Integrand size = 26, antiderivative size = 195 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac {21935 \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1458}-\frac {408665 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5832 \sqrt {7}} \]
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Time = 0.06 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {99, 154, 159, 163, 56, 222, 95, 210} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {21935 \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1458}-\frac {408665 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{5832 \sqrt {7}}-\frac {10385 \sqrt {1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac {2075}{72} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {48625 \sqrt {1-2 x} \sqrt {5 x+3}}{1944} \]
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Rule 56
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx \\ & = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (-\frac {2005}{4}-2050 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx \\ & = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {1}{162} \int \frac {\left (\frac {109865}{8}-56025 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx \\ & = \frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac {\int \frac {\left (\frac {19665}{2}-291750 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{1944} \\ & = -\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {\int \frac {-468735-1316100 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{11664} \\ & = -\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {408665 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{11664}-\frac {109675 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2916} \\ & = -\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {408665 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{5832}-\frac {\left (21935 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1458} \\ & = -\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac {21935 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1458}-\frac {408665 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5832 \sqrt {7}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (-107984-391014 x-420531 x^2-93420 x^3+32400 x^4\right )}{(2+3 x)^3}+307090 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-408665 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{40824} \]
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Time = 3.88 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (32400 x^{4}-93420 x^{3}-420531 x^{2}-391014 x -107984\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1944 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {21935 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{5832}-\frac {408665 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{81648}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(148\) |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (8291430 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-11033955 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-1360800 x^{4} \sqrt {-10 x^{2}-x +3}+16582860 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-22067910 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+3923640 x^{3} \sqrt {-10 x^{2}-x +3}+11055240 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -14711940 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +17662302 x^{2} \sqrt {-10 x^{2}-x +3}+2456720 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3269320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+16422588 x \sqrt {-10 x^{2}-x +3}+4535328 \sqrt {-10 x^{2}-x +3}\right )}{81648 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(287\) |
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Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {307090 \, \sqrt {5} \sqrt {2} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 408665 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (32400 \, x^{4} - 93420 \, x^{3} - 420531 \, x^{2} - 391014 \, x - 107984\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{81648 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{4}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {185}{882} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {16075}{1764} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {189865}{31752} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {6347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3528 \, {\left (3 \, x + 2\right )}} + \frac {41225}{2268} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21935}{5832} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {408665}{81648} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {191965}{13608} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (145) = 290\).
Time = 0.59 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.14 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {81733}{163296} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{486} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 329 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {21935}{5832} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, {\left (2803 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1982400 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 411208000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{324 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^4} \,d x \]
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