Optimal. Leaf size=194 \[ \frac {13441711767 \sqrt {1-2 x} \sqrt {3+5 x}}{655360000}+\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {147858829437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{655360000 \sqrt {10}} \]
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Rubi [A]
time = 0.04, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {147858829437 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{655360000 \sqrt {10}}-\frac {3}{80} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac {9 (1-2 x)^{5/2} (16120 x+25043) (5 x+3)^{7/2}}{448000}-\frac {306029 (1-2 x)^{5/2} (5 x+3)^{5/2}}{256000}-\frac {3366319 (1-2 x)^{5/2} (5 x+3)^{3/2}}{819200}-\frac {37029509 (1-2 x)^{5/2} \sqrt {5 x+3}}{3276800}+\frac {407324599 (1-2 x)^{3/2} \sqrt {5 x+3}}{65536000}+\frac {13441711767 \sqrt {1-2 x} \sqrt {5 x+3}}{655360000} \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 (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx &=-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{80} \int \left (-389-\frac {1209 x}{2}\right ) (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {306029 \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx}{25600}\\ &=-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {3366319 \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx}{102400}\\ &=-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {111088527 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{1638400}\\ &=-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {407324599 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{6553600}\\ &=\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {13441711767 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{131072000}\\ &=\frac {13441711767 \sqrt {1-2 x} \sqrt {3+5 x}}{655360000}+\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {147858829437 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1310720000}\\ &=\frac {13441711767 \sqrt {1-2 x} \sqrt {3+5 x}}{655360000}+\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {147858829437 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{655360000 \sqrt {5}}\\ &=\frac {13441711767 \sqrt {1-2 x} \sqrt {3+5 x}}{655360000}+\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {147858829437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{655360000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 98, normalized size = 0.51 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (348124735143+164830264125 x-2940919441340 x^2-6478964823200 x^3-2775816752000 x^4+10623066880000 x^5+19953945600000 x^6+14294016000000 x^7+3870720000000 x^8\right )-1035011806059 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{45875200000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 172, normalized size = 0.89
method | result | size |
risch | \(\frac {\left (774144000000 x^{7}+2394316800000 x^{6}+2554199040000 x^{5}+592093952000 x^{4}-910419721600 x^{3}-749541131680 x^{2}-138459209260 x +116041578381\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4587520000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {147858829437 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{13107200000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(123\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-15482880000000 \sqrt {-10 x^{2}-x +3}\, x^{7}-47886336000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-51083980800000 x^{5} \sqrt {-10 x^{2}-x +3}-11841879040000 x^{4} \sqrt {-10 x^{2}-x +3}+18208394432000 x^{3} \sqrt {-10 x^{2}-x +3}+14990822633600 x^{2} \sqrt {-10 x^{2}-x +3}+1035011806059 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2769184185200 x \sqrt {-10 x^{2}-x +3}-2320831567620 \sqrt {-10 x^{2}-x +3}\right )}{91750400000 \sqrt {-10 x^{2}-x +3}}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 133, normalized size = 0.69 \begin {gather*} -\frac {27}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} - \frac {2187}{448} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} - \frac {100119}{17920} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {5653247}{1792000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3366319}{409600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {3366319}{8192000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1221973797}{32768000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {147858829437}{13107200000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1221973797}{655360000} \, \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 = 1.43, size = 92, normalized size = 0.47 \begin {gather*} -\frac {1}{4587520000} \, {\left (774144000000 \, x^{7} + 2394316800000 \, x^{6} + 2554199040000 \, x^{5} + 592093952000 \, x^{4} - 910419721600 \, x^{3} - 749541131680 \, x^{2} - 138459209260 \, x + 116041578381\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {147858829437}{13107200000} \, \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. 545 vs.
\(2 (143) = 286\).
time = 1.87, size = 545, normalized size = 2.81 \begin {gather*} -\frac {9}{229376000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 599\right )} {\left (5 \, x + 3\right )} + 175163\right )} {\left (5 \, x + 3\right )} - 4295993\right )} {\left (5 \, x + 3\right )} + 265620213\right )} {\left (5 \, x + 3\right )} - 2676516549\right )} {\left (5 \, x + 3\right )} + 35390483373\right )} {\left (5 \, x + 3\right )} - 164483997363\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 309625826895 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {297}{71680000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1851}{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 {13943}{960000000} \, \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 {883}{9600000} \, \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 {669}{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 {81}{100} \, \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 {108}{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 )}^3\,{\left (5\,x+3\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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