Optimal. Leaf size=40 \[ -\frac {25}{36} (1-2 x)^{9/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {121}{20} (1-2 x)^{5/2} \]
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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} \[ -\frac {25}{36} (1-2 x)^{9/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {121}{20} (1-2 x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (3+5 x)^2 \, dx &=\int \left (\frac {121}{4} (1-2 x)^{3/2}-\frac {55}{2} (1-2 x)^{5/2}+\frac {25}{4} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {121}{20} (1-2 x)^{5/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {25}{36} (1-2 x)^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.58 \[ -\frac {1}{315} (1-2 x)^{5/2} \left (875 x^2+1600 x+887\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 29, normalized size = 0.72 \[ -\frac {1}{315} \, {\left (3500 \, x^{4} + 2900 \, x^{3} - 1977 \, x^{2} - 1948 \, x + 887\right )} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 49, normalized size = 1.22 \[ -\frac {25}{36} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {55}{14} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {121}{20} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]
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 \[ -\frac {\left (875 x^{2}+1600 x +887\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{315} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 28, normalized size = 0.70 \[ -\frac {25}{36} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {55}{14} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {121}{20} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]
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 \[ -\frac {{\left (1-2\,x\right )}^{5/2}\,\left (9900\,x+875\,{\left (2\,x-1\right )}^2+2673\right )}{1260} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.56, size = 236, normalized size = 5.90 \[ \begin {cases} - \frac {20 \sqrt {5} i \left (x + \frac {3}{5}\right )^{4} \sqrt {10 x - 5}}{9} + \frac {220 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{63} - \frac {121 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{525} - \frac {2662 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{7875} - \frac {29282 \sqrt {5} i \sqrt {10 x - 5}}{39375} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\- \frac {20 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{4}}{9} + \frac {220 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{63} - \frac {121 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{525} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{7875} - \frac {29282 \sqrt {5} \sqrt {5 - 10 x}}{39375} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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