Optimal. Leaf size=53 \[ \frac {847}{24 (1-2 x)^{3/2}}-\frac {1133}{8 \sqrt {1-2 x}}-\frac {505}{8} \sqrt {1-2 x}+\frac {25}{8} (1-2 x)^{3/2} \]
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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78}
\begin {gather*} \frac {25}{8} (1-2 x)^{3/2}-\frac {505}{8} \sqrt {1-2 x}-\frac {1133}{8 \sqrt {1-2 x}}+\frac {847}{24 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(2+3 x) (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {847}{8 (1-2 x)^{5/2}}-\frac {1133}{8 (1-2 x)^{3/2}}+\frac {505}{8 \sqrt {1-2 x}}-\frac {75}{8} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {847}{24 (1-2 x)^{3/2}}-\frac {1133}{8 \sqrt {1-2 x}}-\frac {505}{8} \sqrt {1-2 x}+\frac {25}{8} (1-2 x)^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {499-1551 x+645 x^2+75 x^3}{3 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 38, normalized size = 0.72
method | result | size |
gosper | \(-\frac {75 x^{3}+645 x^{2}-1551 x +499}{3 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(25\) |
trager | \(-\frac {\left (75 x^{3}+645 x^{2}-1551 x +499\right ) \sqrt {1-2 x}}{3 \left (-1+2 x \right )^{2}}\) | \(32\) |
risch | \(\frac {75 x^{3}+645 x^{2}-1551 x +499}{3 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(32\) |
derivativedivides | \(\frac {847}{24 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {25 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {1133}{8 \sqrt {1-2 x}}-\frac {505 \sqrt {1-2 x}}{8}\) | \(38\) |
default | \(\frac {847}{24 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {25 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {1133}{8 \sqrt {1-2 x}}-\frac {505 \sqrt {1-2 x}}{8}\) | \(38\) |
meijerg | \(-\frac {12 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {29 \sqrt {\pi }-\frac {29 \sqrt {\pi }\, \left (-24 x +8\right )}{8 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {70 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{3 \sqrt {\pi }}+\frac {50 \sqrt {\pi }-\frac {25 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{64 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 33, normalized size = 0.62 \begin {gather*} \frac {25}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {505}{8} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (309 \, x - 116\right )}}{12 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 36, normalized size = 0.68 \begin {gather*} -\frac {{\left (75 \, x^{3} + 645 \, x^{2} - 1551 \, x + 499\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 88, normalized size = 1.66 \begin {gather*} - \frac {75 x^{3} \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} - \frac {645 x^{2} \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} + \frac {1551 x \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} - \frac {499 \sqrt {1 - 2 x}}{12 x^{2} - 12 x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.85, size = 40, normalized size = 0.75 \begin {gather*} \frac {25}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {505}{8} \, \sqrt {-2 \, x + 1} - \frac {11 \, {\left (309 \, x - 116\right )}}{12 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 38, normalized size = 0.72 \begin {gather*} \frac {1515\,{\left (2\,x-1\right )}^2-6798\,x+75\,{\left (2\,x-1\right )}^3+2552}{\sqrt {1-2\,x}\,\left (48\,x-24\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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