Optimal. Leaf size=71 \[ \frac{(5 x+3)^{3/2}}{\sqrt{1-2 x}}+\frac{15}{4} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{33}{4} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
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Rubi [A] time = 0.0148253, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {47, 50, 54, 216} \[ \frac{(5 x+3)^{3/2}}{\sqrt{1-2 x}}+\frac{15}{4} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{33}{4} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{15}{2} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=\frac{15}{4} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{165}{8} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{15}{4} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{1}{4} \left (33 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=\frac{15}{4} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{33}{4} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}
Mathematica [C] time = 0.0086639, size = 39, normalized size = 0.55 \[ \frac{11 \sqrt{\frac{11}{2}} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{2 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 3+5\,x \right ) ^{{\frac{3}{2}}} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.39006, size = 84, normalized size = 1.18 \begin{align*} -\frac{33}{16} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{4 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76912, size = 240, normalized size = 3.38 \begin{align*} \frac{33 \, \sqrt{5} \sqrt{2}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 4 \,{\left (10 \, x - 27\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{16 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.23271, size = 144, normalized size = 2.03 \begin{align*} \begin{cases} \frac{25 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{2 \sqrt{10 x - 5}} - \frac{165 i \sqrt{x + \frac{3}{5}}}{4 \sqrt{10 x - 5}} + \frac{33 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{8} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{33 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{8} - \frac{25 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{2 \sqrt{5 - 10 x}} + \frac{165 \sqrt{x + \frac{3}{5}}}{4 \sqrt{5 - 10 x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21592, size = 78, normalized size = 1.1 \begin{align*} -\frac{33}{8} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{20 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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