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 = 2.51594, size = 132, normalized size = 0.99 \begin{align*} \frac{27}{200} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{8829}{7000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{107109}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{144681}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{164}{859375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{6}{3125} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59659, size = 263, normalized size = 1.98 \begin{align*} \frac{1148 \, \sqrt{55}{\left (5 \, x + 3\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (1181250 \, x^{5} + 3864375 \, x^{4} + 4760100 \, x^{3} + 2225760 \, x^{2} - 1133340 \, x - 862072\right )} \sqrt{-2 \, x + 1}}{6015625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 126.374, size = 226, normalized size = 1.7 \begin{align*} \frac{27 \left (1 - 2 x\right )^{\frac{9}{2}}}{200} - \frac{8829 \left (1 - 2 x\right )^{\frac{7}{2}}}{7000} + \frac{107109 \left (1 - 2 x\right )^{\frac{5}{2}}}{25000} - \frac{144681 \left (1 - 2 x\right )^{\frac{3}{2}}}{25000} + \frac{6 \sqrt{1 - 2 x}}{3125} - \frac{44 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{15625} + \frac{326 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right )}{15625} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.69322, size = 165, normalized size = 1.24 \begin{align*} \frac{27}{200} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{8829}{7000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{107109}{25000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{144681}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{164}{859375} \, \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{6}{3125} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{15625 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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