Optimal. Leaf size=157 \[ \frac {374762311 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {4122385421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200000 \sqrt {10}} \]
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Rubi [A]
time = 0.04, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {102, 158, 152,
52, 56, 222} \begin {gather*} \frac {4122385421 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200000 \sqrt {10}}-\frac {1}{20} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^3-\frac {333 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2}{2000}-\frac {7 (1-2 x)^{3/2} (5 x+3)^{3/2} (140652 x+231223)}{640000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {5 x+3}}{5120000}+\frac {374762311 \sqrt {1-2 x} \sqrt {5 x+3}}{51200000} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 158
Rule 222
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x} \, dx &=-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {1}{60} \int \left (-312-\frac {999 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx\\ &=-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}+\frac {\int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \left (\frac {77385}{2}+\frac {246141 x}{4}\right ) \, dx}{3000}\\ &=-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {34069301 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{1280000}\\ &=-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {374762311 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{10240000}\\ &=\frac {374762311 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {4122385421 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{102400000}\\ &=\frac {374762311 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {4122385421 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{51200000 \sqrt {5}}\\ &=\frac {374762311 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {4122385421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 88, normalized size = 0.56 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-1554368817-2453268875 x+4635831460 x^2+15564360800 x^3+20355408000 x^4+13167360000 x^5+3456000000 x^6\right )-4122385421 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{512000000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 138, normalized size = 0.88
method | result | size |
risch | \(-\frac {\left (691200000 x^{5}+2218752000 x^{4}+2739830400 x^{3}+1468973920 x^{2}+45781940 x -518122939\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{51200000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {4122385421 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1024000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(113\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (13824000000 x^{5} \sqrt {-10 x^{2}-x +3}+44375040000 x^{4} \sqrt {-10 x^{2}-x +3}+54796608000 x^{3} \sqrt {-10 x^{2}-x +3}+29379478400 x^{2} \sqrt {-10 x^{2}-x +3}+4122385421 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+915638800 x \sqrt {-10 x^{2}-x +3}-10362458780 \sqrt {-10 x^{2}-x +3}\right )}{1024000000 \sqrt {-10 x^{2}-x +3}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 104, normalized size = 0.66 \begin {gather*} -\frac {27}{20} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {8397}{2000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {853821}{160000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2300801}{640000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {34069301}{2560000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {4122385421}{1024000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {34069301}{51200000} \, \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 = 1.18, size = 82, normalized size = 0.52 \begin {gather*} \frac {1}{51200000} \, {\left (691200000 \, x^{5} + 2218752000 \, x^{4} + 2739830400 \, x^{3} + 1468973920 \, x^{2} + 45781940 \, x - 518122939\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {4122385421}{1024000000} \, \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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 356 vs.
\(2 (118) = 236\).
time = 0.61, size = 356, normalized size = 2.27 \begin {gather*} \frac {27}{2560000000} \, \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 {441}{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 {9}{50000} \, \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 {47}{5000} \, \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 {23}{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 {24}{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 = 15.54, size = 1056, normalized size = 6.73 \begin {gather*} \frac {4122385421\,\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 )}{256000000}-\frac {\frac {2152576553931\,{\left (\sqrt {1-2\,x}-1\right )}^5}{2441406250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {13952215351\,{\left (\sqrt {1-2\,x}-1\right )}^3}{244140625000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {792814579\,\left (\sqrt {1-2\,x}-1\right )}{3051757812500\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {3076029438827\,{\left (\sqrt {1-2\,x}-1\right )}^7}{976562500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {11028639133187\,{\left (\sqrt {1-2\,x}-1\right )}^9}{195312500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {3093660904217\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{15625000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}-\frac {3093660904217\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{6250000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}+\frac {11028639133187\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{12500000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}-\frac {3076029438827\,{\left (\sqrt {1-2\,x}-1\right )}^{17}}{10000000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{17}}-\frac {2152576553931\,{\left (\sqrt {1-2\,x}-1\right )}^{19}}{4000000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{19}}+\frac {13952215351\,{\left (\sqrt {1-2\,x}-1\right )}^{21}}{64000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{21}}+\frac {792814579\,{\left (\sqrt {1-2\,x}-1\right )}^{23}}{128000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{23}}+\frac {2228224\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{244140625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {5308416\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{48828125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}-\frac {389513216\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{244140625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {4601470976\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{244140625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {1286299648\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{244140625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {5601267712\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{48828125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {321574912\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{9765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {287591936\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{16}}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}-\frac {6086144\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{18}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{18}}+\frac {20736\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{20}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{20}}+\frac {2176\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{22}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{22}}}{\frac {24576\,{\left (\sqrt {1-2\,x}-1\right )}^2}{48828125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {67584\,{\left (\sqrt {1-2\,x}-1\right )}^4}{9765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {22528\,{\left (\sqrt {1-2\,x}-1\right )}^6}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {25344\,{\left (\sqrt {1-2\,x}-1\right )}^8}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {101376\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {59136\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {25344\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {1584\,{\left (\sqrt {1-2\,x}-1\right )}^{16}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {352\,{\left (\sqrt {1-2\,x}-1\right )}^{18}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{18}}+\frac {264\,{\left (\sqrt {1-2\,x}-1\right )}^{20}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{20}}+\frac {24\,{\left (\sqrt {1-2\,x}-1\right )}^{22}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{22}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{24}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{24}}+\frac {4096}{244140625}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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