Optimal. Leaf size=138 \[ \frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}+\frac {1331 (1-2 x)^{3/2} \sqrt {3+5 x}}{6400}+\frac {121 (1-2 x)^{5/2} \sqrt {3+5 x}}{1600}-\frac {33}{160} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {483153 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {483153 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64000 \sqrt {10}}-\frac {1}{10} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {33}{160} \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {121 \sqrt {5 x+3} (1-2 x)^{5/2}}{1600}+\frac {1331 \sqrt {5 x+3} (1-2 x)^{3/2}}{6400}+\frac {43923 \sqrt {5 x+3} \sqrt {1-2 x}}{64000} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx &=-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {33}{20} \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\\ &=-\frac {33}{160} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {363}{320} \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {121 (1-2 x)^{5/2} \sqrt {3+5 x}}{1600}-\frac {33}{160} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {1331}{640} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {1331 (1-2 x)^{3/2} \sqrt {3+5 x}}{6400}+\frac {121 (1-2 x)^{5/2} \sqrt {3+5 x}}{1600}-\frac {33}{160} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {43923 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{12800}\\ &=\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}+\frac {1331 (1-2 x)^{3/2} \sqrt {3+5 x}}{6400}+\frac {121 (1-2 x)^{5/2} \sqrt {3+5 x}}{1600}-\frac {33}{160} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {483153 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{128000}\\ &=\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}+\frac {1331 (1-2 x)^{3/2} \sqrt {3+5 x}}{6400}+\frac {121 (1-2 x)^{5/2} \sqrt {3+5 x}}{1600}-\frac {33}{160} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {483153 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{64000 \sqrt {5}}\\ &=\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}+\frac {1331 (1-2 x)^{3/2} \sqrt {3+5 x}}{6400}+\frac {121 (1-2 x)^{5/2} \sqrt {3+5 x}}{1600}-\frac {33}{160} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {483153 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 83, normalized size = 0.60 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (89019+497625 x+49780 x^2-1261600 x^3+144000 x^4+1280000 x^5\right )-483153 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{640000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 120, normalized size = 0.87
method | result | size |
risch | \(-\frac {\left (256000 x^{4}-124800 x^{3}-177440 x^{2}+116420 x +29673\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{64000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {483153 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
default | \(\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{25}+\frac {11 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{200}+\frac {121 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{2000}-\frac {1331 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{16000}-\frac {43923 \sqrt {1-2 x}\, \sqrt {3+5 x}}{64000}+\frac {483153 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 84, normalized size = 0.61 \begin {gather*} \frac {1}{25} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {11}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {11}{800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3993}{3200} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {483153}{1280000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3993}{64000} \, \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 = 0.52, size = 77, normalized size = 0.56 \begin {gather*} \frac {1}{64000} \, {\left (256000 \, x^{4} - 124800 \, x^{3} - 177440 \, x^{2} + 116420 \, x + 29673\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {483153}{1280000} \, \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 = 34.75, size = 309, normalized size = 2.24 \begin {gather*} \begin {cases} \frac {40 i \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {10 x - 5}} - \frac {319 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {10 x - 5}} + \frac {8833 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{40 \sqrt {10 x - 5}} - \frac {171699 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{1600 \sqrt {10 x - 5}} - \frac {14641 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{6400 \sqrt {10 x - 5}} + \frac {483153 i \sqrt {x + \frac {3}{5}}}{64000 \sqrt {10 x - 5}} - \frac {483153 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{640000} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {483153 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{640000} - \frac {40 \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {5 - 10 x}} + \frac {319 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {5 - 10 x}} - \frac {8833 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{40 \sqrt {5 - 10 x}} + \frac {171699 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{1600 \sqrt {5 - 10 x}} + \frac {14641 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{6400 \sqrt {5 - 10 x}} - \frac {483153 \sqrt {x + \frac {3}{5}}}{64000 \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. 275 vs.
\(2 (99) = 198\).
time = 1.29, size = 275, normalized size = 1.99 \begin {gather*} \frac {1}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{480000} \, \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 {59}{120000} \, \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 {3}{1000} \, \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 {9}{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 {\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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