Optimal. Leaf size=79 \[ -\frac {405 (3-8 x) \sqrt {3 x-4 x^2}}{32768}-\frac {15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac {1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac {3645 \sin ^{-1}\left (1-\frac {8 x}{3}\right )}{131072} \]
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Rubi [A]
time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {626, 633, 222}
\begin {gather*} -\frac {3645 \text {ArcSin}\left (1-\frac {8 x}{3}\right )}{131072}-\frac {1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac {15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac {405 (3-8 x) \sqrt {3 x-4 x^2}}{32768} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 626
Rule 633
Rubi steps
\begin {align*} \int \left (3 x-4 x^2\right )^{5/2} \, dx &=-\frac {1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}+\frac {15}{32} \int \left (3 x-4 x^2\right )^{3/2} \, dx\\ &=-\frac {15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac {1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}+\frac {405 \int \sqrt {3 x-4 x^2} \, dx}{2048}\\ &=-\frac {405 (3-8 x) \sqrt {3 x-4 x^2}}{32768}-\frac {15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac {1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}+\frac {3645 \int \frac {1}{\sqrt {3 x-4 x^2}} \, dx}{65536}\\ &=-\frac {405 (3-8 x) \sqrt {3 x-4 x^2}}{32768}-\frac {15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac {1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac {1215 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,3-8 x\right )}{131072}\\ &=-\frac {405 (3-8 x) \sqrt {3 x-4 x^2}}{32768}-\frac {15 (3-8 x) \left (3 x-4 x^2\right )^{3/2}}{1024}-\frac {1}{48} (3-8 x) \left (3 x-4 x^2\right )^{5/2}-\frac {3645 \sin ^{-1}\left (1-\frac {8 x}{3}\right )}{131072}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 92, normalized size = 1.16 \begin {gather*} \frac {\sqrt {-x (-3+4 x)} \left (2 \sqrt {x} \sqrt {-3+4 x} \left (-3645-3240 x-3456 x^2+248832 x^3-491520 x^4+262144 x^5\right )+10935 \log \left (-2 \sqrt {x}+\sqrt {-3+4 x}\right )\right )}{196608 \sqrt {x} \sqrt {-3+4 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 64, normalized size = 0.81
method | result | size |
risch | \(-\frac {\left (262144 x^{5}-491520 x^{4}+248832 x^{3}-3456 x^{2}-3240 x -3645\right ) x \left (-3+4 x \right )}{98304 \sqrt {-x \left (-3+4 x \right )}}+\frac {3645 \arcsin \left (-1+\frac {8 x}{3}\right )}{131072}\) | \(53\) |
default | \(-\frac {15 \left (3-8 x \right ) \left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}{1024}-\frac {\left (3-8 x \right ) \left (-4 x^{2}+3 x \right )^{\frac {5}{2}}}{48}+\frac {3645 \arcsin \left (-1+\frac {8 x}{3}\right )}{131072}-\frac {405 \left (3-8 x \right ) \sqrt {-4 x^{2}+3 x}}{32768}\) | \(64\) |
meijerg | \(-\frac {10935 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {3}\, \left (-\frac {1835008}{243} x^{5}+\frac {1146880}{81} x^{4}-7168 x^{3}+\frac {896}{9} x^{2}+\frac {280}{3} x +105\right ) \sqrt {-\frac {4 x}{3}+1}}{30240}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 \sqrt {3}\, \sqrt {x}}{3}\right )}{192}\right )}{1024 \sqrt {\pi }}\) | \(67\) |
trager | \(\left (\frac {8}{3} x^{5}-5 x^{4}+\frac {81}{32} x^{3}-\frac {9}{256} x^{2}-\frac {135}{4096} x -\frac {1215}{32768}\right ) \sqrt {-4 x^{2}+3 x}-\frac {3645 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (8 x \RootOf \left (\textit {\_Z}^{2}+1\right )+4 \sqrt {-4 x^{2}+3 x}-3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{131072}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 90, normalized size = 1.14 \begin {gather*} \frac {1}{6} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}} x - \frac {1}{16} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}} + \frac {15}{128} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}} x - \frac {45}{1024} \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}} + \frac {405}{4096} \, \sqrt {-4 \, x^{2} + 3 \, x} x - \frac {1215}{32768} \, \sqrt {-4 \, x^{2} + 3 \, x} - \frac {3645}{131072} \, \arcsin \left (-\frac {8}{3} \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 58, normalized size = 0.73 \begin {gather*} \frac {1}{98304} \, {\left (262144 \, x^{5} - 491520 \, x^{4} + 248832 \, x^{3} - 3456 \, x^{2} - 3240 \, x - 3645\right )} \sqrt {-4 \, x^{2} + 3 \, x} - \frac {3645}{65536} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- 4 x^{2} + 3 x\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.01, size = 47, normalized size = 0.59 \begin {gather*} \frac {1}{98304} \, {\left (8 \, {\left (16 \, {\left (8 \, {\left (32 \, {\left (8 \, x - 15\right )} x + 243\right )} x - 27\right )} x - 405\right )} x - 3645\right )} \sqrt {-4 \, x^{2} + 3 \, x} + \frac {3645}{131072} \, \arcsin \left (\frac {8}{3} \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 63, normalized size = 0.80 \begin {gather*} \frac {3645\,\mathrm {asin}\left (\frac {8\,x}{3}-1\right )}{131072}+\frac {15\,\left (4\,x-\frac {3}{2}\right )\,{\left (3\,x-4\,x^2\right )}^{3/2}}{512}+\frac {\left (4\,x-\frac {3}{2}\right )\,{\left (3\,x-4\,x^2\right )}^{5/2}}{24}+\frac {405\,\left (\frac {x}{2}-\frac {3}{16}\right )\,\sqrt {3\,x-4\,x^2}}{2048} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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