Optimal. Leaf size=89 \[ -\frac{2960 \sqrt{1-2 x}}{43923 \sqrt{5 x+3}}+\frac{296}{3993 \sqrt{5 x+3} \sqrt{1-2 x}}+\frac{74}{1815 \sqrt{5 x+3} (1-2 x)^{3/2}}-\frac{2}{165 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]
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Rubi [A] time = 0.015575, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ -\frac{2960 \sqrt{1-2 x}}{43923 \sqrt{5 x+3}}+\frac{296}{3993 \sqrt{5 x+3} \sqrt{1-2 x}}+\frac{74}{1815 \sqrt{5 x+3} (1-2 x)^{3/2}}-\frac{2}{165 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=-\frac{2}{165 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{37}{55} \int \frac{1}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2}{165 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{74}{1815 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{148}{363} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2}{165 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{74}{1815 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{296}{3993 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{1480 \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{3993}\\ &=-\frac{2}{165 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{74}{1815 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{296}{3993 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{2960 \sqrt{1-2 x}}{43923 \sqrt{3+5 x}}\\ \end{align*}
Mathematica [A] time = 0.0134908, size = 37, normalized size = 0.42 \[ \frac{-59200 x^3-8880 x^2+26418 x+5728}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.4 \begin{align*} -{\frac{59200\,{x}^{3}+8880\,{x}^{2}-26418\,x-5728}{43923} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09827, size = 80, normalized size = 0.9 \begin{align*} \frac{5920 \, x}{43923 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{296}{43923 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{74 \, x}{363 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{40}{363 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74271, size = 158, normalized size = 1.78 \begin{align*} -\frac{2 \,{\left (29600 \, x^{3} + 4440 \, x^{2} - 13209 \, x - 2864\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43923 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, 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 [B] time = 2.16091, size = 223, normalized size = 2.51 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{702768 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{7 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{5324 \, \sqrt{5 \, x + 3}} - \frac{8 \,{\left (181 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1188 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1098075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{231 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{43923 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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