Optimal. Leaf size=209 \[ \frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {1729615 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^3}+\frac {302171615 \sqrt {1-2 x} \sqrt {3+5 x}}{338688 (2+3 x)^2}+\frac {31603880465 \sqrt {1-2 x} \sqrt {3+5 x}}{4741632 (2+3 x)}-\frac {13391796605 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \]
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Rubi [A]
time = 0.05, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 154, 156,
12, 95, 210} \begin {gather*} -\frac {13391796605 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}}+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac {31603880465 \sqrt {5 x+3} \sqrt {1-2 x}}{4741632 (3 x+2)}+\frac {302171615 \sqrt {5 x+3} \sqrt {1-2 x}}{338688 (3 x+2)^2}+\frac {1729615 \sqrt {5 x+3} \sqrt {1-2 x}}{12096 (3 x+2)^3}+\frac {21199 \sqrt {5 x+3} \sqrt {1-2 x}}{864 (3 x+2)^4}+\frac {497 \sqrt {5 x+3} \sqrt {1-2 x}}{108 (3 x+2)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 154
Rule 156
Rule 210
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 \sqrt {3+5 x}} \, dx &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {\left (\frac {487}{2}-256 x\right ) \sqrt {1-2 x}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}-\frac {1}{270} \int \frac {-\frac {121615}{4}+47140 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}-\frac {\int \frac {-\frac {30857925}{8}+\frac {11129475 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{7560}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {1729615 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^3}-\frac {\int \frac {-\frac {5733084525}{16}+\frac {908047875 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{158760}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {1729615 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^3}+\frac {302171615 \sqrt {1-2 x} \sqrt {3+5 x}}{338688 (2+3 x)^2}-\frac {\int \frac {-\frac {683095555275}{32}+\frac {158640097875 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{2222640}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {1729615 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^3}+\frac {302171615 \sqrt {1-2 x} \sqrt {3+5 x}}{338688 (2+3 x)^2}+\frac {31603880465 \sqrt {1-2 x} \sqrt {3+5 x}}{4741632 (2+3 x)}-\frac {\int -\frac {37965743375175}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{15558480}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {1729615 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^3}+\frac {302171615 \sqrt {1-2 x} \sqrt {3+5 x}}{338688 (2+3 x)^2}+\frac {31603880465 \sqrt {1-2 x} \sqrt {3+5 x}}{4741632 (2+3 x)}+\frac {13391796605 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {1729615 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^3}+\frac {302171615 \sqrt {1-2 x} \sqrt {3+5 x}}{338688 (2+3 x)^2}+\frac {31603880465 \sqrt {1-2 x} \sqrt {3+5 x}}{4741632 (2+3 x)}+\frac {13391796605 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616}\\ &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {497 \sqrt {1-2 x} \sqrt {3+5 x}}{108 (2+3 x)^5}+\frac {21199 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {1729615 \sqrt {1-2 x} \sqrt {3+5 x}}{12096 (2+3 x)^3}+\frac {302171615 \sqrt {1-2 x} \sqrt {3+5 x}}{338688 (2+3 x)^2}+\frac {31603880465 \sqrt {1-2 x} \sqrt {3+5 x}}{4741632 (2+3 x)}-\frac {13391796605 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 91, normalized size = 0.44 \begin {gather*} \frac {1331 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (120549503808+890768460368 x+2634024494432 x^2+3896029345680 x^3+2882422865340 x^4+853304772555 x^5\right )}{1331 (2+3 x)^6}-30184365 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{3687936} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs.
\(2(164)=328\).
time = 0.11, size = 346, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (853304772555 x^{5}+2882422865340 x^{4}+3896029345680 x^{3}+2634024494432 x^{2}+890768460368 x +120549503808\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{526848 \left (2+3 x \right )^{6} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {13391796605 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (29287859175135 \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) \sqrt {7}\, x^{6}+117151436700540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+195252394500900 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+11946266815770 x^{5} \sqrt {-10 x^{2}-x +3}+173557684000800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+40353920114760 x^{4} \sqrt {-10 x^{2}-x +3}+86778842000400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+54544410839520 x^{3} \sqrt {-10 x^{2}-x +3}+23141024533440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +36876342922048 x^{2} \sqrt {-10 x^{2}-x +3}+2571224948160 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+12470758445152 x \sqrt {-10 x^{2}-x +3}+1687693053312 \sqrt {-10 x^{2}-x +3}\right )}{7375872 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 230, normalized size = 1.10 \begin {gather*} \frac {13391796605}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{54 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {469 \, \sqrt {-10 \, x^{2} - x + 3}}{108 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {21199 \, \sqrt {-10 \, x^{2} - x + 3}}{864 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1729615 \, \sqrt {-10 \, x^{2} - x + 3}}{12096 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {302171615 \, \sqrt {-10 \, x^{2} - x + 3}}{338688 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {31603880465 \, \sqrt {-10 \, x^{2} - x + 3}}{4741632 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.72, size = 146, normalized size = 0.70 \begin {gather*} -\frac {40175389815 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (853304772555 \, x^{5} + 2882422865340 \, x^{4} + 3896029345680 \, x^{3} + 2634024494432 \, x^{2} + 890768460368 \, x + 120549503808\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7375872 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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. 484 vs.
\(2 (164) = 328\).
time = 0.85, size = 484, normalized size = 2.32 \begin {gather*} \frac {2678359321}{4917248} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {6655 \, \sqrt {10} {\left (20305527 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 17887837240 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 7599643632000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 1749282956467200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 210267345272320000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {10389680589926400000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {41558722359705600000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{263424 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \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 = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^7\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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