Integrand size = 26, antiderivative size = 86 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {123 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \]
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Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {105, 157, 12, 95, 210} \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {123 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}}-\frac {58 \sqrt {5 x+3}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)} \]
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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 \sqrt {1-2 x} (2+3 x)}+\frac {1}{7} \int \frac {\frac {1}{2}-30 x}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {2}{539} \int -\frac {1353}{4 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {123}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {123}{49} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {58 \sqrt {3+5 x}}{539 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {123 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {7 (115-174 x) \sqrt {3+5 x}-1353 \sqrt {7-14 x} (2+3 x) \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3773 \sqrt {1-2 x} (2+3 x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(160\) vs. \(2(65)=130\).
Time = 1.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.87
method | result | size |
default | \(\frac {\left (8118 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1353 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -2706 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2436 x \sqrt {-10 x^{2}-x +3}-1610 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{7546 \left (2+3 x \right ) \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(161\) |
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Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {1353 \, \sqrt {7} {\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (174 \, x - 115\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7546 \, {\left (6 \, x^{2} + x - 2\right )}} \]
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\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2} \sqrt {5 x + 3}}\, dx \]
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\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (65) = 130\).
Time = 0.36 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.55 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {123}{6860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{2695 \, {\left (2 \, x - 1\right )}} + \frac {198 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{49 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
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Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3}} \,d x \]
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