Optimal. Leaf size=53 \[ \frac{25}{8} (1-2 x)^{3/2}-\frac{505}{8} \sqrt{1-2 x}-\frac{1133}{8 \sqrt{1-2 x}}+\frac{847}{24 (1-2 x)^{3/2}} \]
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Rubi [A] time = 0.0100809, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {77} \[ \frac{25}{8} (1-2 x)^{3/2}-\frac{505}{8} \sqrt{1-2 x}-\frac{1133}{8 \sqrt{1-2 x}}+\frac{847}{24 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(2+3 x) (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{847}{8 (1-2 x)^{5/2}}-\frac{1133}{8 (1-2 x)^{3/2}}+\frac{505}{8 \sqrt{1-2 x}}-\frac{75}{8} \sqrt{1-2 x}\right ) \, dx\\ &=\frac{847}{24 (1-2 x)^{3/2}}-\frac{1133}{8 \sqrt{1-2 x}}-\frac{505}{8} \sqrt{1-2 x}+\frac{25}{8} (1-2 x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0110197, size = 28, normalized size = 0.53 \[ -\frac{75 x^3+645 x^2-1551 x+499}{3 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{75\,{x}^{3}+645\,{x}^{2}-1551\,x+499}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.70761, size = 45, normalized size = 0.85 \begin{align*} \frac{25}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{505}{8} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (309 \, x - 116\right )}}{12 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51311, size = 99, normalized size = 1.87 \begin{align*} -\frac{{\left (75 \, x^{3} + 645 \, x^{2} - 1551 \, x + 499\right )} \sqrt{-2 \, x + 1}}{3 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.69489, size = 102, normalized size = 1.92 \begin{align*} \frac{75 x^{3}}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{645 x^{2}}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} - \frac{1551 x}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{499}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.08672, size = 54, normalized size = 1.02 \begin{align*} \frac{25}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{505}{8} \, \sqrt{-2 \, x + 1} - \frac{11 \,{\left (309 \, x - 116\right )}}{12 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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