Integrand size = 26, antiderivative size = 166 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}-\frac {41307885 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
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Time = 0.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {105, 156, 157, 12, 95, 210} \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {41307885 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {5 x+3}}-\frac {21891025 \sqrt {1-2 x}}{90552 (5 x+3)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (3 x+2) (5 x+3)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}} \]
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Rule 12
Rule 95
Rule 105
Rule 156
Rule 157
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {1}{21} \int \frac {\frac {165}{2}-120 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx \\ & = \frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{294} \int \frac {\frac {40335}{4}-14625 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx \\ & = \frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {\int \frac {\frac {7422495}{8}-1190025 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{2058} \\ & = -\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {\frac {837775605}{16}-\frac {197019225 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{33957} \\ & = -\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}+\frac {2 \int \frac {44984286765}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{373527} \\ & = -\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}+\frac {41307885 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488} \\ & = -\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}+\frac {41307885 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744} \\ & = -\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}-\frac {41307885 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (50617099616+314968389410 x+734310313245 x^2+760212086400 x^3+294889892625 x^4\right )}{996072 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {41307885 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(127)=254\).
Time = 3.96 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.80
method | result | size |
default | \(\frac {\left (10121464522125 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+32388686470800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+41430528110565 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+4128458496750 x^{4} \sqrt {-10 x^{2}-x +3}+26480750142330 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+10642969209600 x^{3} \sqrt {-10 x^{2}-x +3}+8457045911820 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +10280344385430 x^{2} \sqrt {-10 x^{2}-x +3}+1079622882360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4409557451740 x \sqrt {-10 x^{2}-x +3}+708639394624 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{13945008 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(298\) |
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Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {14994762255 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (294889892625 \, x^{4} + 760212086400 \, x^{3} + 734310313245 \, x^{2} + 314968389410 \, x + 50617099616\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{13945008 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{4} \sqrt {-2 \, x + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (127) = 254\).
Time = 0.49 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.59 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {125}{5808} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {8261577}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {8125}{121} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {1485 \, \sqrt {10} {\left (13759 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 6614720 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {818950720 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {3275802880 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1372 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
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Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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