Integrand size = 26, antiderivative size = 151 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}-\frac {78045 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \]
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Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 156, 12, 95, 210} \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {78045 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}}+\frac {57595 \sqrt {1-2 x} \sqrt {5 x+3}}{197568 (3 x+2)}+\frac {85 \sqrt {1-2 x} \sqrt {5 x+3}}{14112 (3 x+2)^2}-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{504 (3 x+2)^3}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4} \]
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {1}{84} \int \frac {-\frac {793}{2}-670 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx \\ & = \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}-\frac {\int \frac {-\frac {8225}{4}-3010 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{1764} \\ & = \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}-\frac {\int \frac {-\frac {126455}{8}+\frac {2975 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{24696} \\ & = \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}-\frac {\int -\frac {4916835}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{172872} \\ & = \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}+\frac {78045 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{43904} \\ & = \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}+\frac {78045 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{21952} \\ & = \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}-\frac {78045 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.54 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {121 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (48240+226348 x+346760 x^2+172785 x^3\right )}{121 (2+3 x)^4}-645 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{153664} \]
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Time = 1.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (172785 x^{3}+346760 x^{2}+226348 x +48240\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{21952 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {78045 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (6321645 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+16857720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+16857720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+2418990 x^{3} \sqrt {-10 x^{2}-x +3}+7492320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +4854640 x^{2} \sqrt {-10 x^{2}-x +3}+1248720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3168872 x \sqrt {-10 x^{2}-x +3}+675360 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
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Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {78045 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (172785 \, x^{3} + 346760 \, x^{2} + 226348 \, x + 48240\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{5}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {78045}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{84 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {43 \, \sqrt {-10 \, x^{2} - x + 3}}{504 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {85 \, \sqrt {-10 \, x^{2} - x + 3}}{14112 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {57595 \, \sqrt {-10 \, x^{2} - x + 3}}{197568 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).
Time = 0.54 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {15609}{614656} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {605 \, \sqrt {10} {\left (129 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 132440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 21026880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {2510681600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {10042726400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{10976 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
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Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^5} \,d x \]
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