Optimal. Leaf size=150 \[ -\frac {1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {(5 x+3)^{3/2} (63120 x+88987) (1-2 x)^{5/2}}{160000}-\frac {339983 \sqrt {5 x+3} (1-2 x)^{5/2}}{384000}+\frac {3739813 \sqrt {5 x+3} (1-2 x)^{3/2}}{7680000}+\frac {41137943 \sqrt {5 x+3} \sqrt {1-2 x}}{25600000}+\frac {452517373 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25600000 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac {1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {(5 x+3)^{3/2} (63120 x+88987) (1-2 x)^{5/2}}{160000}-\frac {339983 \sqrt {5 x+3} (1-2 x)^{5/2}}{384000}+\frac {3739813 \sqrt {5 x+3} (1-2 x)^{3/2}}{7680000}+\frac {41137943 \sqrt {5 x+3} \sqrt {1-2 x}}{25600000}+\frac {452517373 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25600000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx &=-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {1}{60} \int \left (-249-\frac {789 x}{2}\right ) (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x} \, dx\\ &=-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {339983 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{64000}\\ &=-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {3739813 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{768000}\\ &=\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {41137943 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{5120000}\\ &=\frac {41137943 \sqrt {1-2 x} \sqrt {3+5 x}}{25600000}+\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {452517373 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{51200000}\\ &=\frac {41137943 \sqrt {1-2 x} \sqrt {3+5 x}}{25600000}+\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {452517373 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25600000 \sqrt {5}}\\ &=\frac {41137943 \sqrt {1-2 x} \sqrt {3+5 x}}{25600000}+\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {452517373 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25600000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 84, normalized size = 0.56 \[ \frac {10 \sqrt {5 x+3} \left (1382400000 x^6+1810944000 x^5-634003200 x^4-1555668160 x^3-125580440 x^2+537385502 x-81405921\right )+1357552119 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{768000000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 82, normalized size = 0.55 \[ -\frac {1}{76800000} \, {\left (691200000 \, x^{5} + 1251072000 \, x^{4} + 308534400 \, x^{3} - 623566880 \, x^{2} - 374573660 \, x + 81405921\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {452517373}{512000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 356, normalized size = 2.37 \[ -\frac {9}{1280000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {189}{320000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {111}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {23}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{20} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {12}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 138, normalized size = 0.92 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-13824000000 \sqrt {-10 x^{2}-x +3}\, x^{5}-25021440000 \sqrt {-10 x^{2}-x +3}\, x^{4}-6170688000 \sqrt {-10 x^{2}-x +3}\, x^{3}+12471337600 \sqrt {-10 x^{2}-x +3}\, x^{2}+7491473200 \sqrt {-10 x^{2}-x +3}\, x +1357552119 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1628118420 \sqrt {-10 x^{2}-x +3}\right )}{1536000000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 104, normalized size = 0.69 \[ \frac {9}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {1539}{1000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {41427}{80000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {385939}{960000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3739813}{1280000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {452517373}{512000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3739813}{25600000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 124.09, size = 695, normalized size = 4.63 \[ \frac {22 \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}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{15625} + \frac {194 \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}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{15625} + \frac {558 \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}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{15625} + \frac {486 \sqrt {5} \left (\begin {cases} \frac {161051 \sqrt {2} \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}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{15625} - \frac {108 \sqrt {5} \left (\begin {cases} \frac {1771561 \sqrt {2} \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}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{15625} \]
Verification of antiderivative is not currently implemented for this CAS.
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