Optimal. Leaf size=172 \[ \frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000000 \sqrt {10}} \]
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Rubi [A]
time = 0.04, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {102, 152, 52,
56, 222} \begin {gather*} \frac {3735929329 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{256000000 \sqrt {10}}-\frac {3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {3 (5 x+3)^{3/2} (26700 x+33857) (1-2 x)^{7/2}}{280000}-\frac {255169 \sqrt {5 x+3} (1-2 x)^{7/2}}{640000}+\frac {2806859 \sqrt {5 x+3} (1-2 x)^{5/2}}{19200000}+\frac {30875449 \sqrt {5 x+3} (1-2 x)^{3/2}}{76800000}+\frac {339629939 \sqrt {5 x+3} \sqrt {1-2 x}}{256000000} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 222
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x} \, dx &=-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {1}{70} \int \left (-253-\frac {801 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx\\ &=-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {255169 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{80000}\\ &=-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {2806859 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{1280000}\\ &=\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {30875449 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{7680000}\\ &=\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {339629939 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{51200000}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{512000000}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{256000000 \sqrt {5}}\\ &=\frac {339629939 \sqrt {1-2 x} \sqrt {3+5 x}}{256000000}+\frac {30875449 (1-2 x)^{3/2} \sqrt {3+5 x}}{76800000}+\frac {2806859 (1-2 x)^{5/2} \sqrt {3+5 x}}{19200000}-\frac {255169 (1-2 x)^{7/2} \sqrt {3+5 x}}{640000}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac {3735929329 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 93, normalized size = 0.54 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-2037835593+86060710125 x+178815054340 x^2-205753776800 x^3-582885168000 x^4+30032640000 x^5+736128000000 x^6+414720000000 x^7\right )-78454515909 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{53760000000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 155, normalized size = 0.90
method | result | size |
risch | \(-\frac {\left (82944000000 x^{6}+97459200000 x^{5}-52468992000 x^{4}-85095638400 x^{3}+9906627680 x^{2}+29819034260 x -679278531\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5376000000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {3735929329 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5120000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(118\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1658880000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+1949184000000 x^{5} \sqrt {-10 x^{2}-x +3}-1049379840000 x^{4} \sqrt {-10 x^{2}-x +3}-1701912768000 x^{3} \sqrt {-10 x^{2}-x +3}+198132553600 x^{2} \sqrt {-10 x^{2}-x +3}+78454515909 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+596380685200 x \sqrt {-10 x^{2}-x +3}-13585570620 \sqrt {-10 x^{2}-x +3}\right )}{107520000000 \sqrt {-10 x^{2}-x +3}}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.82, size = 121, normalized size = 0.70 \begin {gather*} -\frac {54}{35} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {1161}{700} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {47529}{70000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {5697497}{5600000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {5531929}{67200000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30875449}{12800000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3735929329}{5120000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30875449}{256000000} \, \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 = 0.68, size = 87, normalized size = 0.51 \begin {gather*} \frac {1}{5376000000} \, {\left (82944000000 \, x^{6} + 97459200000 \, x^{5} - 52468992000 \, x^{4} - 85095638400 \, x^{3} + 9906627680 \, x^{2} + 29819034260 \, x - 679278531\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3735929329}{5120000000} \, \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 = 110.18, size = 1051, normalized size = 6.11 \begin {gather*} \frac {242 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (- \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{121} + \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}\right )}{32} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{78125} + \frac {418 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (- \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{8} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{15625} + \frac {46 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \left (- \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{16} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{625} + \frac {846 \sqrt {5} \left (\begin {cases} \frac {161051 \sqrt {2} \cdot \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} - \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{7744} - \frac {3 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{256}\right )}{32} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{15625} - \frac {432 \sqrt {5} \left (\begin {cases} \frac {1771561 \sqrt {2} \cdot \left (\frac {4 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} + \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (- 20 x - 1\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}{85034928} - \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{15488} - \frac {13 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{14992384} + \frac {21 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{1024}\right )}{64} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{15625} + \frac {216 \sqrt {5} \left (\begin {cases} \frac {19487171 \sqrt {2} \left (- \frac {4 \sqrt {2} \left (5 - 10 x\right )^{\frac {7}{2}} \left (5 x + 3\right )^{\frac {7}{2}}}{136410197} + \frac {6 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} + \frac {5 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (- 20 x - 1\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}{170069856} - \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{30976} - \frac {25 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{29984768} + \frac {33 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{2048}\right )}{128} & \text {for}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{78125} \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. 446 vs.
\(2 (127) = 254\).
time = 0.56, size = 446, normalized size = 2.59 \begin {gather*} \frac {9}{89600000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{400000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {33}{320000000} \, \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 {17}{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 {77}{60000} \, \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 {13}{500} \, \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 {12}{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 {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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