Optimal. Leaf size=143 \[ \frac {498883 \sqrt {1-2 x} \sqrt {3+5 x}}{640000}+\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {5487713 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{640000 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 143, 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 = {92, 81, 52, 56,
222} \begin {gather*} \frac {5487713 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640000 \sqrt {10}}-\frac {3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac {4123 \sqrt {5 x+3} (1-2 x)^{5/2}}{9600}+\frac {45353 \sqrt {5 x+3} (1-2 x)^{3/2}}{192000}+\frac {498883 \sqrt {5 x+3} \sqrt {1-2 x}}{640000} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 81
Rule 92
Rule 222
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx &=-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}-\frac {1}{50} \int \left (-182-\frac {567 x}{2}\right ) (1-2 x)^{3/2} \sqrt {3+5 x} \, dx\\ &=-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {4123 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{1600}\\ &=-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {45353 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{19200}\\ &=\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {498883 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{128000}\\ &=\frac {498883 \sqrt {1-2 x} \sqrt {3+5 x}}{640000}+\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {5487713 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1280000}\\ &=\frac {498883 \sqrt {1-2 x} \sqrt {3+5 x}}{640000}+\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {5487713 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{640000 \sqrt {5}}\\ &=\frac {498883 \sqrt {1-2 x} \sqrt {3+5 x}}{640000}+\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {5487713 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{640000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 83, normalized size = 0.58 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (1146303-12706875 x-33786140 x^2+6152800 x^3+57168000 x^4+34560000 x^5\right )-16463139 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{19200000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 121, normalized size = 0.85
method | result | size |
risch | \(\frac {\left (6912000 x^{4}+7286400 x^{3}-3141280 x^{2}-4872460 x +382101\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1920000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {5487713 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{12800000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-138240000 x^{4} \sqrt {-10 x^{2}-x +3}-145728000 x^{3} \sqrt {-10 x^{2}-x +3}+62825600 x^{2} \sqrt {-10 x^{2}-x +3}+16463139 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+97449200 x \sqrt {-10 x^{2}-x +3}-7642020 \sqrt {-10 x^{2}-x +3}\right )}{38400000 \sqrt {-10 x^{2}-x +3}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 87, normalized size = 0.61 \begin {gather*} \frac {9}{25} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {687}{2000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2159}{24000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {45353}{32000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {5487713}{12800000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {45353}{640000} \, \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 = 1.57, size = 77, normalized size = 0.54 \begin {gather*} -\frac {1}{1920000} \, {\left (6912000 \, x^{4} + 7286400 \, x^{3} - 3141280 \, x^{2} - 4872460 \, x + 382101\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {5487713}{12800000} \, \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 = 34.52, size = 571, normalized size = 3.99 \begin {gather*} \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}\: \sqrt {5 x + 3} > - \frac {\sqrt {22}}{2} \wedge \sqrt {5 x + 3} < \frac {\sqrt {22}}{2} \end {cases}\right )}{3125} + \frac {128 \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 )}{3125} + \frac {174 \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 )}{3125} - \frac {36 \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 )}{3125} \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. 275 vs.
\(2 (104) = 208\).
time = 1.63, size = 275, normalized size = 1.92 \begin {gather*} -\frac {3}{32000000} \, \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 {43}{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 {1}{4800} \, \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 {2}{125} \, \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 )} \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 )}^{3/2}\,{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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