Integrand size = 26, antiderivative size = 224 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=-\frac {754386765 \sqrt {1-2 x}}{6272 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {270667969 \sqrt {1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac {20529722435 \sqrt {1-2 x}}{18816 \sqrt {3+5 x}}-\frac {46975917593 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}} \]
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Time = 0.07 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {100, 154, 156, 157, 12, 95, 210} \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=-\frac {46975917593 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}}+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}+\frac {20529722435 \sqrt {1-2 x}}{18816 \sqrt {5 x+3}}+\frac {270667969 \sqrt {1-2 x}}{18816 (3 x+2) (5 x+3)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (3 x+2)^3 (5 x+3)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (3 x+2)^4 (5 x+3)^{3/2}}-\frac {754386765 \sqrt {1-2 x}}{6272 (5 x+3)^{3/2}} \]
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Rule 12
Rule 95
Rule 100
Rule 154
Rule 156
Rule 157
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1}{15} \int \frac {\left (\frac {561}{2}-330 x\right ) \sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}-\frac {1}{180} \int \frac {-\frac {177903}{4}+72435 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {57169035}{8}+11131890 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx}{3780} \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {14435442945}{16}+\frac {5244260175 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx}{52920} \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {270667969 \sqrt {1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {2658860407665}{32}+\frac {426302051175 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{370440} \\ & = -\frac {754386765 \sqrt {1-2 x}}{6272 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {270667969 \sqrt {1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac {\int \frac {-\frac {300110253247035}{64}+\frac {70576653799575 x}{16}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{6112260} \\ & = -\frac {754386765 \sqrt {1-2 x}}{6272 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {270667969 \sqrt {1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac {20529722435 \sqrt {1-2 x}}{18816 \sqrt {3+5 x}}-\frac {\int -\frac {16114383891514755}{128 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{33617430} \\ & = -\frac {754386765 \sqrt {1-2 x}}{6272 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {270667969 \sqrt {1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac {20529722435 \sqrt {1-2 x}}{18816 \sqrt {3+5 x}}+\frac {46975917593 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{12544} \\ & = -\frac {754386765 \sqrt {1-2 x}}{6272 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {270667969 \sqrt {1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac {20529722435 \sqrt {1-2 x}}{18816 \sqrt {3+5 x}}+\frac {46975917593 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{6272} \\ & = -\frac {754386765 \sqrt {1-2 x}}{6272 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {270667969 \sqrt {1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac {20529722435 \sqrt {1-2 x}}{18816 \sqrt {3+5 x}}-\frac {46975917593 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}} \\ \end{align*}
Time = 8.21 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=\frac {395136 (1-2 x)^{7/2}+3252816 (1-2 x)^{7/2} (2+3 x)+(2+3 x)^2 \left (29407896 (1-2 x)^{7/2}+(2+3 x) \left (324091386 (1-2 x)^{7/2}+4270537963 (2+3 x) \left (3 (1-2 x)^{5/2}-55 (2+3 x) \left (-\sqrt {1-2 x} (62+107 x)+21 \sqrt {7} (3+5 x)^{3/2} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )\right )\right )\right )}{4609920 (2+3 x)^5 (3+5 x)^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(393\) vs. \(2(173)=346\).
Time = 1.16 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.76
method | result | size |
default | \(\frac {\left (4280680490662125 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}+19405751557668300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+37689013564451865 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+1746052893096750 \sqrt {-10 x^{2}-x +3}\, x^{6}+40650610289102550 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+6829311689562600 x^{5} \sqrt {-10 x^{2}-x +3}+26297118668561400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+11125554365281230 x^{4} \sqrt {-10 x^{2}-x +3}+10203169301199600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+9662658051124260 x^{3} \sqrt {-10 x^{2}-x +3}+2198472943352400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +4718679545989416 x^{2} \sqrt {-10 x^{2}-x +3}+202935964001760 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1228469050319504 x \sqrt {-10 x^{2}-x +3}+133202515888064 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{1317120 \left (2+3 x \right )^{5} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(394\) |
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Time = 0.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=-\frac {704638763895 \, \sqrt {7} {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (124718063792625 \, x^{6} + 487807977825900 \, x^{5} + 794682454662945 \, x^{4} + 690189860794590 \, x^{3} + 337048538999244 \, x^{2} + 87747789308536 \, x + 9514465420576\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1317120 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (173) = 346\).
Time = 0.33 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.91 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=\frac {46975917593}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {20529722435 \, x}{9408 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {21434986553}{18816 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {2211170555 \, x}{4032 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{405 \, {\left (243 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} + 810 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + 1080 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 720 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 240 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 32 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {43561}{1080 \, {\left (81 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 96 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 16 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {2438681}{6480 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {110694619}{25920 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1309509421}{17280 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {21497905297}{72576 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (173) = 346\).
Time = 0.73 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=-\frac {275}{48} \, \sqrt {10} {\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 {46975917593}{878080} \, \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 )} + 27775 \, \sqrt {10} {\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 )} + \frac {11 \, \sqrt {10} {\left (3277500437 \, {\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} + 3147123544880 \, {\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} + 1168996576419840 \, {\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} + 196941720284288000 \, {\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 {12621260024737280000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {50485040098949120000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3136 \, {\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 )}^{5}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^6\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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