Optimal. Leaf size=53 \[ \frac {125}{72} (1-2 x)^{9/2}-\frac {825}{56} (1-2 x)^{7/2}+\frac {363}{8} (1-2 x)^{5/2}-\frac {1331}{24} (1-2 x)^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 53, 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} \begin {gather*} \frac {125}{72} (1-2 x)^{9/2}-\frac {825}{56} (1-2 x)^{7/2}+\frac {363}{8} (1-2 x)^{5/2}-\frac {1331}{24} (1-2 x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (3+5 x)^3 \, dx &=\int \left (\frac {1331}{8} \sqrt {1-2 x}-\frac {1815}{8} (1-2 x)^{3/2}+\frac {825}{8} (1-2 x)^{5/2}-\frac {125}{8} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {1331}{24} (1-2 x)^{3/2}+\frac {363}{8} (1-2 x)^{5/2}-\frac {825}{56} (1-2 x)^{7/2}+\frac {125}{72} (1-2 x)^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{63} (1-2 x)^{3/2} \left (875 x^3+2400 x^2+2661 x+1454\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 40, normalized size = 0.75 \begin {gather*} \frac {1}{504} \left (875 (1-2 x)^3-7425 (1-2 x)^2+22869 (1-2 x)-27951\right ) (1-2 x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 29, normalized size = 0.55 \begin {gather*} \frac {1}{63} \, {\left (1750 \, x^{4} + 3925 \, x^{3} + 2922 \, x^{2} + 247 \, x - 1454\right )} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 58, normalized size = 1.09 \begin {gather*} \frac {125}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {825}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {363}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1331}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {\left (875 x^{3}+2400 x^{2}+2661 x +1454\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{63} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 37, normalized size = 0.70 \begin {gather*} \frac {125}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {825}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {363}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1331}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {363\,{\left (1-2\,x\right )}^{5/2}}{8}-\frac {1331\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {825\,{\left (1-2\,x\right )}^{7/2}}{56}+\frac {125\,{\left (1-2\,x\right )}^{9/2}}{72} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.04, size = 236, normalized size = 4.45 \begin {gather*} \begin {cases} \frac {50 \sqrt {5} i \left (x + \frac {3}{5}\right )^{4} \sqrt {10 x - 5}}{9} - \frac {55 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{63} - \frac {121 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{105} - \frac {2662 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{1575} - \frac {29282 \sqrt {5} i \sqrt {10 x - 5}}{7875} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {50 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{4}}{9} - \frac {55 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{63} - \frac {121 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{105} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{1575} - \frac {29282 \sqrt {5} \sqrt {5 - 10 x}}{7875} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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