Optimal. Leaf size=88 \[ -\frac{2589 \sqrt{1-2 x}}{13310 (5 x+3)}-\frac{613 \sqrt{1-2 x}}{605 (5 x+3)^2}+\frac{49}{22 \sqrt{1-2 x} (5 x+3)^2}-\frac{2589 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6655 \sqrt{55}} \]
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Rubi [A] time = 0.0225122, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 51, 63, 206} \[ -\frac{2589 \sqrt{1-2 x}}{13310 (5 x+3)}-\frac{613 \sqrt{1-2 x}}{605 (5 x+3)^2}+\frac{49}{22 \sqrt{1-2 x} (5 x+3)^2}-\frac{2589 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6655 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)^2}-\frac{1}{22} \int \frac{-431+99 x}{\sqrt{1-2 x} (3+5 x)^3} \, dx\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)^2}-\frac{613 \sqrt{1-2 x}}{605 (3+5 x)^2}+\frac{2589 \int \frac{1}{\sqrt{1-2 x} (3+5 x)^2} \, dx}{1210}\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)^2}-\frac{613 \sqrt{1-2 x}}{605 (3+5 x)^2}-\frac{2589 \sqrt{1-2 x}}{13310 (3+5 x)}+\frac{2589 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{13310}\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)^2}-\frac{613 \sqrt{1-2 x}}{605 (3+5 x)^2}-\frac{2589 \sqrt{1-2 x}}{13310 (3+5 x)}-\frac{2589 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{13310}\\ &=\frac{49}{22 \sqrt{1-2 x} (3+5 x)^2}-\frac{613 \sqrt{1-2 x}}{605 (3+5 x)^2}-\frac{2589 \sqrt{1-2 x}}{13310 (3+5 x)}-\frac{2589 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6655 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0380928, size = 58, normalized size = 0.66 \[ \frac{\frac{55 \left (25890 x^2+29561 x+8392\right )}{\sqrt{1-2 x} (5 x+3)^2}-5178 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{732050} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 57, normalized size = 0.7 \begin{align*}{\frac{98}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{50}{1331\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{139}{50} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1551}{250}\sqrt{1-2\,x}} \right ) }-{\frac{2589\,\sqrt{55}}{366025}{\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 = 3.58111, size = 112, normalized size = 1.27 \begin{align*} \frac{2589}{732050} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{12945 \,{\left (2 \, x - 1\right )}^{2} + 110902 \, x + 3839}{6655 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67631, size = 248, normalized size = 2.82 \begin{align*} \frac{2589 \, \sqrt{55}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (25890 \, x^{2} + 29561 \, x + 8392\right )} \sqrt{-2 \, x + 1}}{732050 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.40094, size = 104, normalized size = 1.18 \begin{align*} \frac{2589}{732050} \, \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{98}{1331 \, \sqrt{-2 \, x + 1}} + \frac{695 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1551 \, \sqrt{-2 \, x + 1}}{26620 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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