Optimal. Leaf size=120 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{10 (5 x+3)^2}-\frac{131 \sqrt{1-2 x} (3 x+2)^3}{550 (5 x+3)}+\frac{1428 \sqrt{1-2 x} (3 x+2)^2}{6875}-\frac{21 (704-375 x) \sqrt{1-2 x}}{68750}-\frac{12803 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]
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Rubi [A] time = 0.037464, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 149, 153, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{10 (5 x+3)^2}-\frac{131 \sqrt{1-2 x} (3 x+2)^3}{550 (5 x+3)}+\frac{1428 \sqrt{1-2 x} (3 x+2)^2}{6875}-\frac{21 (704-375 x) \sqrt{1-2 x}}{68750}-\frac{12803 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 153
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)^4}{(3+5 x)^3} \, dx &=-\frac{\sqrt{1-2 x} (2+3 x)^4}{10 (3+5 x)^2}+\frac{1}{10} \int \frac{(10-27 x) (2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x} (2+3 x)^4}{10 (3+5 x)^2}-\frac{131 \sqrt{1-2 x} (2+3 x)^3}{550 (3+5 x)}+\frac{1}{550} \int \frac{(847-2856 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{1428 \sqrt{1-2 x} (2+3 x)^2}{6875}-\frac{\sqrt{1-2 x} (2+3 x)^4}{10 (3+5 x)^2}-\frac{131 \sqrt{1-2 x} (2+3 x)^3}{550 (3+5 x)}-\frac{\int \frac{(2+3 x) (-8078+7875 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{13750}\\ &=-\frac{21 (704-375 x) \sqrt{1-2 x}}{68750}+\frac{1428 \sqrt{1-2 x} (2+3 x)^2}{6875}-\frac{\sqrt{1-2 x} (2+3 x)^4}{10 (3+5 x)^2}-\frac{131 \sqrt{1-2 x} (2+3 x)^3}{550 (3+5 x)}+\frac{12803 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{68750}\\ &=-\frac{21 (704-375 x) \sqrt{1-2 x}}{68750}+\frac{1428 \sqrt{1-2 x} (2+3 x)^2}{6875}-\frac{\sqrt{1-2 x} (2+3 x)^4}{10 (3+5 x)^2}-\frac{131 \sqrt{1-2 x} (2+3 x)^3}{550 (3+5 x)}-\frac{12803 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{68750}\\ &=-\frac{21 (704-375 x) \sqrt{1-2 x}}{68750}+\frac{1428 \sqrt{1-2 x} (2+3 x)^2}{6875}-\frac{\sqrt{1-2 x} (2+3 x)^4}{10 (3+5 x)^2}-\frac{131 \sqrt{1-2 x} (2+3 x)^3}{550 (3+5 x)}-\frac{12803 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.069112, size = 68, normalized size = 0.57 \[ \frac{\frac{55 \sqrt{1-2 x} \left (445500 x^4+1103850 x^3+506880 x^2-200305 x-121976\right )}{(5 x+3)^2}-25606 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3781250} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.6 \begin{align*}{\frac{81}{1250} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{369}{1250} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{108}{3125}\sqrt{1-2\,x}}+{\frac{4}{125\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{263}{220} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{53}{20}\sqrt{1-2\,x}} \right ) }-{\frac{12803\,\sqrt{55}}{1890625}{\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 = 1.73919, size = 136, normalized size = 1.13 \begin{align*} \frac{81}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{369}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{12803}{3781250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{108}{3125} \, \sqrt{-2 \, x + 1} + \frac{263 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 583 \, \sqrt{-2 \, x + 1}}{6875 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67665, size = 269, normalized size = 2.24 \begin{align*} \frac{12803 \, \sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (445500 \, x^{4} + 1103850 \, x^{3} + 506880 \, x^{2} - 200305 \, x - 121976\right )} \sqrt{-2 \, x + 1}}{3781250 \,{\left (25 \, x^{2} + 30 \, x + 9\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 = 2.64006, size = 138, normalized size = 1.15 \begin{align*} \frac{81}{1250} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{369}{1250} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{12803}{3781250} \, \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{108}{3125} \, \sqrt{-2 \, x + 1} + \frac{263 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 583 \, \sqrt{-2 \, x + 1}}{27500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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