Optimal. Leaf size=76 \[ \frac{76}{1331 \sqrt{1-2 x}}-\frac{1}{55 (1-2 x)^{3/2} (5 x+3)}+\frac{76}{1815 (1-2 x)^{3/2}}-\frac{76 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
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Rubi [A] time = 0.019041, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ \frac{76}{1331 \sqrt{1-2 x}}-\frac{1}{55 (1-2 x)^{3/2} (5 x+3)}+\frac{76}{1815 (1-2 x)^{3/2}}-\frac{76 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
Antiderivative was successfully verified.
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Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx &=-\frac{1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac{38}{55} \int \frac{1}{(1-2 x)^{5/2} (3+5 x)} \, dx\\ &=\frac{76}{1815 (1-2 x)^{3/2}}-\frac{1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac{38}{121} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac{76}{1815 (1-2 x)^{3/2}}+\frac{76}{1331 \sqrt{1-2 x}}-\frac{1}{55 (1-2 x)^{3/2} (3+5 x)}+\frac{190 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac{76}{1815 (1-2 x)^{3/2}}+\frac{76}{1331 \sqrt{1-2 x}}-\frac{1}{55 (1-2 x)^{3/2} (3+5 x)}-\frac{190 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1331}\\ &=\frac{76}{1815 (1-2 x)^{3/2}}+\frac{76}{1331 \sqrt{1-2 x}}-\frac{1}{55 (1-2 x)^{3/2} (3+5 x)}-\frac{76 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}
Mathematica [C] time = 0.0115061, size = 46, normalized size = 0.61 \[ -\frac{33-76 (5 x+3) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )}{1815 (1-2 x)^{3/2} (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 54, normalized size = 0.7 \begin{align*}{\frac{14}{363} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{74}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{1331}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{76\,\sqrt{55}}{14641}{\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 = 2.09552, size = 100, normalized size = 1.32 \begin{align*} \frac{38}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (570 \,{\left (2 \, x - 1\right )}^{2} + 1672 \, x - 1683\right )}}{3993 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03041, size = 258, normalized size = 3.39 \begin{align*} \frac{114 \, \sqrt{11} \sqrt{5}{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \,{\left (2280 \, x^{2} - 608 \, x - 1113\right )} \sqrt{-2 \, x + 1}}{43923 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 = 2.0891, size = 104, normalized size = 1.37 \begin{align*} \frac{38}{14641} \, \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{4 \,{\left (111 \, x - 94\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{5 \, \sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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