Integrand size = 26, antiderivative size = 108 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {190 \sqrt {3+5 x}}{1617 (1-2 x)^{3/2}}-\frac {4390 \sqrt {3+5 x}}{124509 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}-\frac {405 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \]
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Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {105, 157, 12, 95, 210} \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {405 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{343 \sqrt {7}}-\frac {4390 \sqrt {5 x+3}}{124509 \sqrt {1-2 x}}+\frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {190 \sqrt {5 x+3}}{1617 (1-2 x)^{3/2}} \]
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Rule 12
Rule 95
Rule 105
Rule 157
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {3 \sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {1}{7} \int \frac {-\frac {35}{2}-60 x}{(1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {190 \sqrt {3+5 x}}{1617 (1-2 x)^{3/2}}+\frac {3 \sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}-\frac {2 \int \frac {-\frac {655}{4}+1425 x}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{1617} \\ & = -\frac {190 \sqrt {3+5 x}}{1617 (1-2 x)^{3/2}}-\frac {4390 \sqrt {3+5 x}}{124509 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {4 \int \frac {147015}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{124509} \\ & = -\frac {190 \sqrt {3+5 x}}{1617 (1-2 x)^{3/2}}-\frac {4390 \sqrt {3+5 x}}{124509 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {405}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {190 \sqrt {3+5 x}}{1617 (1-2 x)^{3/2}}-\frac {4390 \sqrt {3+5 x}}{124509 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {405}{343} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {190 \sqrt {3+5 x}}{1617 (1-2 x)^{3/2}}-\frac {4390 \sqrt {3+5 x}}{124509 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}-\frac {405 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {-7 \sqrt {3+5 x} \left (15321-39500 x+26340 x^2\right )-147015 \sqrt {7-14 x} \left (-2+x+6 x^2\right ) \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{871563 (1-2 x)^{3/2} (2+3 x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(81)=162\).
Time = 1.25 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.94
method | result | size |
default | \(\frac {\left (1764180 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-588060 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-735075 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +368760 x^{2} \sqrt {-10 x^{2}-x +3}+294030 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-553000 x \sqrt {-10 x^{2}-x +3}+214494 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{1743126 \left (2+3 x \right ) \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(209\) |
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Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {147015 \, \sqrt {7} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (26340 \, x^{2} - 39500 \, x + 15321\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1743126 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
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\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2} \sqrt {5 x + 3}}\, dx \]
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\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (81) = 162\).
Time = 0.37 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.15 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {81}{9604} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {594 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{343 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} - \frac {8 \, {\left (536 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3333 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{3112725 \, {\left (2 \, x - 1\right )}^{2}} \]
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Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3}} \,d x \]
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