Optimal. Leaf size=67 \[ -\frac{160 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x}}{363 (5 x+3)^{3/2}}+\frac{2}{11 (5 x+3)^{3/2} \sqrt{1-2 x}} \]
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Rubi [A] time = 0.0102942, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{160 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}-\frac{40 \sqrt{1-2 x}}{363 (5 x+3)^{3/2}}+\frac{2}{11 (5 x+3)^{3/2} \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{20}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x}}{363 (3+5 x)^{3/2}}+\frac{80}{363} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{40 \sqrt{1-2 x}}{363 (3+5 x)^{3/2}}-\frac{160 \sqrt{1-2 x}}{3993 \sqrt{3+5 x}}\\ \end{align*}
Mathematica [A] time = 0.0072954, size = 32, normalized size = 0.48 \[ \frac{2 \left (800 x^2+520 x-97\right )}{3993 \sqrt{1-2 x} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 27, normalized size = 0.4 \begin{align*}{\frac{1600\,{x}^{2}+1040\,x-194}{3993} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.16982, size = 86, normalized size = 1.28 \begin{align*} \frac{320 \, x}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{16}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{33 \,{\left (5 \, \sqrt{-10 \, x^{2} - x + 3} x + 3 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5157, size = 122, normalized size = 1.82 \begin{align*} -\frac{2 \,{\left (800 \, x^{2} + 520 \, x - 97\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\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 [A] time = 22.9272, size = 230, normalized size = 3.43 \begin{align*} \begin{cases} - \frac{1600 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} + \frac{880 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} + \frac{242 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{1600 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} + \frac{880 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} + \frac{242 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{- 219615 x + 199650 \left (x + \frac{3}{5}\right )^{2} - 131769} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.41198, size = 205, normalized size = 3.06 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{63888 \,{\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 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{6655 \,{\left (2 \, x - 1\right )}} + \frac{{\left (\frac{21 \, \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}}}{3993 \,{\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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