Optimal. Leaf size=138 \[ \frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}} \]
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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 = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222}
\begin {gather*} \frac {3674891 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}}-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac {251}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {2761 (1-2 x)^{3/2} (5 x+3)^{3/2}}{1920}-\frac {30371 (1-2 x)^{3/2} \sqrt {5 x+3}}{5120}+\frac {334081 \sqrt {1-2 x} \sqrt {5 x+3}}{51200} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 81
Rule 222
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx &=-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {251}{100} \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx\\ &=-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {2761}{320} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {30371 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{1280}\\ &=-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {334081 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{10240}\\ &=\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{102400}\\ &=\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{51200 \sqrt {5}}\\ &=\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 83, normalized size = 0.60 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-3762261-4115415 x+16522420 x^2+37765600 x^3+33936000 x^4+11520000 x^5\right )-11024673 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1536000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 121, normalized size = 0.88
method | result | size |
risch | \(-\frac {\left (2304000 x^{4}+5404800 x^{3}+4310240 x^{2}+718340 x -1254087\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{153600 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {3674891 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1024000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (46080000 x^{4} \sqrt {-10 x^{2}-x +3}+108096000 x^{3} \sqrt {-10 x^{2}-x +3}+86204800 x^{2} \sqrt {-10 x^{2}-x +3}+11024673 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+14366800 x \sqrt {-10 x^{2}-x +3}-25081740 \sqrt {-10 x^{2}-x +3}\right )}{3072000 \sqrt {-10 x^{2}-x +3}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 87, normalized size = 0.63 \begin {gather*} -\frac {3}{2} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {539}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1121}{384} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30371}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3674891}{1024000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30371}{51200} \, \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 = 0.76, size = 77, normalized size = 0.56 \begin {gather*} \frac {1}{153600} \, {\left (2304000 \, x^{4} + 5404800 \, x^{3} + 4310240 \, x^{2} + 718340 \, x - 1254087\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3674891}{1024000} \, \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 [A]
time = 39.66, size = 570, normalized size = 4.13 \begin {gather*} - \frac {847 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{16} + \frac {1133 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{16} - \frac {505 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{16} + \frac {75 \sqrt {2} \left (\begin {cases} \frac {161051 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{322102} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{7744} - \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{256}\right )}{3125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{16} \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 = 2.19, size = 275, normalized size = 1.99 \begin {gather*} \frac {1}{2560000} \, \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 {37}{384000} \, \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 {57}{8000} \, \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 {351}{2000} \, \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}{25} \, \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 (3\,x+2\right )\,{\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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