Integrand size = 26, antiderivative size = 130 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {3105 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \]
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Time = 0.04 (sec) , antiderivative size = 130, 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 = {105, 157, 12, 95, 210} \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {3105 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {5 x+3}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}-\frac {190}{1617 (1-2 x)^{3/2} \sqrt {5 x+3}} \]
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Rule 12
Rule 95
Rule 105
Rule 157
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {1}{7} \int \frac {-\frac {5}{2}-90 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}} \, dx \\ & = -\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}-\frac {2 \int \frac {-\frac {3785}{4}+2850 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx}{1617} \\ & = -\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {299105}{8}-\frac {28725 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{124509} \\ & = -\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}-\frac {8 \int \frac {12398265}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1369599} \\ & = -\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}-\frac {3105}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}-\frac {3105}{343} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {190}{1617 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {3830}{124509 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1840225 \sqrt {1-2 x}}{1369599 \sqrt {3+5 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {3105 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {-3499599+8760465 x+7613680 x^2-22082700 x^3}{1369599 (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}}+\frac {3105 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(97)=194\).
Time = 4.14 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.98
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \left (743895900 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+198372240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-458735805 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+309157800 x^{3} \sqrt {-10 x^{2}-x +3}-61991325 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -106591520 x^{2} \sqrt {-10 x^{2}-x +3}+74389590 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-122646510 x \sqrt {-10 x^{2}-x +3}+48994386 \sqrt {-10 x^{2}-x +3}\right )}{19174386 \left (2+3 x \right ) \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(257\) |
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Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {12398265 \, \sqrt {7} {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (22082700 \, x^{3} - 7613680 \, x^{2} - 8760465 \, x + 3499599\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{19174386 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \]
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\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (97) = 194\).
Time = 0.38 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {621}{9604} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {125}{2662} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {1782 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{343 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} - \frac {32 \, {\left (373 \, \sqrt {5} {\left (5 \, x + 3\right )} - 2244 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{34239975 \, {\left (2 \, x - 1\right )}^{2}} \]
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Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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