Integrand size = 26, antiderivative size = 188 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx=\frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac {81733 \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5832}+\frac {21935 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2916} \]
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Time = 0.06 (sec) , antiderivative size = 188, 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} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx=\frac {81733 \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5832}+\frac {21935 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2916}+\frac {575}{162} \sqrt {1-2 x} (5 x+3)^{5/2}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac {785}{36} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {34145 \sqrt {1-2 x} \sqrt {5 x+3}}{1944} \]
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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}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx \\ & = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {1}{18} \int \frac {\left (-\frac {355}{4}-2875 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx \\ & = \frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {1}{810} \int \frac {\left (\frac {202525}{4}-211950 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx \\ & = -\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac {\int \frac {(84825-1024350 x) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{9720} \\ & = \frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {\int \frac {-2551200-6129975 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{58320} \\ & = \frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {153545 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5832}+\frac {408665 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{11664} \\ & = \frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {153545 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2916}+\frac {\left (81733 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5832} \\ & = \frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac {81733 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5832}+\frac {21935 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2916} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx=\frac {\frac {6 \sqrt {1-2 x} \left (159612+627622 x+697863 x^2+71715 x^3-80100 x^4+108000 x^5\right )}{(2+3 x)^2 \sqrt {3+5 x}}-81733 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+87740 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{11664} \]
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Time = 1.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (21600 x^{4}-28980 x^{3}+31731 x^{2}+120534 x +53204\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1944 \left (2+3 x \right )^{2} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {81733 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{23328}+\frac {21935 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{5832}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(148\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (259200 x^{4} \sqrt {-10 x^{2}-x +3}+735597 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-789660 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-347760 x^{3} \sqrt {-10 x^{2}-x +3}+980796 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1052880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +380772 x^{2} \sqrt {-10 x^{2}-x +3}+326932 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-350960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1446408 x \sqrt {-10 x^{2}-x +3}+638448 \sqrt {-10 x^{2}-x +3}\right )}{23328 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\) | \(242\) |
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Time = 0.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx=-\frac {81733 \, \sqrt {5} \sqrt {2} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 87740 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 12 \, {\left (21600 \, x^{4} - 28980 \, x^{3} + 31731 \, x^{2} + 120534 \, x + 53204\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{23328 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{3}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx=\frac {5}{21} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {925}{126} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {10135}{2268} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{28 \, {\left (3 \, x + 2\right )}} - \frac {925}{81} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {81733}{23328} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {21935}{5832} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {20825}{1944} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (138) = 276\).
Time = 0.55 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.96 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx=-\frac {4387}{11664} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{3240} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 155 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 5245 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {81733}{23328} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {77 \, {\left (263 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 92120 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{486 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^3} \,d x \]
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