Optimal. Leaf size=106 \[ \frac {47761 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {525371 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000 \sqrt {10}} \]
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Rubi [A]
time = 0.02, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {525371 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64000 \sqrt {10}}-\frac {3}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2-\frac {21 (1-2 x)^{3/2} \sqrt {5 x+3} (216 x+335)}{6400}+\frac {47761 \sqrt {1-2 x} \sqrt {5 x+3}}{64000} \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 \frac {\sqrt {1-2 x} (2+3 x)^3}{\sqrt {3+5 x}} \, dx &=-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{40} \int \frac {\left (-175-\frac {567 x}{2}\right ) \sqrt {1-2 x} (2+3 x)}{\sqrt {3+5 x}} \, dx\\ &=-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {47761 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{12800}\\ &=\frac {47761 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {525371 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{128000}\\ &=\frac {47761 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {525371 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{64000 \sqrt {5}}\\ &=\frac {47761 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {525371 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000 \sqrt {10}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 78, normalized size = 0.74 \begin {gather*} \frac {2 \sqrt {5-10 x} \sqrt {3+5 x} \left (-41789+76140 x+162720 x^2+86400 x^3\right )+525371 i \sqrt {2} \log \left (\sqrt {5-10 x}-i \sqrt {6+10 x}\right )}{128000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 104, normalized size = 0.98
method | result | size |
risch | \(-\frac {\left (86400 x^{3}+162720 x^{2}+76140 x -41789\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{64000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {525371 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1728000 x^{3} \sqrt {-10 x^{2}-x +3}+3254400 x^{2} \sqrt {-10 x^{2}-x +3}+525371 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1522800 x \sqrt {-10 x^{2}-x +3}-835780 \sqrt {-10 x^{2}-x +3}\right )}{1280000 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 73, normalized size = 0.69 \begin {gather*} -\frac {27}{200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {525371}{1280000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {963}{4000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {21663}{16000} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {887}{12800} \, \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 = 0.73, size = 72, normalized size = 0.68 \begin {gather*} \frac {1}{64000} \, {\left (86400 \, x^{3} + 162720 \, x^{2} + 76140 \, x - 41789\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {525371}{1280000} \, \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 = 18.20, size = 544, normalized size = 5.13 \begin {gather*} - \frac {343 \sqrt {2} \left (\begin {cases} \frac {11 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{2}\right )}{25} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} + \frac {441 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (\frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{968} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} - \frac {189 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} + \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{1936} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} + \frac {27 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{3872} + \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{3125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} \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. 203 vs.
\(2 (79) = 158\).
time = 1.42, size = 203, normalized size = 1.92 \begin {gather*} \frac {9}{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 {9}{20000} \, \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 {9}{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 {4}{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 [B]
time = 11.18, size = 708, normalized size = 6.68 \begin {gather*} \frac {525371\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{320000}-\frac {\frac {204911\,{\left (\sqrt {1-2\,x}-1\right )}^3}{39062500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {498629\,\left (\sqrt {1-2\,x}-1\right )}{97656250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {116866071\,{\left (\sqrt {1-2\,x}-1\right )}^5}{78125000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {264903917\,{\left (\sqrt {1-2\,x}-1\right )}^7}{31250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {264903917\,{\left (\sqrt {1-2\,x}-1\right )}^9}{12500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {116866071\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{5000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}-\frac {204911\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{400000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}+\frac {498629\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{160000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {2048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {86016\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {623616\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {1223168\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {155904\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}-\frac {5376\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {32\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}}{\frac {1024\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {224\,{\left (\sqrt {1-2\,x}-1\right )}^8}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {448\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {112\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {16\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{16}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {256}{390625}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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