Optimal. Leaf size=121 \[ \frac {14443 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {158873 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800 \sqrt {10}} \]
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Rubi [A]
time = 0.02, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {158873 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800 \sqrt {10}}-\frac {3}{40} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {37}{160} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {1313 \sqrt {5 x+3} (1-2 x)^{3/2}}{1280}+\frac {14443 \sqrt {5 x+3} \sqrt {1-2 x}}{12800} \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 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx &=-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}-\frac {1}{40} \int \left (-178-\frac {555 x}{2}\right ) \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {1313}{320} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {14443 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2560}\\ &=\frac {14443 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {158873 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {14443 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {158873 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{12800 \sqrt {5}}\\ &=\frac {14443 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {158873 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 78, normalized size = 0.64 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-39981+865 x+267540 x^2+344800 x^3+144000 x^4\right )-158873 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{128000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 104, normalized size = 0.86
method | result | size |
risch | \(-\frac {\left (28800 x^{3}+51680 x^{2}+22500 x -13327\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{12800 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {158873 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (576000 x^{3} \sqrt {-10 x^{2}-x +3}+1033600 x^{2} \sqrt {-10 x^{2}-x +3}+158873 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+450000 x \sqrt {-10 x^{2}-x +3}-266540 \sqrt {-10 x^{2}-x +3}\right )}{256000 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 70, normalized size = 0.58 \begin {gather*} -\frac {9}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {61}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1313}{640} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {158873}{256000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1313}{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 = 0.80, size = 72, normalized size = 0.60 \begin {gather*} \frac {1}{12800} \, {\left (28800 \, x^{3} + 51680 \, x^{2} + 22500 \, x - 13327\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {158873}{256000} \, \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 = 228.97, size = 376, normalized size = 3.11 \begin {gather*} - \frac {49 \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 )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} + \frac {21 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{4} - \frac {9 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\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 {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\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} \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.
\(2 (88) = 176\).
time = 0.54, size = 203, normalized size = 1.68 \begin {gather*} \frac {3}{640000} \, \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 {29}{40000} \, \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 {7}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.53, size = 708, normalized size = 5.85 \begin {gather*} \frac {158873\,\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 )}{64000}-\frac {\frac {38070349\,{\left (\sqrt {1-2\,x}-1\right )}^5}{15625000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {3947\,{\left (\sqrt {1-2\,x}-1\right )}^3}{7812500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {148327\,\left (\sqrt {1-2\,x}-1\right )}{19531250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {88735647\,{\left (\sqrt {1-2\,x}-1\right )}^7}{6250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {88735647\,{\left (\sqrt {1-2\,x}-1\right )}^9}{2500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {38070349\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{1000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {3947\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{80000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}+\frac {148327\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{32000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {16384\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {137728\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1014272\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {364288\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {253568\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}-\frac {8608\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {256\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{25\,{\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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