Optimal. Leaf size=53 \[ \frac{75}{56} (1-2 x)^{7/2}-\frac{101}{8} (1-2 x)^{5/2}+\frac{1133}{24} (1-2 x)^{3/2}-\frac{847}{8} \sqrt{1-2 x} \]
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Rubi [A] time = 0.0106217, 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} \[ \frac{75}{56} (1-2 x)^{7/2}-\frac{101}{8} (1-2 x)^{5/2}+\frac{1133}{24} (1-2 x)^{3/2}-\frac{847}{8} \sqrt{1-2 x} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(2+3 x) (3+5 x)^2}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{847}{8 \sqrt{1-2 x}}-\frac{1133}{8} \sqrt{1-2 x}+\frac{505}{8} (1-2 x)^{3/2}-\frac{75}{8} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac{847}{8} \sqrt{1-2 x}+\frac{1133}{24} (1-2 x)^{3/2}-\frac{101}{8} (1-2 x)^{5/2}+\frac{75}{56} (1-2 x)^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0101598, size = 28, normalized size = 0.53 \[ -\frac{1}{21} \sqrt{1-2 x} \left (225 x^3+723 x^2+1091 x+1469\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 0.5 \begin{align*} -{\frac{225\,{x}^{3}+723\,{x}^{2}+1091\,x+1469}{21}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01415, size = 50, normalized size = 0.94 \begin{align*} \frac{75}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{101}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{1133}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{847}{8} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52172, size = 78, normalized size = 1.47 \begin{align*} -\frac{1}{21} \,{\left (225 \, x^{3} + 723 \, x^{2} + 1091 \, x + 1469\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.4082, size = 46, normalized size = 0.87 \begin{align*} \frac{75 \left (1 - 2 x\right )^{\frac{7}{2}}}{56} - \frac{101 \left (1 - 2 x\right )^{\frac{5}{2}}}{8} + \frac{1133 \left (1 - 2 x\right )^{\frac{3}{2}}}{24} - \frac{847 \sqrt{1 - 2 x}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.60995, size = 69, normalized size = 1.3 \begin{align*} -\frac{75}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{101}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{1133}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{847}{8} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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