Integrand size = 26, antiderivative size = 144 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}+\frac {463881 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}} \]
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Time = 0.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 156, 157, 12, 95, 210} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=\frac {463881 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}}-\frac {608185 \sqrt {1-2 x}}{504 \sqrt {5 x+3}}+\frac {13409 \sqrt {1-2 x}}{168 (3 x+2) \sqrt {5 x+3}}+\frac {77 \sqrt {1-2 x}}{12 (3 x+2)^2 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}} \]
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 157
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {1}{9} \int \frac {\frac {275}{2}-198 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx \\ & = \frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1}{126} \int \frac {\frac {50743}{4}-16170 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx \\ & = \frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}+\frac {1}{882} \int \frac {\frac {5986981}{8}-\frac {1407945 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx \\ & = -\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {321469533}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{4851} \\ & = -\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}-\frac {463881}{112} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}-\frac {463881}{56} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}+\frac {463881 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}} \\ \end{align*}
Time = 2.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=\frac {1}{392} \left (-\frac {7 \sqrt {1-2 x} \left (521968+2378026 x+3608883 x^2+1824555 x^3\right )}{(2+3 x)^3 \sqrt {3+5 x}}-463881 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-463881 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(111)=222\).
Time = 5.01 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.74
method | result | size |
default | \(-\frac {\left (62623935 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+162822231 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+158647302 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+25543770 x^{3} \sqrt {-10 x^{2}-x +3}+68654388 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +50524362 x^{2} \sqrt {-10 x^{2}-x +3}+11133144 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+33292364 x \sqrt {-10 x^{2}-x +3}+7307552 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{784 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(250\) |
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Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=\frac {463881 \, \sqrt {7} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1824555 \, x^{3} + 3608883 \, x^{2} + 2378026 \, x + 521968\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{784 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.47 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {463881}{784} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {608185 \, x}{252 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {635003}{504 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{27 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {1561}{108 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {4367}{24 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (111) = 222\).
Time = 0.48 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.56 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {463881}{7840} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {55}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {11 \, \sqrt {10} {\left (33989 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 15023680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1769566400 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {7078265600 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{28 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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