Optimal. Leaf size=128 \[ \frac {3558401 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000}-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {39142411 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1280000 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 128, 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 = {102, 152, 52,
56, 222} \begin {gather*} \frac {39142411 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1280000 \sqrt {10}}-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac {21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac {323491 (1-2 x)^{3/2} \sqrt {5 x+3}}{128000}+\frac {3558401 \sqrt {1-2 x} \sqrt {5 x+3}}{1280000} \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 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx &=-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {1}{50} \int \left (-245-\frac {777 x}{2}\right ) \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx\\ &=-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {323491 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{32000}\\ &=-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {3558401 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{256000}\\ &=\frac {3558401 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000}-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {39142411 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2560000}\\ &=\frac {3558401 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000}-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {39142411 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1280000 \sqrt {5}}\\ &=\frac {3558401 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000}-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {39142411 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1280000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 83, normalized size = 0.65 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-12847047-12404125 x+59852860 x^2+126832800 x^3+107568000 x^4+34560000 x^5\right )-39142411 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{12800000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 121, normalized size = 0.95
method | result | size |
risch | \(-\frac {\left (6912000 x^{4}+17366400 x^{3}+14946720 x^{2}+3002540 x -4282349\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1280000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {39142411 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{25600000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (138240000 x^{4} \sqrt {-10 x^{2}-x +3}+347328000 x^{3} \sqrt {-10 x^{2}-x +3}+298934400 x^{2} \sqrt {-10 x^{2}-x +3}+39142411 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+60050800 x \sqrt {-10 x^{2}-x +3}-85646980 \sqrt {-10 x^{2}-x +3}\right )}{25600000 \sqrt {-10 x^{2}-x +3}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 87, normalized size = 0.68 \begin {gather*} -\frac {27}{50} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {5211}{4000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {19191}{16000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {323491}{64000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {39142411}{25600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {323491}{1280000} \, \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 = 1.19, size = 77, normalized size = 0.60 \begin {gather*} \frac {1}{1280000} \, {\left (6912000 \, x^{4} + 17366400 \, x^{3} + 14946720 \, x^{2} + 3002540 \, x - 4282349\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {39142411}{25600000} \, \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. 275 vs.
\(2 (95) = 190\).
time = 0.59, size = 275, normalized size = 2.15 \begin {gather*} \frac {9}{64000000} \, \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 {117}{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 {57}{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 {37}{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 {12}{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 = 13.16, size = 881, normalized size = 6.88 \begin {gather*} \frac {\frac {22297589\,\left (\sqrt {1-2\,x}-1\right )}{12207031250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {369826027\,{\left (\sqrt {1-2\,x}-1\right )}^3}{4882812500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {4945417109\,{\left (\sqrt {1-2\,x}-1\right )}^5}{2441406250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {1598593169\,{\left (\sqrt {1-2\,x}-1\right )}^7}{195312500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {914901953\,{\left (\sqrt {1-2\,x}-1\right )}^9}{156250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {914901953\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{62500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}-\frac {1598593169\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{12500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}+\frac {4945417109\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{25000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}-\frac {369826027\,{\left (\sqrt {1-2\,x}-1\right )}^{17}}{8000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{17}}-\frac {22297589\,{\left (\sqrt {1-2\,x}-1\right )}^{19}}{3200000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{19}}-\frac {8192\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {90112\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{9765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {8316928\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{9765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}-\frac {216457216\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{9765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {58587136\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{1953125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}-\frac {54114304\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {519808\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {1408\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{16}}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}-\frac {32\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{18}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{18}}}{\frac {1024\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {2304\,{\left (\sqrt {1-2\,x}-1\right )}^4}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {3072\,{\left (\sqrt {1-2\,x}-1\right )}^6}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {2688\,{\left (\sqrt {1-2\,x}-1\right )}^8}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {8064\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {672\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {192\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {36\,{\left (\sqrt {1-2\,x}-1\right )}^{16}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {4\,{\left (\sqrt {1-2\,x}-1\right )}^{18}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{18}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{20}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{20}}+\frac {1024}{9765625}}+\frac {39142411\,\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 )}{6400000} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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