Optimal. Leaf size=116 \[ -\frac {3}{40} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {181}{480} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac {1991 (1-2 x)^{3/2} \sqrt {5 x+3}}{1280}+\frac {21901 \sqrt {1-2 x} \sqrt {5 x+3}}{12800}+\frac {240911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac {3}{40} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {181}{480} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac {1991 (1-2 x)^{3/2} \sqrt {5 x+3}}{1280}+\frac {21901 \sqrt {1-2 x} \sqrt {5 x+3}}{12800}+\frac {240911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800 \sqrt {10}} \]
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)^{3/2} \, dx &=-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {181}{80} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1991}{320} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {21901 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2560}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{12800 \sqrt {5}}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 74, normalized size = 0.64 \[ \frac {722733 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (288000 x^4+347200 x^3-46840 x^2-226154 x+63387\right )}{384000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 72, normalized size = 0.62 \[ \frac {1}{38400} \, {\left (144000 \, x^{3} + 245600 \, x^{2} + 99380 \, x - 63387\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {240911}{256000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.00, size = 203, normalized size = 1.75 \[ \frac {1}{128000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {7}{6000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {87}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 104, normalized size = 0.90 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (2880000 \sqrt {-10 x^{2}-x +3}\, x^{3}+4912000 \sqrt {-10 x^{2}-x +3}\, x^{2}+1987600 \sqrt {-10 x^{2}-x +3}\, x +722733 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1267740 \sqrt {-10 x^{2}-x +3}\right )}{768000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 70, normalized size = 0.60 \[ -\frac {3}{8} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {289}{480} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1991}{640} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {240911}{256000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1991}{12800} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \sqrt {1-2\,x}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.59, size = 314, normalized size = 2.71 \[ - \frac {77 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{8} + \frac {17 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{2} - \frac {15 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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