Optimal. Leaf size=116 \[ \frac {1331}{512} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {14641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{512 \sqrt {10}} \]
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Rubi [A]
time = 0.02, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {52, 56, 222}
\begin {gather*} \frac {14641 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{512 \sqrt {10}}-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {5 x+3}+\frac {1331}{512} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx &=-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {55}{16} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {605}{64} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {6655}{512} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {1331}{512} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {14641 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1024}\\ &=\frac {1331}{512} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {14641 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{512 \sqrt {5}}\\ &=\frac {1331}{512} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {14641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{512 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 78, normalized size = 0.67 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-12015-2517 x+75740 x^2+106400 x^3+48000 x^4\right )-43923 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{15360 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 104, normalized size = 0.90
method | result | size |
risch | \(-\frac {\left (9600 x^{3}+15520 x^{2}+5836 x -4005\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1536 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {14641 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10240 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
default | \(-\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{8}-\frac {55 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{96}-\frac {605 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{256}+\frac {1331 \sqrt {1-2 x}\, \sqrt {3+5 x}}{512}+\frac {14641 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10240 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 70, normalized size = 0.60 \begin {gather*} -\frac {5}{8} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {91}{96} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {605}{128} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {14641}{10240} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {121}{512} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.95, size = 72, normalized size = 0.62 \begin {gather*} \frac {1}{1536} \, {\left (9600 \, x^{3} + 15520 \, x^{2} + 5836 \, x - 4005\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {14641}{10240} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 21.54, size = 270, normalized size = 2.33 \begin {gather*} \begin {cases} \frac {125 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {10 x - 5}} - \frac {1925 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{24 \sqrt {10 x - 5}} - \frac {605 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{192 \sqrt {10 x - 5}} - \frac {6655 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{768 \sqrt {10 x - 5}} + \frac {14641 i \sqrt {x + \frac {3}{5}}}{512 \sqrt {10 x - 5}} - \frac {14641 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{5120} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {14641 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{5120} - \frac {125 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {5 - 10 x}} + \frac {1925 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{24 \sqrt {5 - 10 x}} + \frac {605 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{192 \sqrt {5 - 10 x}} + \frac {6655 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{768 \sqrt {5 - 10 x}} - \frac {14641 \sqrt {x + \frac {3}{5}}}{512 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 203 vs.
\(2 (83) = 166\).
time = 1.71, size = 203, normalized size = 1.75 \begin {gather*} \frac {1}{76800} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{1600} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{400} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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