Integrand size = 26, antiderivative size = 172 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {1994287 \sqrt {1-2 x} \sqrt {3+5 x}}{3110400}-\frac {14557 \sqrt {1-2 x} (3+5 x)^{3/2}}{28800}+\frac {4783 \sqrt {1-2 x} (3+5 x)^{5/2}}{32400}+\frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {109715471 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{9331200 \sqrt {10}}+\frac {98}{729} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 172, 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 = {103, 159, 163, 56, 222, 95, 210} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {109715471 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{9331200 \sqrt {10}}+\frac {98}{729} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}+\frac {37}{360} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac {4783 \sqrt {1-2 x} (5 x+3)^{5/2}}{32400}-\frac {14557 \sqrt {1-2 x} (5 x+3)^{3/2}}{28800}-\frac {1994287 \sqrt {1-2 x} \sqrt {5 x+3}}{3110400} \]
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Rule 56
Rule 95
Rule 103
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{15} \int \frac {\left (-50-\frac {185 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{2+3 x} \, dx \\ & = \frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{900} \int \frac {\left (-\frac {4705}{2}-\frac {23915 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx \\ & = \frac {4783 \sqrt {1-2 x} (3+5 x)^{5/2}}{32400}+\frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {\int \frac {\left (\frac {30935}{4}-\frac {1965195 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx}{40500} \\ & = -\frac {14557 \sqrt {1-2 x} (3+5 x)^{3/2}}{28800}+\frac {4783 \sqrt {1-2 x} (3+5 x)^{5/2}}{32400}+\frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {15459435}{8}+\frac {29914305 x}{16}\right )}{\sqrt {1-2 x} (2+3 x)} \, dx}{486000} \\ & = -\frac {1994287 \sqrt {1-2 x} \sqrt {3+5 x}}{3110400}-\frac {14557 \sqrt {1-2 x} (3+5 x)^{3/2}}{28800}+\frac {4783 \sqrt {1-2 x} (3+5 x)^{5/2}}{32400}+\frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {\int \frac {-\frac {526625355}{16}-\frac {1645732065 x}{32}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2916000} \\ & = -\frac {1994287 \sqrt {1-2 x} \sqrt {3+5 x}}{3110400}-\frac {14557 \sqrt {1-2 x} (3+5 x)^{3/2}}{28800}+\frac {4783 \sqrt {1-2 x} (3+5 x)^{5/2}}{32400}+\frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {343}{729} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {109715471 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{18662400} \\ & = -\frac {1994287 \sqrt {1-2 x} \sqrt {3+5 x}}{3110400}-\frac {14557 \sqrt {1-2 x} (3+5 x)^{3/2}}{28800}+\frac {4783 \sqrt {1-2 x} (3+5 x)^{5/2}}{32400}+\frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {686}{729} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {109715471 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{9331200 \sqrt {5}} \\ & = -\frac {1994287 \sqrt {1-2 x} \sqrt {3+5 x}}{3110400}-\frac {14557 \sqrt {1-2 x} (3+5 x)^{3/2}}{28800}+\frac {4783 \sqrt {1-2 x} (3+5 x)^{5/2}}{32400}+\frac {37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {109715471 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{9331200 \sqrt {10}}+\frac {98}{729} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {30 \sqrt {1-2 x} \left (6495351+36344365 x+6699780 x^2-95229600 x^3+3024000 x^4+103680000 x^5\right )-109715471 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+12544000 \sqrt {7} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{93312000 \sqrt {3+5 x}} \]
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Time = 1.17 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {\left (20736000 x^{4}-11836800 x^{3}-11943840 x^{2}+8506260 x +2165117\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3110400 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {109715471 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{186624000}+\frac {49 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{729}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(141\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1244160000 x^{4} \sqrt {-10 x^{2}-x +3}-710208000 x^{3} \sqrt {-10 x^{2}-x +3}-716630400 x^{2} \sqrt {-10 x^{2}-x +3}+109715471 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-12544000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+510375600 x \sqrt {-10 x^{2}-x +3}+129907020 \sqrt {-10 x^{2}-x +3}\right )}{186624000 \sqrt {-10 x^{2}-x +3}}\) | \(149\) |
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Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {1}{3110400} \, {\left (20736000 \, x^{4} - 11836800 \, x^{3} - 11943840 \, x^{2} + 8506260 \, x + 2165117\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} + \frac {49}{729} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {109715471}{186624000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{3 x + 2}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {1}{15} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {37}{72} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {787}{12960} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {79439}{51840} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {109715471}{186624000} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {49}{729} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {865517}{3110400} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Time = 0.42 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.23 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {49}{7290} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{77760000} \, {\left (12 \, {\left (8 \, {\left (36 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} - 713 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 112817 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 655065 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 9971435 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {109715471}{186624000} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{3\,x+2} \,d x \]
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