Optimal. Leaf size=45 \[ -\frac{206 \sqrt{1-2 x}}{1815 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{165 (5 x+3)^{3/2}} \]
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Rubi [A] time = 0.0056482, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {78, 37} \[ -\frac{206 \sqrt{1-2 x}}{1815 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{165 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{165 (3+5 x)^{3/2}}+\frac{103}{165} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{165 (3+5 x)^{3/2}}-\frac{206 \sqrt{1-2 x}}{1815 \sqrt{3+5 x}}\\ \end{align*}
Mathematica [A] time = 0.0078725, size = 27, normalized size = 0.6 \[ -\frac{2 \sqrt{1-2 x} (103 x+64)}{363 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 0.5 \begin{align*} -{\frac{206\,x+128}{363}\sqrt{1-2\,x} \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 = 2.25692, size = 65, normalized size = 1.44 \begin{align*} -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{165 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{206 \, \sqrt{-10 \, x^{2} - x + 3}}{1815 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61772, size = 95, normalized size = 2.11 \begin{align*} -\frac{2 \,{\left (103 \, x + 64\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3 x + 2}{\sqrt{1 - 2 x} \left (5 x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.57601, size = 170, normalized size = 3.78 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{145200 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{69 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{12100 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{207 \, \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}}}{9075 \,{\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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