Optimal. Leaf size=40 \[ -\frac {25}{28} (1-2 x)^{7/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {121}{12} (1-2 x)^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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} \[ -\frac {25}{28} (1-2 x)^{7/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {121}{12} (1-2 x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (3+5 x)^2 \, dx &=\int \left (\frac {121}{4} \sqrt {1-2 x}-\frac {55}{2} (1-2 x)^{3/2}+\frac {25}{4} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac {121}{12} (1-2 x)^{3/2}+\frac {11}{2} (1-2 x)^{5/2}-\frac {25}{28} (1-2 x)^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.58 \[ -\frac {1}{21} (1-2 x)^{3/2} \left (75 x^2+156 x+115\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 24, normalized size = 0.60 \[ \frac {1}{21} \, {\left (150 \, x^{3} + 237 \, x^{2} + 74 \, x - 115\right )} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 42, normalized size = 1.05 \[ \frac {25}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {11}{2} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {121}{12} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.50 \[ -\frac {\left (75 x^{2}+156 x +115\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 28, normalized size = 0.70 \[ -\frac {25}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {11}{2} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {121}{12} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 23, normalized size = 0.58 \[ -\frac {{\left (1-2\,x\right )}^{3/2}\,\left (924\,x+75\,{\left (2\,x-1\right )}^2+385\right )}{84} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.43, size = 187, normalized size = 4.68 \[ \begin {cases} \frac {10 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{7} - \frac {11 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{35} - \frac {242 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{525} - \frac {2662 \sqrt {5} i \sqrt {10 x - 5}}{2625} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {10 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{7} - \frac {11 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{35} - \frac {242 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{525} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x}}{2625} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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