Optimal. Leaf size=138 \[ -\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac {251}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {2761 (1-2 x)^{3/2} (5 x+3)^{3/2}}{1920}-\frac {30371 (1-2 x)^{3/2} \sqrt {5 x+3}}{5120}+\frac {334081 \sqrt {1-2 x} \sqrt {5 x+3}}{51200}+\frac {3674891 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}} \]
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Rubi [A] time = 0.04, 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 = {80, 50, 54, 216} \[ -\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac {251}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {2761 (1-2 x)^{3/2} (5 x+3)^{3/2}}{1920}-\frac {30371 (1-2 x)^{3/2} \sqrt {5 x+3}}{5120}+\frac {334081 \sqrt {1-2 x} \sqrt {5 x+3}}{51200}+\frac {3674891 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 216
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 \operatorname {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 = 0.09, size = 79, normalized size = 0.57 \[ \frac {11024673 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (4608000 x^5+8505600 x^4+3215680 x^3-2873560 x^2-3226514 x+1254087\right )}{1536000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 77, normalized size = 0.56 \[ \frac {1}{153600} \, {\left (2304000 \, x^{4} + 5404800 \, x^{3} + 4310240 \, x^{2} + 718340 \, x - 1254087\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3674891}{1024000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.09, size = 275, normalized size = 1.99 \[ \frac {1}{2560000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {37}{384000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {57}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {351}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 121, normalized size = 0.88 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (46080000 \sqrt {-10 x^{2}-x +3}\, x^{4}+108096000 \sqrt {-10 x^{2}-x +3}\, x^{3}+86204800 \sqrt {-10 x^{2}-x +3}\, x^{2}+14366800 \sqrt {-10 x^{2}-x +3}\, x +11024673 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-25081740 \sqrt {-10 x^{2}-x +3}\right )}{3072000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 87, normalized size = 0.63 \[ -\frac {3}{2} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {539}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1121}{384} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30371}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3674891}{1024000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30371}{51200} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \sqrt {1-2\,x}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 73.28, size = 488, normalized size = 3.54 \[ - \frac {847 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{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} \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{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} \left (20 x + 1\right )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \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}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{16} + \frac {75 \sqrt {2} \left (\begin {cases} \frac {161051 \sqrt {5} \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} \left (20 x + 1\right )}{7744} - \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \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}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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