Optimal. Leaf size=40 \[ -\frac{5}{4} (1-2 x)^{5/2}+\frac{55}{6} (1-2 x)^{3/2}-\frac{121}{4} \sqrt{1-2 x} \]
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Rubi [A] time = 0.0075216, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ -\frac{5}{4} (1-2 x)^{5/2}+\frac{55}{6} (1-2 x)^{3/2}-\frac{121}{4} \sqrt{1-2 x} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(3+5 x)^2}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{121}{4 \sqrt{1-2 x}}-\frac{55}{2} \sqrt{1-2 x}+\frac{25}{4} (1-2 x)^{3/2}\right ) \, dx\\ &=-\frac{121}{4} \sqrt{1-2 x}+\frac{55}{6} (1-2 x)^{3/2}-\frac{5}{4} (1-2 x)^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0072686, size = 23, normalized size = 0.57 \[ -\frac{1}{3} \sqrt{1-2 x} \left (15 x^2+40 x+67\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{\frac{15\,{x}^{2}+40\,x+67}{3}\sqrt{1-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.14306, size = 38, normalized size = 0.95 \begin{align*} -\frac{5}{4} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{55}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{121}{4} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53232, size = 57, normalized size = 1.42 \begin{align*} -\frac{1}{3} \,{\left (15 \, x^{2} + 40 \, x + 67\right )} \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.2059, size = 134, normalized size = 3.35 \begin{align*} \begin{cases} - \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5} - \frac{22 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{15} - \frac{242 \sqrt{5} i \sqrt{10 x - 5}}{75} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2} - \frac{22 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{15} - \frac{242 \sqrt{5} \sqrt{5 - 10 x}}{75} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.83239, size = 47, normalized size = 1.18 \begin{align*} -\frac{5}{4} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{55}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{121}{4} \, \sqrt{-2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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