Optimal. Leaf size=165 \[ -\frac {1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac {7189 \sqrt {5 x+3} (1-2 x)^{7/2}}{32000}+\frac {79079 \sqrt {5 x+3} (1-2 x)^{5/2}}{960000}+\frac {869869 \sqrt {5 x+3} (1-2 x)^{3/2}}{3840000}+\frac {9568559 \sqrt {5 x+3} \sqrt {1-2 x}}{12800000}+\frac {105254149 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800000 \sqrt {10}} \]
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Rubi [A] time = 0.05, antiderivative size = 165, 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 = {90, 80, 50, 54, 216} \[ -\frac {1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac {7189 \sqrt {5 x+3} (1-2 x)^{7/2}}{32000}+\frac {79079 \sqrt {5 x+3} (1-2 x)^{5/2}}{960000}+\frac {869869 \sqrt {5 x+3} (1-2 x)^{3/2}}{3840000}+\frac {9568559 \sqrt {5 x+3} \sqrt {1-2 x}}{12800000}+\frac {105254149 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 90
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx &=-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}-\frac {1}{60} \int \left (-186-\frac {579 x}{2}\right ) (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\\ &=-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {7189 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{4000}\\ &=-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {79079 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{64000}\\ &=\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {869869 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{384000}\\ &=\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {9568559 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2560000}\\ &=\frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{25600000}\\ &=\frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{12800000 \sqrt {5}}\\ &=\frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 0.51 \[ \frac {315762447 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (460800000 x^6-41472000 x^5-568838400 x^4+114662080 x^3+266309720 x^2-83915726 x-9303927\right )}{384000000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 82, normalized size = 0.50 \[ \frac {1}{38400000} \, {\left (230400000 \, x^{5} + 94464000 \, x^{4} - 237187200 \, x^{3} - 61262560 \, x^{2} + 102523580 \, x + 9303927\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {105254149}{256000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.32, size = 356, normalized size = 2.16 \[ \frac {3}{640000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {7}{40000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {79}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {89}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{250} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {6}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 138, normalized size = 0.84 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (4608000000 \sqrt {-10 x^{2}-x +3}\, x^{5}+1889280000 \sqrt {-10 x^{2}-x +3}\, x^{4}-4743744000 \sqrt {-10 x^{2}-x +3}\, x^{3}-1225251200 \sqrt {-10 x^{2}-x +3}\, x^{2}+2050471600 \sqrt {-10 x^{2}-x +3}\, x +315762447 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+186078540 \sqrt {-10 x^{2}-x +3}\right )}{768000000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 104, normalized size = 0.63 \[ -\frac {3}{5} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {93}{500} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {18251}{40000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {27893}{480000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {869869}{640000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {105254149}{256000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {869869}{12800000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 129.59, size = 695, normalized size = 4.21 \[ \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}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{15625} + \frac {1364 \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 {1658 \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 {744 \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 {72 \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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